diff --git a/.cirrus.yml b/.cirrus.yml
index 35a9a4536726c3..a2e7f36d1fa3d0 100644
--- a/.cirrus.yml
+++ b/.cirrus.yml
@@ -4,10 +4,10 @@ env:
   # Specific warnings can be disabled with -Wno-error=foo.
   # -pedantic-errors is not equivalent to -Werror=pedantic and thus not implied by -Werror according to the GCC manual.
   WERROR_CFLAGS: -Werror -pedantic-errors
-  MAKEFLAGS: -j2
+  MAKEFLAGS: -j4
   BUILD: check
   ### secp256k1 config
-  STATICPRECOMPUTATION: yes
+  ECMULTWINDOW: auto
   ECMULTGENPRECISION: auto
   ASM: no
   WIDEMUL: auto
@@ -23,6 +23,8 @@ env:
   BENCH: yes
   SECP256K1_BENCH_ITERS: 2
   CTIMETEST: yes
+  # Compile and run the tests
+  EXAMPLES: yes
 
 cat_logs_snippet: &CAT_LOGS
   always:
@@ -50,28 +52,32 @@ merge_base_script_snippet: &MERGE_BASE
     - git config --global user.name "ci"
     - git merge FETCH_HEAD  # Merge base to detect silent merge conflicts
 
-task:
-  name: "x86_64: Linux (Debian stable)"
+linux_container_snippet: &LINUX_CONTAINER
   container:
     dockerfile: ci/linux-debian.Dockerfile
     # Reduce number of CPUs to be able to do more builds in parallel.
     cpu: 1
+    # Gives us more CPUs for free if they're available.
+    greedy: true
     # More than enough for our scripts.
     memory: 1G
+
+task:
+  name: "x86_64: Linux (Debian stable)"
+  << : *LINUX_CONTAINER
   matrix: &ENV_MATRIX
     - env: {WIDEMUL:  int64,  RECOVERY: yes}
-    - env: {WIDEMUL:  int64,                 ECDH: yes, EXPERIMENTAL: yes, SCHNORRSIG: yes}
+    - env: {WIDEMUL:  int64,                 ECDH: yes, SCHNORRSIG: yes}
     - env: {WIDEMUL: int128}
-    - env: {WIDEMUL: int128,  RECOVERY: yes,            EXPERIMENTAL: yes, SCHNORRSIG: yes}
-    - env: {WIDEMUL: int128,                 ECDH: yes, EXPERIMENTAL: yes, SCHNORRSIG: yes}
+    - env: {WIDEMUL: int128,  RECOVERY: yes,            SCHNORRSIG: yes}
+    - env: {WIDEMUL: int128,                 ECDH: yes, SCHNORRSIG: yes}
     - env: {WIDEMUL: int128,  ASM: x86_64}
-    - env: {                  RECOVERY: yes,            EXPERIMENTAL: yes, SCHNORRSIG: yes}
-    - env: {                  STATICPRECOMPUTATION: no}
+    - env: {                  RECOVERY: yes,            SCHNORRSIG: yes}
     - env: {BUILD: distcheck, WITH_VALGRIND: no, CTIMETEST: no, BENCH: no}
     - env: {CPPFLAGS: -DDETERMINISTIC}
     - env: {CFLAGS: -O0, CTIMETEST: no}
-    - env: { ECMULTGENPRECISION: 2 }
-    - env: { ECMULTGENPRECISION: 8 }
+    - env: { ECMULTGENPRECISION: 2, ECMULTWINDOW: 2 }
+    - env: { ECMULTGENPRECISION: 8, ECMULTWINDOW: 4 }
   matrix:
     - env:
         CC: gcc
@@ -84,15 +90,11 @@ task:
 
 task:
   name: "i686: Linux (Debian stable)"
-  container:
-    dockerfile: ci/linux-debian.Dockerfile
-    cpu: 1
-    memory: 1G
+  << : *LINUX_CONTAINER
   env:
     HOST: i686-linux-gnu
     ECDH: yes
     RECOVERY: yes
-    EXPERIMENTAL: yes
     SCHNORRSIG: yes
   matrix:
     - env:
@@ -134,8 +136,10 @@ task:
   ##   - rm /tmp/.com.apple.dt.CommandLineTools.installondemand.in-progress
   ##
   brew_valgrind_pre_script:
+    # Retry a few times because this tends to fail randomly.
+    - for i in {1..5}; do brew update && break || sleep 15; done
     - brew config
-    - brew tap --shallow LouisBrunner/valgrind
+    - brew tap LouisBrunner/valgrind
     # Fetch valgrind source but don't build it yet.
     - brew fetch --HEAD LouisBrunner/valgrind/valgrind
   brew_valgrind_cache:
@@ -165,10 +169,7 @@ task:
 
 task:
   name: "s390x (big-endian): Linux (Debian stable, QEMU)"
-  container:
-    dockerfile: ci/linux-debian.Dockerfile
-    cpu: 1
-    memory: 1G
+  << : *LINUX_CONTAINER
   env:
     WRAPPER_CMD: qemu-s390x
     SECP256K1_TEST_ITERS: 16
@@ -176,7 +177,6 @@ task:
     WITH_VALGRIND: no
     ECDH: yes
     RECOVERY: yes
-    EXPERIMENTAL: yes
     SCHNORRSIG: yes
     CTIMETEST: no
   << : *MERGE_BASE
@@ -188,10 +188,7 @@ task:
 
 task:
   name: "ARM32: Linux (Debian stable, QEMU)"
-  container:
-    dockerfile: ci/linux-debian.Dockerfile
-    cpu: 1
-    memory: 1G
+  << : *LINUX_CONTAINER
   env:
     WRAPPER_CMD: qemu-arm
     SECP256K1_TEST_ITERS: 16
@@ -199,12 +196,11 @@ task:
     WITH_VALGRIND: no
     ECDH: yes
     RECOVERY: yes
-    EXPERIMENTAL: yes
     SCHNORRSIG: yes
     CTIMETEST: no
   matrix:
     - env: {}
-    - env: {ASM: arm}
+    - env: {EXPERIMENTAL: yes, ASM: arm}
   << : *MERGE_BASE
   test_script:
     - ./ci/cirrus.sh
@@ -212,10 +208,7 @@ task:
 
 task:
   name: "ARM64: Linux (Debian stable, QEMU)"
-  container:
-    dockerfile: ci/linux-debian.Dockerfile
-    cpu: 1
-    memory: 1G
+  << : *LINUX_CONTAINER
   env:
     WRAPPER_CMD: qemu-aarch64
     SECP256K1_TEST_ITERS: 16
@@ -223,7 +216,6 @@ task:
     WITH_VALGRIND: no
     ECDH: yes
     RECOVERY: yes
-    EXPERIMENTAL: yes
     SCHNORRSIG: yes
     CTIMETEST: no
   << : *MERGE_BASE
@@ -233,10 +225,7 @@ task:
 
 task:
   name: "ppc64le: Linux (Debian stable, QEMU)"
-  container:
-    dockerfile: ci/linux-debian.Dockerfile
-    cpu: 1
-    memory: 1G
+  << : *LINUX_CONTAINER
   env:
     WRAPPER_CMD: qemu-ppc64le
     SECP256K1_TEST_ITERS: 16
@@ -244,7 +233,6 @@ task:
     WITH_VALGRIND: no
     ECDH: yes
     RECOVERY: yes
-    EXPERIMENTAL: yes
     SCHNORRSIG: yes
     CTIMETEST: no
   << : *MERGE_BASE
@@ -254,10 +242,7 @@ task:
 
 task:
   name: "x86_64 (mingw32-w64): Windows (Debian stable, Wine)"
-  container:
-    dockerfile: ci/linux-debian.Dockerfile
-    cpu: 1
-    memory: 1G
+  << : *LINUX_CONTAINER
   env:
     WRAPPER_CMD: wine64-stable
     SECP256K1_TEST_ITERS: 16
@@ -265,7 +250,6 @@ task:
     WITH_VALGRIND: no
     ECDH: yes
     RECOVERY: yes
-    EXPERIMENTAL: yes
     SCHNORRSIG: yes
     CTIMETEST: no
   << : *MERGE_BASE
@@ -275,23 +259,23 @@ task:
 
 # Sanitizers
 task:
-  container:
-    dockerfile: ci/linux-debian.Dockerfile
-    cpu: 1
-    memory: 2G
+  << : *LINUX_CONTAINER
   env:
     ECDH: yes
     RECOVERY: yes
-    EXPERIMENTAL: yes
     SCHNORRSIG: yes
     CTIMETEST: no
   matrix:
     - name: "Valgrind (memcheck)"
+      container:
+        cpu: 2
       env:
         # The `--error-exitcode` is required to make the test fail if valgrind found errors, otherwise it'll return 0 (https://www.valgrind.org/docs/manual/manual-core.html)
         WRAPPER_CMD: "valgrind --error-exitcode=42"
         SECP256K1_TEST_ITERS: 2
     - name: "UBSan, ASan, LSan"
+      container:
+        memory: 2G
       env:
         CFLAGS: "-fsanitize=undefined,address -g"
         UBSAN_OPTIONS: "print_stacktrace=1:halt_on_error=1"
@@ -302,11 +286,10 @@ task:
   matrix:
     - env:
         ASM: auto
-        STATICPRECOMPUTATION: yes
     - env:
         ASM: no
-        STATICPRECOMPUTATION: no
         ECMULTGENPRECISION: 2
+        ECMULTWINDOW: 2
   matrix:
     - env:
         CC: clang
@@ -320,17 +303,13 @@ task:
 
 task:
   name: "C++ -fpermissive"
-  container:
-    dockerfile: ci/linux-debian.Dockerfile
-    cpu: 1
-    memory: 1G
+  << : *LINUX_CONTAINER
   env:
     # ./configure correctly errors out when given CC=g++.
     # We hack around this by passing CC=g++ only to make.
     CC: gcc
-    MAKEFLAGS: -j2 CC=g++ CFLAGS=-fpermissive\ -g
+    MAKEFLAGS: -j4 CC=g++ CFLAGS=-fpermissive\ -g
     WERROR_CFLAGS:
-    EXPERIMENTAL: yes
     ECDH: yes
     RECOVERY: yes
     SCHNORRSIG: yes
@@ -338,3 +317,10 @@ task:
   test_script:
     - ./ci/cirrus.sh
   << : *CAT_LOGS
+
+task:
+  name: "sage prover"
+  << : *LINUX_CONTAINER
+  test_script:
+    - cd sage
+    - sage prove_group_implementations.sage
diff --git a/.gitattributes b/.gitattributes
index a0fa567da81b58..30efb2244fee74 100644
--- a/.gitattributes
+++ b/.gitattributes
@@ -1,2 +1,2 @@
-src/ecmult_static_pre_g.h linguist-generated
-src/ecmult_gen_static_prec_table.h linguist-generated
+src/precomputed_ecmult.c linguist-generated
+src/precomputed_ecmult_gen.c linguist-generated
diff --git a/.gitignore b/.gitignore
index 22cd50050135e7..d88627d72ec677 100644
--- a/.gitignore
+++ b/.gitignore
@@ -3,14 +3,19 @@ bench_ecmult
 bench_internal
 tests
 exhaustive_tests
-gen_ecmult_gen_static_prec_table
-gen_ecmult_static_pre_g
+precompute_ecmult_gen
+precompute_ecmult
 valgrind_ctime_test
+ecdh_example
+ecdsa_example
+schnorr_example
 *.exe
 *.so
 *.a
 *.csv
 !.gitignore
+*.log
+*.trs
 
 Makefile
 configure
@@ -41,6 +46,7 @@ coverage.*.html
 
 src/libsecp256k1-config.h
 src/libsecp256k1-config.h.in
+build-aux/ar-lib
 build-aux/config.guess
 build-aux/config.sub
 build-aux/depcomp
diff --git a/Makefile.am b/Makefile.am
index 7ea29bc6e3fd58..51c5960301a49e 100644
--- a/Makefile.am
+++ b/Makefile.am
@@ -26,12 +26,14 @@ noinst_HEADERS += src/eckey.h
 noinst_HEADERS += src/eckey_impl.h
 noinst_HEADERS += src/ecmult.h
 noinst_HEADERS += src/ecmult_impl.h
+noinst_HEADERS += src/ecmult_compute_table.h
+noinst_HEADERS += src/ecmult_compute_table_impl.h
 noinst_HEADERS += src/ecmult_const.h
 noinst_HEADERS += src/ecmult_const_impl.h
 noinst_HEADERS += src/ecmult_gen.h
 noinst_HEADERS += src/ecmult_gen_impl.h
-noinst_HEADERS += src/ecmult_gen_prec.h
-noinst_HEADERS += src/ecmult_gen_prec_impl.h
+noinst_HEADERS += src/ecmult_gen_compute_table.h
+noinst_HEADERS += src/ecmult_gen_compute_table_impl.h
 noinst_HEADERS += src/field_10x26.h
 noinst_HEADERS += src/field_10x26_impl.h
 noinst_HEADERS += src/field_5x52.h
@@ -42,6 +44,8 @@ noinst_HEADERS += src/modinv32.h
 noinst_HEADERS += src/modinv32_impl.h
 noinst_HEADERS += src/modinv64.h
 noinst_HEADERS += src/modinv64_impl.h
+noinst_HEADERS += src/precomputed_ecmult.h
+noinst_HEADERS += src/precomputed_ecmult_gen.h
 noinst_HEADERS += src/assumptions.h
 noinst_HEADERS += src/util.h
 noinst_HEADERS += src/scratch.h
@@ -59,13 +63,19 @@ noinst_HEADERS += contrib/lax_der_parsing.h
 noinst_HEADERS += contrib/lax_der_parsing.c
 noinst_HEADERS += contrib/lax_der_privatekey_parsing.h
 noinst_HEADERS += contrib/lax_der_privatekey_parsing.c
+noinst_HEADERS += examples/random.h
+
+PRECOMPUTED_LIB = libsecp256k1_precomputed.la
+noinst_LTLIBRARIES = $(PRECOMPUTED_LIB)
+libsecp256k1_precomputed_la_SOURCES =  src/precomputed_ecmult.c src/precomputed_ecmult_gen.c
+libsecp256k1_precomputed_la_CPPFLAGS = $(SECP_INCLUDES)
 
 if USE_EXTERNAL_ASM
 COMMON_LIB = libsecp256k1_common.la
-noinst_LTLIBRARIES = $(COMMON_LIB)
 else
 COMMON_LIB =
 endif
+noinst_LTLIBRARIES += $(COMMON_LIB)
 
 pkgconfigdir = $(libdir)/pkgconfig
 pkgconfig_DATA = libsecp256k1.pc
@@ -78,8 +88,8 @@ endif
 
 libsecp256k1_la_SOURCES = src/secp256k1.c
 libsecp256k1_la_CPPFLAGS = -I$(top_srcdir)/include -I$(top_srcdir)/src $(SECP_INCLUDES)
-libsecp256k1_la_LIBADD = $(SECP_LIBS) $(COMMON_LIB)
-libsecp256k1_la_LDFLAGS = -no-undefined
+libsecp256k1_la_LIBADD = $(SECP_LIBS) $(COMMON_LIB) $(PRECOMPUTED_LIB)
+libsecp256k1_la_LDFLAGS = -no-undefined -version-info $(LIB_VERSION_CURRENT):$(LIB_VERSION_REVISION):$(LIB_VERSION_AGE)
 
 if VALGRIND_ENABLED
 libsecp256k1_la_CPPFLAGS += -DVALGRIND
@@ -91,10 +101,10 @@ noinst_PROGRAMS += bench bench_internal bench_ecmult
 bench_SOURCES = src/bench.c
 bench_LDADD = libsecp256k1.la $(SECP_LIBS) $(SECP_TEST_LIBS) $(COMMON_LIB)
 bench_internal_SOURCES = src/bench_internal.c
-bench_internal_LDADD = $(SECP_LIBS) $(COMMON_LIB)
+bench_internal_LDADD = $(SECP_LIBS) $(COMMON_LIB) $(PRECOMPUTED_LIB)
 bench_internal_CPPFLAGS = $(SECP_INCLUDES)
 bench_ecmult_SOURCES = src/bench_ecmult.c
-bench_ecmult_LDADD = $(SECP_LIBS) $(COMMON_LIB)
+bench_ecmult_LDADD = $(SECP_LIBS) $(COMMON_LIB) $(PRECOMPUTED_LIB)
 bench_ecmult_CPPFLAGS = $(SECP_INCLUDES)
 endif
 
@@ -112,7 +122,7 @@ endif
 if !ENABLE_COVERAGE
 tests_CPPFLAGS += -DVERIFY
 endif
-tests_LDADD = $(SECP_LIBS) $(SECP_TEST_LIBS) $(COMMON_LIB)
+tests_LDADD = $(SECP_LIBS) $(SECP_TEST_LIBS) $(COMMON_LIB) $(PRECOMPUTED_LIB)
 tests_LDFLAGS = -static
 TESTS += tests
 endif
@@ -124,22 +134,57 @@ exhaustive_tests_CPPFLAGS = $(SECP_INCLUDES)
 if !ENABLE_COVERAGE
 exhaustive_tests_CPPFLAGS += -DVERIFY
 endif
+# Note: do not include $(PRECOMPUTED_LIB) in exhaustive_tests (it uses runtime-generated tables).
 exhaustive_tests_LDADD = $(SECP_LIBS) $(COMMON_LIB)
 exhaustive_tests_LDFLAGS = -static
 TESTS += exhaustive_tests
 endif
 
+if USE_EXAMPLES
+noinst_PROGRAMS += ecdsa_example
+ecdsa_example_SOURCES = examples/ecdsa.c
+ecdsa_example_CPPFLAGS = -I$(top_srcdir)/include
+ecdsa_example_LDADD = libsecp256k1.la
+ecdsa_example_LDFLAGS = -static
+if BUILD_WINDOWS
+ecdsa_example_LDFLAGS += -lbcrypt
+endif
+TESTS += ecdsa_example
+if ENABLE_MODULE_ECDH
+noinst_PROGRAMS += ecdh_example
+ecdh_example_SOURCES = examples/ecdh.c
+ecdh_example_CPPFLAGS = -I$(top_srcdir)/include
+ecdh_example_LDADD = libsecp256k1.la
+ecdh_example_LDFLAGS = -static
+if BUILD_WINDOWS
+ecdh_example_LDFLAGS += -lbcrypt
+endif
+TESTS += ecdh_example
+endif
+if ENABLE_MODULE_SCHNORRSIG
+noinst_PROGRAMS += schnorr_example
+schnorr_example_SOURCES = examples/schnorr.c
+schnorr_example_CPPFLAGS = -I$(top_srcdir)/include
+schnorr_example_LDADD = libsecp256k1.la
+schnorr_example_LDFLAGS = -static
+if BUILD_WINDOWS
+schnorr_example_LDFLAGS += -lbcrypt
+endif
+TESTS += schnorr_example
+endif
+endif
+
 ### Precomputed tables
-EXTRA_PROGRAMS = gen_ecmult_static_pre_g gen_ecmult_gen_static_prec_table
+EXTRA_PROGRAMS = precompute_ecmult precompute_ecmult_gen
 CLEANFILES = $(EXTRA_PROGRAMS)
 
-gen_ecmult_static_pre_g_SOURCES = src/gen_ecmult_static_pre_g.c
-gen_ecmult_static_pre_g_CPPFLAGS = $(SECP_INCLUDES)
-gen_ecmult_static_pre_g_LDADD = $(SECP_LIBS) $(COMMON_LIB)
+precompute_ecmult_SOURCES = src/precompute_ecmult.c
+precompute_ecmult_CPPFLAGS = $(SECP_INCLUDES)
+precompute_ecmult_LDADD = $(SECP_LIBS) $(COMMON_LIB)
 
-gen_ecmult_gen_static_prec_table_SOURCES = src/gen_ecmult_gen_static_prec_table.c
-gen_ecmult_gen_static_prec_table_CPPFLAGS = $(SECP_INCLUDES)
-gen_ecmult_gen_static_prec_table_LDADD = $(SECP_LIBS) $(COMMON_LIB)
+precompute_ecmult_gen_SOURCES = src/precompute_ecmult_gen.c
+precompute_ecmult_gen_CPPFLAGS = $(SECP_INCLUDES)
+precompute_ecmult_gen_LDADD = $(SECP_LIBS) $(COMMON_LIB)
 
 # See Automake manual, Section "Errors with distclean".
 # We don't list any dependencies for the prebuilt files here because
@@ -147,15 +192,14 @@ gen_ecmult_gen_static_prec_table_LDADD = $(SECP_LIBS) $(COMMON_LIB)
 # build by a normal user) depends on mtimes, and thus is very fragile.
 # This means that rebuilds of the prebuilt files always need to be
 # forced by deleting them, e.g., by invoking `make clean-precomp`.
-src/ecmult_static_pre_g.h:
-	$(MAKE) $(AM_MAKEFLAGS) gen_ecmult_static_pre_g$(EXEEXT)
-	./gen_ecmult_static_pre_g$(EXEEXT)
-src/ecmult_gen_static_prec_table.h:
-	$(MAKE) $(AM_MAKEFLAGS) gen_ecmult_gen_static_prec_table$(EXEEXT)
-	./gen_ecmult_gen_static_prec_table$(EXEEXT)
-
-PRECOMP = src/ecmult_gen_static_prec_table.h src/ecmult_static_pre_g.h
-noinst_HEADERS += $(PRECOMP)
+src/precomputed_ecmult.c:
+	$(MAKE) $(AM_MAKEFLAGS) precompute_ecmult$(EXEEXT)
+	./precompute_ecmult$(EXEEXT)
+src/precomputed_ecmult_gen.c:
+	$(MAKE) $(AM_MAKEFLAGS) precompute_ecmult_gen$(EXEEXT)
+	./precompute_ecmult_gen$(EXEEXT)
+
+PRECOMP = src/precomputed_ecmult_gen.c src/precomputed_ecmult.c
 precomp: $(PRECOMP)
 
 # Ensure the prebuilt files will be build first (only if they don't exist,
diff --git a/README.md b/README.md
index 5fc07dd4fa4684..f5db915e83e7b9 100644
--- a/README.md
+++ b/README.md
@@ -17,9 +17,7 @@ Features:
 * Suitable for embedded systems.
 * Optional module for public key recovery.
 * Optional module for ECDH key exchange.
-* Optional module for Schnorr signatures according to [BIP-340](https://github.com/bitcoin/bips/blob/master/bip-0340.mediawiki) (experimental).
-
-Experimental features have not received enough scrutiny to satisfy the standard of quality of this library but are made available for testing and review by the community. The APIs of these features should not be considered stable.
+* Optional module for Schnorr signatures according to [BIP-340](https://github.com/bitcoin/bips/blob/master/bip-0340.mediawiki).
 
 Implementation details
 ----------------------
@@ -35,6 +33,7 @@ Implementation details
   * Optimized implementation of arithmetic modulo the curve's field size (2^256 - 0x1000003D1).
     * Using 5 52-bit limbs (including hand-optimized assembly for x86_64, by Diederik Huys).
     * Using 10 26-bit limbs (including hand-optimized assembly for 32-bit ARM, by Wladimir J. van der Laan).
+      * This is an experimental feature that has not received enough scrutiny to satisfy the standard of quality of this library but is made available for testing and review by the community.
 * Scalar operations
   * Optimized implementation without data-dependent branches of arithmetic modulo the curve's order.
     * Using 4 64-bit limbs (relying on __int128 support in the compiler).
@@ -69,6 +68,16 @@ libsecp256k1 is built using autotools:
     $ make check  # run the test suite
     $ sudo make install  # optional
 
+To compile optional modules (such as Schnorr signatures), you need to run `./configure` with additional flags (such as `--enable-module-schnorrsig`). Run `./configure --help` to see the full list of available flags.
+
+Usage examples
+-----------
+  Usage examples can be found in the [examples](examples) directory. To compile them you need to configure with `--enable-examples`.
+  * [ECDSA example](examples/ecdsa.c)
+  * [Schnorr signatures example](examples/schnorr.c)
+  * [Deriving a shared secret (ECDH) example](examples/ecdh.c)
+  To compile the Schnorr signature and ECDH examples, you also need to configure with `--enable-module-schnorrsig` and `--enable-module-ecdh`.
+
 Test coverage
 -----------
 
diff --git a/build-aux/m4/bitcoin_secp.m4 b/build-aux/m4/bitcoin_secp.m4
index c14d09fa1b1447..dda770e001c2d7 100644
--- a/build-aux/m4/bitcoin_secp.m4
+++ b/build-aux/m4/bitcoin_secp.m4
@@ -38,3 +38,16 @@ AC_DEFUN([SECP_TRY_APPEND_CFLAGS], [
   unset flag_works
   AC_SUBST($2)
 ])
+
+dnl SECP_SET_DEFAULT(VAR, default, default-dev-mode)
+dnl Set VAR to default or default-dev-mode, depending on whether dev mode is enabled
+AC_DEFUN([SECP_SET_DEFAULT], [
+  if test "${enable_dev_mode+set}" != set; then
+    AC_MSG_ERROR([[Set enable_dev_mode before calling SECP_SET_DEFAULT]])
+  fi
+  if test x"$enable_dev_mode" = x"yes"; then
+    $1="$3"
+  else
+    $1="$2"
+  fi
+])
diff --git a/ci/cirrus.sh b/ci/cirrus.sh
index e27b34782e17f3..b85f012d3f3c08 100755
--- a/ci/cirrus.sh
+++ b/ci/cirrus.sh
@@ -15,9 +15,11 @@ valgrind --version || true
 ./configure \
     --enable-experimental="$EXPERIMENTAL" \
     --with-test-override-wide-multiply="$WIDEMUL" --with-asm="$ASM" \
-    --enable-ecmult-static-precomputation="$STATICPRECOMPUTATION" --with-ecmult-gen-precision="$ECMULTGENPRECISION" \
+    --with-ecmult-window="$ECMULTWINDOW" \
+    --with-ecmult-gen-precision="$ECMULTGENPRECISION" \
     --enable-module-ecdh="$ECDH" --enable-module-recovery="$RECOVERY" \
     --enable-module-schnorrsig="$SCHNORRSIG" \
+    --enable-examples="$EXAMPLES" \
     --with-valgrind="$WITH_VALGRIND" \
     --host="$HOST" $EXTRAFLAGS
 
diff --git a/ci/linux-debian.Dockerfile b/ci/linux-debian.Dockerfile
index fdba12aa00b474..5cccbb5565f032 100644
--- a/ci/linux-debian.Dockerfile
+++ b/ci/linux-debian.Dockerfile
@@ -19,7 +19,8 @@ RUN apt-get install --no-install-recommends --no-upgrade -y \
         gcc-arm-linux-gnueabihf libc6-dev-armhf-cross libc6-dbg:armhf \
         gcc-aarch64-linux-gnu libc6-dev-arm64-cross libc6-dbg:arm64 \
         gcc-powerpc64le-linux-gnu libc6-dev-ppc64el-cross libc6-dbg:ppc64el \
-        wine gcc-mingw-w64-x86-64
+        wine gcc-mingw-w64-x86-64 \
+        sagemath
 
 # Run a dummy command in wine to make it set up configuration
 RUN wine64-stable xcopy || true
diff --git a/configure.ac b/configure.ac
index 94feea7bb72d2a..2db59a8ff32e1c 100644
--- a/configure.ac
+++ b/configure.ac
@@ -1,30 +1,47 @@
 AC_PREREQ([2.60])
-AC_INIT([libsecp256k1],[0.1])
+
+# The package (a.k.a. release) version is based on semantic versioning 2.0.0 of
+# the API. All changes in experimental modules are treated as
+# backwards-compatible and therefore at most increase the minor version.
+define(_PKG_VERSION_MAJOR, 0)
+define(_PKG_VERSION_MINOR, 1)
+define(_PKG_VERSION_BUILD, 0)
+define(_PKG_VERSION_IS_RELEASE, false)
+
+# The library version is based on libtool versioning of the ABI. The set of
+# rules for updating the version can be found here:
+# https://www.gnu.org/software/libtool/manual/html_node/Updating-version-info.html
+# All changes in experimental modules are treated as if they don't affect the
+# interface and therefore only increase the revision.
+define(_LIB_VERSION_CURRENT, 0)
+define(_LIB_VERSION_REVISION, 0)
+define(_LIB_VERSION_AGE, 0)
+
+AC_INIT([libsecp256k1],m4_join([.], _PKG_VERSION_MAJOR, _PKG_VERSION_MINOR, _PKG_VERSION_BUILD)m4_if(_PKG_VERSION_IS_RELEASE, [true], [], [-pre]),[https://github.com/bitcoin-core/secp256k1/issues],[libsecp256k1],[https://github.com/bitcoin-core/secp256k1])
+
 AC_CONFIG_AUX_DIR([build-aux])
 AC_CONFIG_MACRO_DIR([build-aux/m4])
 AC_CANONICAL_HOST
 AH_TOP([#ifndef LIBSECP256K1_CONFIG_H])
 AH_TOP([#define LIBSECP256K1_CONFIG_H])
 AH_BOTTOM([#endif /*LIBSECP256K1_CONFIG_H*/])
-AM_INIT_AUTOMAKE([foreign subdir-objects])
 
-LT_INIT([win32-dll])
+# Require Automake 1.11.2 for AM_PROG_AR
+AM_INIT_AUTOMAKE([1.11.2 foreign subdir-objects])
 
 # Make the compilation flags quiet unless V=1 is used.
 m4_ifdef([AM_SILENT_RULES], [AM_SILENT_RULES([yes])])
 
-PKG_PROG_PKG_CONFIG
-
-AC_PATH_TOOL(AR, ar)
-AC_PATH_TOOL(RANLIB, ranlib)
-AC_PATH_TOOL(STRIP, strip)
-
-AM_PROG_CC_C_O
-AC_PROG_CC_C89
+AC_PROG_CC
 if test x"$ac_cv_prog_cc_c89" = x"no"; then
   AC_MSG_ERROR([c89 compiler support required])
 fi
 AM_PROG_AS
+AM_PROG_AR
+
+LT_INIT([win32-dll])
+
+build_windows=no
 
 case $host_os in
   *darwin*)
@@ -49,6 +66,9 @@ case $host_os in
        fi
      fi
    ;;
+   cygwin*|mingw*)
+     build_windows=yes
+   ;;
 esac
 
 # Try if some desirable compiler flags are supported and append them to SECP_CFLAGS.
@@ -91,55 +111,54 @@ SECP_TRY_APPEND_DEFAULT_CFLAGS(SECP_CFLAGS)
 ### Define config arguments
 ###
 
+# In dev mode, we enable all binaries and modules by default but individual options can still be overridden explicitly.
+# Check for dev mode first because SECP_SET_DEFAULT needs enable_dev_mode set.
+AC_ARG_ENABLE(dev_mode, [], [],
+    [enable_dev_mode=no])
+
 AC_ARG_ENABLE(benchmark,
-    AS_HELP_STRING([--enable-benchmark],[compile benchmark [default=yes]]),
-    [use_benchmark=$enableval],
-    [use_benchmark=yes])
+    AS_HELP_STRING([--enable-benchmark],[compile benchmark [default=yes]]), [],
+    [SECP_SET_DEFAULT([enable_benchmark], [yes], [yes])])
 
 AC_ARG_ENABLE(coverage,
-    AS_HELP_STRING([--enable-coverage],[enable compiler flags to support kcov coverage analysis [default=no]]),
-    [enable_coverage=$enableval],
-    [enable_coverage=no])
+    AS_HELP_STRING([--enable-coverage],[enable compiler flags to support kcov coverage analysis [default=no]]), [],
+    [SECP_SET_DEFAULT([enable_coverage], [no], [no])])
 
 AC_ARG_ENABLE(tests,
-    AS_HELP_STRING([--enable-tests],[compile tests [default=yes]]),
-    [use_tests=$enableval],
-    [use_tests=yes])
+    AS_HELP_STRING([--enable-tests],[compile tests [default=yes]]), [],
+    [SECP_SET_DEFAULT([enable_tests], [yes], [yes])])
 
 AC_ARG_ENABLE(experimental,
-    AS_HELP_STRING([--enable-experimental],[allow experimental configure options [default=no]]),
-    [use_experimental=$enableval],
-    [use_experimental=no])
+    AS_HELP_STRING([--enable-experimental],[allow experimental configure options [default=no]]), [],
+    [SECP_SET_DEFAULT([enable_experimental], [no], [yes])])
 
 AC_ARG_ENABLE(exhaustive_tests,
-    AS_HELP_STRING([--enable-exhaustive-tests],[compile exhaustive tests [default=yes]]),
-    [use_exhaustive_tests=$enableval],
-    [use_exhaustive_tests=yes])
+    AS_HELP_STRING([--enable-exhaustive-tests],[compile exhaustive tests [default=yes]]), [],
+    [SECP_SET_DEFAULT([enable_exhaustive_tests], [yes], [yes])])
+
+AC_ARG_ENABLE(examples,
+    AS_HELP_STRING([--enable-examples],[compile the examples [default=no]]), [],
+    [SECP_SET_DEFAULT([enable_examples], [no], [yes])])
 
 AC_ARG_ENABLE(module_ecdh,
-    AS_HELP_STRING([--enable-module-ecdh],[enable ECDH shared secret computation]),
-    [enable_module_ecdh=$enableval],
-    [enable_module_ecdh=no])
+    AS_HELP_STRING([--enable-module-ecdh],[enable ECDH module [default=no]]), [],
+    [SECP_SET_DEFAULT([enable_module_ecdh], [no], [yes])])
 
 AC_ARG_ENABLE(module_recovery,
-    AS_HELP_STRING([--enable-module-recovery],[enable ECDSA pubkey recovery module [default=no]]),
-    [enable_module_recovery=$enableval],
-    [enable_module_recovery=no])
+    AS_HELP_STRING([--enable-module-recovery],[enable ECDSA pubkey recovery module [default=no]]), [],
+    [SECP_SET_DEFAULT([enable_module_recovery], [no], [yes])])
 
 AC_ARG_ENABLE(module_extrakeys,
-    AS_HELP_STRING([--enable-module-extrakeys],[enable extrakeys module (experimental)]),
-    [enable_module_extrakeys=$enableval],
-    [enable_module_extrakeys=no])
+    AS_HELP_STRING([--enable-module-extrakeys],[enable extrakeys module [default=no]]), [],
+    [SECP_SET_DEFAULT([enable_module_extrakeys], [no], [yes])])
 
 AC_ARG_ENABLE(module_schnorrsig,
-    AS_HELP_STRING([--enable-module-schnorrsig],[enable schnorrsig module (experimental)]),
-    [enable_module_schnorrsig=$enableval],
-    [enable_module_schnorrsig=no])
+    AS_HELP_STRING([--enable-module-schnorrsig],[enable schnorrsig module [default=no]]), [],
+    [SECP_SET_DEFAULT([enable_module_schnorrsig], [no], [yes])])
 
 AC_ARG_ENABLE(external_default_callbacks,
-    AS_HELP_STRING([--enable-external-default-callbacks],[enable external default callback functions [default=no]]),
-    [use_external_default_callbacks=$enableval],
-    [use_external_default_callbacks=no])
+    AS_HELP_STRING([--enable-external-default-callbacks],[enable external default callback functions [default=no]]), [],
+    [SECP_SET_DEFAULT([enable_external_default_callbacks], [no], [no])])
 
 # Test-only override of the (autodetected by the C code) "widemul" setting.
 # Legal values are int64 (for [u]int64_t), int128 (for [unsigned] __int128), and auto (the default).
@@ -152,7 +171,7 @@ AC_ARG_WITH([ecmult-window], [AS_HELP_STRING([--with-ecmult-window=SIZE|auto],
 [window size for ecmult precomputation for verification, specified as integer in range [2..24].]
 [Larger values result in possibly better performance at the cost of an exponentially larger precomputed table.]
 [The table will store 2^(SIZE-1) * 64 bytes of data but can be larger in memory due to platform-specific padding and alignment.]
-[A window size larger than 15 will require you delete the prebuilt ecmult_static_pre_g.h file so that it can be rebuilt.]
+[A window size larger than 15 will require you delete the prebuilt precomputed_ecmult.c file so that it can be rebuilt.]
 [For very large window sizes, use "make -j 1" to reduce memory use during compilation.]
 ["auto" is a reasonable setting for desktop machines (currently 15). [default=auto]]
 )],
@@ -229,14 +248,14 @@ else
 fi
 
 # Select assembly optimization
-use_external_asm=no
+enable_external_asm=no
 
 case $set_asm in
 x86_64)
   AC_DEFINE(USE_ASM_X86_64, 1, [Define this symbol to enable x86_64 assembly optimizations])
   ;;
 arm)
-  use_external_asm=yes
+  enable_external_asm=yes
   ;;
 no)
   ;;
@@ -245,7 +264,7 @@ no)
   ;;
 esac
 
-if test x"$use_external_asm" = x"yes"; then
+if test x"$enable_external_asm" = x"yes"; then
   AC_DEFINE(USE_EXTERNAL_ASM, 1, [Define this symbol if an external (non-inline) assembly implementation is used])
 fi
 
@@ -333,7 +352,7 @@ if test x"$enable_module_extrakeys" = x"yes"; then
   AC_DEFINE(ENABLE_MODULE_EXTRAKEYS, 1, [Define this symbol to enable the extrakeys module])
 fi
 
-if test x"$use_external_default_callbacks" = x"yes"; then
+if test x"$enable_external_default_callbacks" = x"yes"; then
   AC_DEFINE(USE_EXTERNAL_DEFAULT_CALLBACKS, 1, [Define this symbol if an external implementation of the default callbacks is used])
 fi
 
@@ -345,16 +364,8 @@ if test x"$enable_experimental" = x"yes"; then
   AC_MSG_NOTICE([******])
   AC_MSG_NOTICE([WARNING: experimental build])
   AC_MSG_NOTICE([Experimental features do not have stable APIs or properties, and may not be safe for production use.])
-  AC_MSG_NOTICE([Building extrakeys module: $enable_module_extrakeys])
-  AC_MSG_NOTICE([Building schnorrsig module: $enable_module_schnorrsig])
   AC_MSG_NOTICE([******])
 else
-  if test x"$enable_module_extrakeys" = x"yes"; then
-    AC_MSG_ERROR([extrakeys module is experimental. Use --enable-experimental to allow.])
-  fi
-  if test x"$enable_module_schnorrsig" = x"yes"; then
-    AC_MSG_ERROR([schnorrsig module is experimental. Use --enable-experimental to allow.])
-  fi
   if test x"$set_asm" = x"arm"; then
     AC_MSG_ERROR([ARM assembly optimization is experimental. Use --enable-experimental to allow.])
   fi
@@ -372,29 +383,30 @@ AC_SUBST(SECP_TEST_LIBS)
 AC_SUBST(SECP_TEST_INCLUDES)
 AC_SUBST(SECP_CFLAGS)
 AM_CONDITIONAL([ENABLE_COVERAGE], [test x"$enable_coverage" = x"yes"])
-AM_CONDITIONAL([USE_TESTS], [test x"$use_tests" != x"no"])
-AM_CONDITIONAL([USE_EXHAUSTIVE_TESTS], [test x"$use_exhaustive_tests" != x"no"])
-AM_CONDITIONAL([USE_BENCHMARK], [test x"$use_benchmark" = x"yes"])
+AM_CONDITIONAL([USE_TESTS], [test x"$enable_tests" != x"no"])
+AM_CONDITIONAL([USE_EXHAUSTIVE_TESTS], [test x"$enable_exhaustive_tests" != x"no"])
+AM_CONDITIONAL([USE_EXAMPLES], [test x"$enable_examples" != x"no"])
+AM_CONDITIONAL([USE_BENCHMARK], [test x"$enable_benchmark" = x"yes"])
 AM_CONDITIONAL([ENABLE_MODULE_ECDH], [test x"$enable_module_ecdh" = x"yes"])
 AM_CONDITIONAL([ENABLE_MODULE_RECOVERY], [test x"$enable_module_recovery" = x"yes"])
 AM_CONDITIONAL([ENABLE_MODULE_EXTRAKEYS], [test x"$enable_module_extrakeys" = x"yes"])
 AM_CONDITIONAL([ENABLE_MODULE_SCHNORRSIG], [test x"$enable_module_schnorrsig" = x"yes"])
-AM_CONDITIONAL([USE_EXTERNAL_ASM], [test x"$use_external_asm" = x"yes"])
+AM_CONDITIONAL([USE_EXTERNAL_ASM], [test x"$enable_external_asm" = x"yes"])
 AM_CONDITIONAL([USE_ASM_ARM], [test x"$set_asm" = x"arm"])
-
-# Make sure nothing new is exported so that we don't break the cache.
-PKGCONFIG_PATH_TEMP="$PKG_CONFIG_PATH"
-unset PKG_CONFIG_PATH
-PKG_CONFIG_PATH="$PKGCONFIG_PATH_TEMP"
+AM_CONDITIONAL([BUILD_WINDOWS], [test "$build_windows" = "yes"])
+AC_SUBST(LIB_VERSION_CURRENT, _LIB_VERSION_CURRENT)
+AC_SUBST(LIB_VERSION_REVISION, _LIB_VERSION_REVISION)
+AC_SUBST(LIB_VERSION_AGE, _LIB_VERSION_AGE)
 
 AC_OUTPUT
 
 echo
 echo "Build Options:"
-echo "  with external callbacks = $use_external_default_callbacks"
-echo "  with benchmarks         = $use_benchmark"
-echo "  with tests              = $use_tests"
+echo "  with external callbacks = $enable_external_default_callbacks"
+echo "  with benchmarks         = $enable_benchmark"
+echo "  with tests              = $enable_tests"
 echo "  with coverage           = $enable_coverage"
+echo "  with examples           = $enable_examples"
 echo "  module ecdh             = $enable_module_ecdh"
 echo "  module recovery         = $enable_module_recovery"
 echo "  module extrakeys        = $enable_module_extrakeys"
diff --git a/doc/CHANGELOG.md b/doc/CHANGELOG.md
new file mode 100644
index 00000000000000..3c4c2e4583756b
--- /dev/null
+++ b/doc/CHANGELOG.md
@@ -0,0 +1,12 @@
+# Changelog
+
+This file is currently only a template for future use.
+
+Each change falls into one of the following categories: Added, Changed, Deprecated, Removed, Fixed or Security.
+
+## [Unreleased]
+
+## [MAJOR.MINOR.PATCH] - YYYY-MM-DD
+
+### Added/Changed/Deprecated/Removed/Fixed/Security
+- [Title with link to Pull Request](https://link-to-pr)
diff --git a/doc/release-process.md b/doc/release-process.md
new file mode 100644
index 00000000000000..a35b8a9db31515
--- /dev/null
+++ b/doc/release-process.md
@@ -0,0 +1,14 @@
+# Release Process
+
+1. Open PR to master that
+   1. adds release notes to `doc/CHANGELOG.md` and
+   2. if this is **not** a patch release, updates `_PKG_VERSION_{MAJOR,MINOR}` and `_LIB_VERSIONS_*` in `configure.ac`
+2. After the PR is merged,
+   * if this is **not** a patch release, create a release branch with name `MAJOR.MINOR`.
+     Make sure that the branch contains the right commits.
+     Create commit on the release branch that sets `_PKG_VERSION_IS_RELEASE` in `configure.ac` to `true`.
+   * if this **is** a patch release, open a pull request with the bugfixes to the `MAJOR.MINOR` branch.
+     Also include the release note commit bump `_PKG_VERSION_BUILD` and `_LIB_VERSIONS_*` in `configure.ac`.
+4. Tag the commit with `git tag -s vMAJOR.MINOR.PATCH`.
+5. Push branch and tag with `git push origin --tags`.
+6. Create a new GitHub release with a link to the corresponding entry in `doc/CHANGELOG.md`.
diff --git a/examples/EXAMPLES_COPYING b/examples/EXAMPLES_COPYING
new file mode 100644
index 00000000000000..0e259d42c99674
--- /dev/null
+++ b/examples/EXAMPLES_COPYING
@@ -0,0 +1,121 @@
+Creative Commons Legal Code
+
+CC0 1.0 Universal
+
+    CREATIVE COMMONS CORPORATION IS NOT A LAW FIRM AND DOES NOT PROVIDE
+    LEGAL SERVICES. DISTRIBUTION OF THIS DOCUMENT DOES NOT CREATE AN
+    ATTORNEY-CLIENT RELATIONSHIP. CREATIVE COMMONS PROVIDES THIS
+    INFORMATION ON AN "AS-IS" BASIS. CREATIVE COMMONS MAKES NO WARRANTIES
+    REGARDING THE USE OF THIS DOCUMENT OR THE INFORMATION OR WORKS
+    PROVIDED HEREUNDER, AND DISCLAIMS LIABILITY FOR DAMAGES RESULTING FROM
+    THE USE OF THIS DOCUMENT OR THE INFORMATION OR WORKS PROVIDED
+    HEREUNDER.
+
+Statement of Purpose
+
+The laws of most jurisdictions throughout the world automatically confer
+exclusive Copyright and Related Rights (defined below) upon the creator
+and subsequent owner(s) (each and all, an "owner") of an original work of
+authorship and/or a database (each, a "Work").
+
+Certain owners wish to permanently relinquish those rights to a Work for
+the purpose of contributing to a commons of creative, cultural and
+scientific works ("Commons") that the public can reliably and without fear
+of later claims of infringement build upon, modify, incorporate in other
+works, reuse and redistribute as freely as possible in any form whatsoever
+and for any purposes, including without limitation commercial purposes.
+These owners may contribute to the Commons to promote the ideal of a free
+culture and the further production of creative, cultural and scientific
+works, or to gain reputation or greater distribution for their Work in
+part through the use and efforts of others.
+
+For these and/or other purposes and motivations, and without any
+expectation of additional consideration or compensation, the person
+associating CC0 with a Work (the "Affirmer"), to the extent that he or she
+is an owner of Copyright and Related Rights in the Work, voluntarily
+elects to apply CC0 to the Work and publicly distribute the Work under its
+terms, with knowledge of his or her Copyright and Related Rights in the
+Work and the meaning and intended legal effect of CC0 on those rights.
+
+1. Copyright and Related Rights. A Work made available under CC0 may be
+protected by copyright and related or neighboring rights ("Copyright and
+Related Rights"). Copyright and Related Rights include, but are not
+limited to, the following:
+
+  i. the right to reproduce, adapt, distribute, perform, display,
+     communicate, and translate a Work;
+ ii. moral rights retained by the original author(s) and/or performer(s);
+iii. publicity and privacy rights pertaining to a person's image or
+     likeness depicted in a Work;
+ iv. rights protecting against unfair competition in regards to a Work,
+     subject to the limitations in paragraph 4(a), below;
+  v. rights protecting the extraction, dissemination, use and reuse of data
+     in a Work;
+ vi. database rights (such as those arising under Directive 96/9/EC of the
+     European Parliament and of the Council of 11 March 1996 on the legal
+     protection of databases, and under any national implementation
+     thereof, including any amended or successor version of such
+     directive); and
+vii. other similar, equivalent or corresponding rights throughout the
+     world based on applicable law or treaty, and any national
+     implementations thereof.
+
+2. Waiver. To the greatest extent permitted by, but not in contravention
+of, applicable law, Affirmer hereby overtly, fully, permanently,
+irrevocably and unconditionally waives, abandons, and surrenders all of
+Affirmer's Copyright and Related Rights and associated claims and causes
+of action, whether now known or unknown (including existing as well as
+future claims and causes of action), in the Work (i) in all territories
+worldwide, (ii) for the maximum duration provided by applicable law or
+treaty (including future time extensions), (iii) in any current or future
+medium and for any number of copies, and (iv) for any purpose whatsoever,
+including without limitation commercial, advertising or promotional
+purposes (the "Waiver"). Affirmer makes the Waiver for the benefit of each
+member of the public at large and to the detriment of Affirmer's heirs and
+successors, fully intending that such Waiver shall not be subject to
+revocation, rescission, cancellation, termination, or any other legal or
+equitable action to disrupt the quiet enjoyment of the Work by the public
+as contemplated by Affirmer's express Statement of Purpose.
+
+3. Public License Fallback. Should any part of the Waiver for any reason
+be judged legally invalid or ineffective under applicable law, then the
+Waiver shall be preserved to the maximum extent permitted taking into
+account Affirmer's express Statement of Purpose. In addition, to the
+extent the Waiver is so judged Affirmer hereby grants to each affected
+person a royalty-free, non transferable, non sublicensable, non exclusive,
+irrevocable and unconditional license to exercise Affirmer's Copyright and
+Related Rights in the Work (i) in all territories worldwide, (ii) for the
+maximum duration provided by applicable law or treaty (including future
+time extensions), (iii) in any current or future medium and for any number
+of copies, and (iv) for any purpose whatsoever, including without
+limitation commercial, advertising or promotional purposes (the
+"License"). The License shall be deemed effective as of the date CC0 was
+applied by Affirmer to the Work. Should any part of the License for any
+reason be judged legally invalid or ineffective under applicable law, such
+partial invalidity or ineffectiveness shall not invalidate the remainder
+of the License, and in such case Affirmer hereby affirms that he or she
+will not (i) exercise any of his or her remaining Copyright and Related
+Rights in the Work or (ii) assert any associated claims and causes of
+action with respect to the Work, in either case contrary to Affirmer's
+express Statement of Purpose.
+
+4. Limitations and Disclaimers.
+
+ a. No trademark or patent rights held by Affirmer are waived, abandoned,
+    surrendered, licensed or otherwise affected by this document.
+ b. Affirmer offers the Work as-is and makes no representations or
+    warranties of any kind concerning the Work, express, implied,
+    statutory or otherwise, including without limitation warranties of
+    title, merchantability, fitness for a particular purpose, non
+    infringement, or the absence of latent or other defects, accuracy, or
+    the present or absence of errors, whether or not discoverable, all to
+    the greatest extent permissible under applicable law.
+ c. Affirmer disclaims responsibility for clearing rights of other persons
+    that may apply to the Work or any use thereof, including without
+    limitation any person's Copyright and Related Rights in the Work.
+    Further, Affirmer disclaims responsibility for obtaining any necessary
+    consents, permissions or other rights required for any use of the
+    Work.
+ d. Affirmer understands and acknowledges that Creative Commons is not a
+    party to this document and has no duty or obligation with respect to
+    this CC0 or use of the Work.
diff --git a/examples/ecdh.c b/examples/ecdh.c
new file mode 100644
index 00000000000000..d7e8add361338b
--- /dev/null
+++ b/examples/ecdh.c
@@ -0,0 +1,127 @@
+/*************************************************************************
+ * Written in 2020-2022 by Elichai Turkel                                *
+ * To the extent possible under law, the author(s) have dedicated all    *
+ * copyright and related and neighboring rights to the software in this  *
+ * file to the public domain worldwide. This software is distributed     *
+ * without any warranty. For the CC0 Public Domain Dedication, see       *
+ * EXAMPLES_COPYING or https://creativecommons.org/publicdomain/zero/1.0 *
+ *************************************************************************/
+
+#include <stdio.h>
+#include <assert.h>
+#include <string.h>
+
+#include <secp256k1.h>
+#include <secp256k1_ecdh.h>
+
+#include "random.h"
+
+
+int main(void) {
+    unsigned char seckey1[32];
+    unsigned char seckey2[32];
+    unsigned char compressed_pubkey1[33];
+    unsigned char compressed_pubkey2[33];
+    unsigned char shared_secret1[32];
+    unsigned char shared_secret2[32];
+    unsigned char randomize[32];
+    int return_val;
+    size_t len;
+    secp256k1_pubkey pubkey1;
+    secp256k1_pubkey pubkey2;
+
+    /* The specification in secp256k1.h states that `secp256k1_ec_pubkey_create`
+     * needs a context object initialized for signing, which is why we create
+     * a context with the SECP256K1_CONTEXT_SIGN flag.
+     * (The docs for `secp256k1_ecdh` don't require any special context, just
+     * some initialized context) */
+    secp256k1_context* ctx = secp256k1_context_create(SECP256K1_CONTEXT_SIGN);
+    if (!fill_random(randomize, sizeof(randomize))) {
+        printf("Failed to generate randomness\n");
+        return 1;
+    }
+    /* Randomizing the context is recommended to protect against side-channel
+     * leakage See `secp256k1_context_randomize` in secp256k1.h for more
+     * information about it. This should never fail. */
+    return_val = secp256k1_context_randomize(ctx, randomize);
+    assert(return_val);
+
+    /*** Key Generation ***/
+
+    /* If the secret key is zero or out of range (bigger than secp256k1's
+     * order), we try to sample a new key. Note that the probability of this
+     * happening is negligible. */
+    while (1) {
+        if (!fill_random(seckey1, sizeof(seckey1)) || !fill_random(seckey2, sizeof(seckey2))) {
+            printf("Failed to generate randomness\n");
+            return 1;
+        }
+        if (secp256k1_ec_seckey_verify(ctx, seckey1) && secp256k1_ec_seckey_verify(ctx, seckey2)) {
+            break;
+        }
+    }
+
+    /* Public key creation using a valid context with a verified secret key should never fail */
+    return_val = secp256k1_ec_pubkey_create(ctx, &pubkey1, seckey1);
+    assert(return_val);
+    return_val = secp256k1_ec_pubkey_create(ctx, &pubkey2, seckey2);
+    assert(return_val);
+
+    /* Serialize pubkey1 in a compressed form (33 bytes), should always return 1 */
+    len = sizeof(compressed_pubkey1);
+    return_val = secp256k1_ec_pubkey_serialize(ctx, compressed_pubkey1, &len, &pubkey1, SECP256K1_EC_COMPRESSED);
+    assert(return_val);
+    /* Should be the same size as the size of the output, because we passed a 33 byte array. */
+    assert(len == sizeof(compressed_pubkey1));
+
+    /* Serialize pubkey2 in a compressed form (33 bytes) */
+    len = sizeof(compressed_pubkey2);
+    return_val = secp256k1_ec_pubkey_serialize(ctx, compressed_pubkey2, &len, &pubkey2, SECP256K1_EC_COMPRESSED);
+    assert(return_val);
+    /* Should be the same size as the size of the output, because we passed a 33 byte array. */
+    assert(len == sizeof(compressed_pubkey2));
+
+    /*** Creating the shared secret ***/
+
+    /* Perform ECDH with seckey1 and pubkey2. Should never fail with a verified
+     * seckey and valid pubkey */
+    return_val = secp256k1_ecdh(ctx, shared_secret1, &pubkey2, seckey1, NULL, NULL);
+    assert(return_val);
+
+    /* Perform ECDH with seckey2 and pubkey1. Should never fail with a verified
+     * seckey and valid pubkey */
+    return_val = secp256k1_ecdh(ctx, shared_secret2, &pubkey1, seckey2, NULL, NULL);
+    assert(return_val);
+
+    /* Both parties should end up with the same shared secret */
+    return_val = memcmp(shared_secret1, shared_secret2, sizeof(shared_secret1));
+    assert(return_val == 0);
+
+    printf("Secret Key1: ");
+    print_hex(seckey1, sizeof(seckey1));
+    printf("Compressed Pubkey1: ");
+    print_hex(compressed_pubkey1, sizeof(compressed_pubkey1));
+    printf("\nSecret Key2: ");
+    print_hex(seckey2, sizeof(seckey2));
+    printf("Compressed Pubkey2: ");
+    print_hex(compressed_pubkey2, sizeof(compressed_pubkey2));
+    printf("\nShared Secret: ");
+    print_hex(shared_secret1, sizeof(shared_secret1));
+
+    /* This will clear everything from the context and free the memory */
+    secp256k1_context_destroy(ctx);
+
+    /* It's best practice to try to clear secrets from memory after using them.
+     * This is done because some bugs can allow an attacker to leak memory, for
+     * example through "out of bounds" array access (see Heartbleed), Or the OS
+     * swapping them to disk. Hence, we overwrite the secret key buffer with zeros.
+     *
+     * TODO: Prevent these writes from being optimized out, as any good compiler
+     * will remove any writes that aren't used. */
+    memset(seckey1, 0, sizeof(seckey1));
+    memset(seckey2, 0, sizeof(seckey2));
+    memset(shared_secret1, 0, sizeof(shared_secret1));
+    memset(shared_secret2, 0, sizeof(shared_secret2));
+
+    return 0;
+}
diff --git a/examples/ecdsa.c b/examples/ecdsa.c
new file mode 100644
index 00000000000000..434c856ba0ff22
--- /dev/null
+++ b/examples/ecdsa.c
@@ -0,0 +1,137 @@
+/*************************************************************************
+ * Written in 2020-2022 by Elichai Turkel                                *
+ * To the extent possible under law, the author(s) have dedicated all    *
+ * copyright and related and neighboring rights to the software in this  *
+ * file to the public domain worldwide. This software is distributed     *
+ * without any warranty. For the CC0 Public Domain Dedication, see       *
+ * EXAMPLES_COPYING or https://creativecommons.org/publicdomain/zero/1.0 *
+ *************************************************************************/
+
+#include <stdio.h>
+#include <assert.h>
+#include <string.h>
+
+#include <secp256k1.h>
+
+#include "random.h"
+
+
+
+int main(void) {
+    /* Instead of signing the message directly, we must sign a 32-byte hash.
+     * Here the message is "Hello, world!" and the hash function was SHA-256.
+     * An actual implementation should just call SHA-256, but this example
+     * hardcodes the output to avoid depending on an additional library.
+     * See https://bitcoin.stackexchange.com/questions/81115/if-someone-wanted-to-pretend-to-be-satoshi-by-posting-a-fake-signature-to-defrau/81116#81116 */
+    unsigned char msg_hash[32] = {
+        0x31, 0x5F, 0x5B, 0xDB, 0x76, 0xD0, 0x78, 0xC4,
+        0x3B, 0x8A, 0xC0, 0x06, 0x4E, 0x4A, 0x01, 0x64,
+        0x61, 0x2B, 0x1F, 0xCE, 0x77, 0xC8, 0x69, 0x34,
+        0x5B, 0xFC, 0x94, 0xC7, 0x58, 0x94, 0xED, 0xD3,
+    };
+    unsigned char seckey[32];
+    unsigned char randomize[32];
+    unsigned char compressed_pubkey[33];
+    unsigned char serialized_signature[64];
+    size_t len;
+    int is_signature_valid;
+    int return_val;
+    secp256k1_pubkey pubkey;
+    secp256k1_ecdsa_signature sig;
+    /* The specification in secp256k1.h states that `secp256k1_ec_pubkey_create` needs
+     * a context object initialized for signing and `secp256k1_ecdsa_verify` needs
+     * a context initialized for verification, which is why we create a context
+     * for both signing and verification with the SECP256K1_CONTEXT_SIGN and
+     * SECP256K1_CONTEXT_VERIFY flags. */
+    secp256k1_context* ctx = secp256k1_context_create(SECP256K1_CONTEXT_SIGN | SECP256K1_CONTEXT_VERIFY);
+    if (!fill_random(randomize, sizeof(randomize))) {
+        printf("Failed to generate randomness\n");
+        return 1;
+    }
+    /* Randomizing the context is recommended to protect against side-channel
+     * leakage See `secp256k1_context_randomize` in secp256k1.h for more
+     * information about it. This should never fail. */
+    return_val = secp256k1_context_randomize(ctx, randomize);
+    assert(return_val);
+
+    /*** Key Generation ***/
+
+    /* If the secret key is zero or out of range (bigger than secp256k1's
+     * order), we try to sample a new key. Note that the probability of this
+     * happening is negligible. */
+    while (1) {
+        if (!fill_random(seckey, sizeof(seckey))) {
+            printf("Failed to generate randomness\n");
+            return 1;
+        }
+        if (secp256k1_ec_seckey_verify(ctx, seckey)) {
+            break;
+        }
+    }
+
+    /* Public key creation using a valid context with a verified secret key should never fail */
+    return_val = secp256k1_ec_pubkey_create(ctx, &pubkey, seckey);
+    assert(return_val);
+
+    /* Serialize the pubkey in a compressed form(33 bytes). Should always return 1. */
+    len = sizeof(compressed_pubkey);
+    return_val = secp256k1_ec_pubkey_serialize(ctx, compressed_pubkey, &len, &pubkey, SECP256K1_EC_COMPRESSED);
+    assert(return_val);
+    /* Should be the same size as the size of the output, because we passed a 33 byte array. */
+    assert(len == sizeof(compressed_pubkey));
+
+    /*** Signing ***/
+
+    /* Generate an ECDSA signature `noncefp` and `ndata` allows you to pass a
+     * custom nonce function, passing `NULL` will use the RFC-6979 safe default.
+     * Signing with a valid context, verified secret key
+     * and the default nonce function should never fail. */
+    return_val = secp256k1_ecdsa_sign(ctx, &sig, msg_hash, seckey, NULL, NULL);
+    assert(return_val);
+
+    /* Serialize the signature in a compact form. Should always return 1
+     * according to the documentation in secp256k1.h. */
+    return_val = secp256k1_ecdsa_signature_serialize_compact(ctx, serialized_signature, &sig);
+    assert(return_val);
+
+
+    /*** Verification ***/
+
+    /* Deserialize the signature. This will return 0 if the signature can't be parsed correctly. */
+    if (!secp256k1_ecdsa_signature_parse_compact(ctx, &sig, serialized_signature)) {
+        printf("Failed parsing the signature\n");
+        return 1;
+    }
+
+    /* Deserialize the public key. This will return 0 if the public key can't be parsed correctly. */
+    if (!secp256k1_ec_pubkey_parse(ctx, &pubkey, compressed_pubkey, sizeof(compressed_pubkey))) {
+        printf("Failed parsing the public key\n");
+        return 1;
+    }
+
+    /* Verify a signature. This will return 1 if it's valid and 0 if it's not. */
+    is_signature_valid = secp256k1_ecdsa_verify(ctx, &sig, msg_hash, &pubkey);
+
+    printf("Is the signature valid? %s\n", is_signature_valid ? "true" : "false");
+    printf("Secret Key: ");
+    print_hex(seckey, sizeof(seckey));
+    printf("Public Key: ");
+    print_hex(compressed_pubkey, sizeof(compressed_pubkey));
+    printf("Signature: ");
+    print_hex(serialized_signature, sizeof(serialized_signature));
+
+
+    /* This will clear everything from the context and free the memory */
+    secp256k1_context_destroy(ctx);
+
+    /* It's best practice to try to clear secrets from memory after using them.
+     * This is done because some bugs can allow an attacker to leak memory, for
+     * example through "out of bounds" array access (see Heartbleed), Or the OS
+     * swapping them to disk. Hence, we overwrite the secret key buffer with zeros.
+     *
+     * TODO: Prevent these writes from being optimized out, as any good compiler
+     * will remove any writes that aren't used. */
+    memset(seckey, 0, sizeof(seckey));
+
+    return 0;
+}
diff --git a/examples/random.h b/examples/random.h
new file mode 100644
index 00000000000000..439226f09f5bf9
--- /dev/null
+++ b/examples/random.h
@@ -0,0 +1,73 @@
+/*************************************************************************
+ * Copyright (c) 2020-2021 Elichai Turkel                                *
+ * Distributed under the CC0 software license, see the accompanying file *
+ * EXAMPLES_COPYING or https://creativecommons.org/publicdomain/zero/1.0 *
+ *************************************************************************/
+
+/*
+ * This file is an attempt at collecting best practice methods for obtaining randomness with different operating systems.
+ * It may be out-of-date. Consult the documentation of the operating system before considering to use the methods below.
+ *
+ * Platform randomness sources:
+ * Linux   -> `getrandom(2)`(`sys/random.h`), if not available `/dev/urandom` should be used. http://man7.org/linux/man-pages/man2/getrandom.2.html, https://linux.die.net/man/4/urandom
+ * macOS   -> `getentropy(2)`(`sys/random.h`), if not available `/dev/urandom` should be used. https://www.unix.com/man-page/mojave/2/getentropy, https://opensource.apple.com/source/xnu/xnu-517.12.7/bsd/man/man4/random.4.auto.html
+ * FreeBSD -> `getrandom(2)`(`sys/random.h`), if not available `kern.arandom` should be used. https://www.freebsd.org/cgi/man.cgi?query=getrandom, https://www.freebsd.org/cgi/man.cgi?query=random&sektion=4
+ * OpenBSD -> `getentropy(2)`(`unistd.h`), if not available `/dev/urandom` should be used. https://man.openbsd.org/getentropy, https://man.openbsd.org/urandom
+ * Windows -> `BCryptGenRandom`(`bcrypt.h`). https://docs.microsoft.com/en-us/windows/win32/api/bcrypt/nf-bcrypt-bcryptgenrandom
+ */
+
+#if defined(_WIN32)
+#include <windows.h>
+#include <ntstatus.h>
+#include <bcrypt.h>
+#elif defined(__linux__) || defined(__APPLE__) || defined(__FreeBSD__)
+#include <sys/random.h>
+#elif defined(__OpenBSD__)
+#include <unistd.h>
+#else
+#error "Couldn't identify the OS"
+#endif
+
+#include <stddef.h>
+#include <limits.h>
+#include <stdio.h>
+
+
+/* Returns 1 on success, and 0 on failure. */
+static int fill_random(unsigned char* data, size_t size) {
+#if defined(_WIN32)
+    NTSTATUS res = BCryptGenRandom(NULL, data, size, BCRYPT_USE_SYSTEM_PREFERRED_RNG);
+    if (res != STATUS_SUCCESS || size > ULONG_MAX) {
+        return 0;
+    } else {
+        return 1;
+    }
+#elif defined(__linux__) || defined(__FreeBSD__)
+    /* If `getrandom(2)` is not available you should fallback to /dev/urandom */
+    ssize_t res = getrandom(data, size, 0);
+    if (res < 0 || (size_t)res != size ) {
+        return 0;
+    } else {
+        return 1;
+    }
+#elif defined(__APPLE__) || defined(__OpenBSD__)
+    /* If `getentropy(2)` is not available you should fallback to either
+     * `SecRandomCopyBytes` or /dev/urandom */
+    int res = getentropy(data, size);
+    if (res == 0) {
+        return 1;
+    } else {
+        return 0;
+    }
+#endif
+    return 0;
+}
+
+static void print_hex(unsigned char* data, size_t size) {
+    size_t i;
+    printf("0x");
+    for (i = 0; i < size; i++) {
+        printf("%02x", data[i]);
+    }
+    printf("\n");
+}
diff --git a/examples/schnorr.c b/examples/schnorr.c
new file mode 100644
index 00000000000000..82eb07d5d7f007
--- /dev/null
+++ b/examples/schnorr.c
@@ -0,0 +1,152 @@
+/*************************************************************************
+ * Written in 2020-2022 by Elichai Turkel                                *
+ * To the extent possible under law, the author(s) have dedicated all    *
+ * copyright and related and neighboring rights to the software in this  *
+ * file to the public domain worldwide. This software is distributed     *
+ * without any warranty. For the CC0 Public Domain Dedication, see       *
+ * EXAMPLES_COPYING or https://creativecommons.org/publicdomain/zero/1.0 *
+ *************************************************************************/
+
+#include <stdio.h>
+#include <assert.h>
+#include <string.h>
+
+#include <secp256k1.h>
+#include <secp256k1_extrakeys.h>
+#include <secp256k1_schnorrsig.h>
+
+#include "random.h"
+
+int main(void) {
+    unsigned char msg[12] = "Hello World!";
+    unsigned char msg_hash[32];
+    unsigned char tag[17] = "my_fancy_protocol";
+    unsigned char seckey[32];
+    unsigned char randomize[32];
+    unsigned char auxiliary_rand[32];
+    unsigned char serialized_pubkey[32];
+    unsigned char signature[64];
+    int is_signature_valid;
+    int return_val;
+    secp256k1_xonly_pubkey pubkey;
+    secp256k1_keypair keypair;
+    /* The specification in secp256k1_extrakeys.h states that `secp256k1_keypair_create`
+     * needs a context object initialized for signing. And in secp256k1_schnorrsig.h
+     * they state that `secp256k1_schnorrsig_verify` needs a context initialized for
+     * verification, which is why we create a context for both signing and verification
+     * with the SECP256K1_CONTEXT_SIGN and SECP256K1_CONTEXT_VERIFY flags. */
+    secp256k1_context* ctx = secp256k1_context_create(SECP256K1_CONTEXT_SIGN | SECP256K1_CONTEXT_VERIFY);
+    if (!fill_random(randomize, sizeof(randomize))) {
+        printf("Failed to generate randomness\n");
+        return 1;
+    }
+    /* Randomizing the context is recommended to protect against side-channel
+     * leakage See `secp256k1_context_randomize` in secp256k1.h for more
+     * information about it. This should never fail. */
+    return_val = secp256k1_context_randomize(ctx, randomize);
+    assert(return_val);
+
+    /*** Key Generation ***/
+
+    /* If the secret key is zero or out of range (bigger than secp256k1's
+     * order), we try to sample a new key. Note that the probability of this
+     * happening is negligible. */
+    while (1) {
+        if (!fill_random(seckey, sizeof(seckey))) {
+            printf("Failed to generate randomness\n");
+            return 1;
+        }
+        /* Try to create a keypair with a valid context, it should only fail if
+         * the secret key is zero or out of range. */
+        if (secp256k1_keypair_create(ctx, &keypair, seckey)) {
+            break;
+        }
+    }
+
+    /* Extract the X-only public key from the keypair. We pass NULL for
+     * `pk_parity` as the parity isn't needed for signing or verification.
+     * `secp256k1_keypair_xonly_pub` supports returning the parity for
+     * other use cases such as tests or verifying Taproot tweaks.
+     * This should never fail with a valid context and public key. */
+    return_val = secp256k1_keypair_xonly_pub(ctx, &pubkey, NULL, &keypair);
+    assert(return_val);
+
+    /* Serialize the public key. Should always return 1 for a valid public key. */
+    return_val = secp256k1_xonly_pubkey_serialize(ctx, serialized_pubkey, &pubkey);
+    assert(return_val);
+
+    /*** Signing ***/
+
+    /* Instead of signing (possibly very long) messages directly, we sign a
+     * 32-byte hash of the message in this example.
+     *
+     * We use secp256k1_tagged_sha256 to create this hash. This function expects
+     * a context-specific "tag", which restricts the context in which the signed
+     * messages should be considered valid. For example, if protocol A mandates
+     * to use the tag "my_fancy_protocol" and protocol B mandates to use the tag
+     * "my_boring_protocol", then signed messages from protocol A will never be
+     * valid in protocol B (and vice versa), even if keys are reused across
+     * protocols. This implements "domain separation", which is considered good
+     * practice. It avoids attacks in which users are tricked into signing a
+     * message that has intended consequences in the intended context (e.g.,
+     * protocol A) but would have unintended consequences if it were valid in
+     * some other context (e.g., protocol B). */
+    return_val = secp256k1_tagged_sha256(ctx, msg_hash, tag, sizeof(tag), msg, sizeof(msg));
+    assert(return_val);
+
+    /* Generate 32 bytes of randomness to use with BIP-340 schnorr signing. */
+    if (!fill_random(auxiliary_rand, sizeof(auxiliary_rand))) {
+        printf("Failed to generate randomness\n");
+        return 1;
+    }
+
+    /* Generate a Schnorr signature.
+     *
+     * We use the secp256k1_schnorrsig_sign32 function that provides a simple
+     * interface for signing 32-byte messages (which in our case is a hash of
+     * the actual message). BIP-340 recommends passing 32 bytes of randomness
+     * to the signing function to improve security against side-channel attacks.
+     * Signing with a valid context, a 32-byte message, a verified keypair, and
+     * any 32 bytes of auxiliary random data should never fail. */
+    return_val = secp256k1_schnorrsig_sign32(ctx, signature, msg_hash, &keypair, auxiliary_rand);
+    assert(return_val);
+
+    /*** Verification ***/
+
+    /* Deserialize the public key. This will return 0 if the public key can't
+     * be parsed correctly */
+    if (!secp256k1_xonly_pubkey_parse(ctx, &pubkey, serialized_pubkey)) {
+        printf("Failed parsing the public key\n");
+        return 1;
+    }
+
+    /* Compute the tagged hash on the received messages using the same tag as the signer. */
+    return_val = secp256k1_tagged_sha256(ctx, msg_hash, tag, sizeof(tag), msg, sizeof(msg));
+    assert(return_val);
+
+    /* Verify a signature. This will return 1 if it's valid and 0 if it's not. */
+    is_signature_valid = secp256k1_schnorrsig_verify(ctx, signature, msg_hash, 32, &pubkey);
+
+
+    printf("Is the signature valid? %s\n", is_signature_valid ? "true" : "false");
+    printf("Secret Key: ");
+    print_hex(seckey, sizeof(seckey));
+    printf("Public Key: ");
+    print_hex(serialized_pubkey, sizeof(serialized_pubkey));
+    printf("Signature: ");
+    print_hex(signature, sizeof(signature));
+
+    /* This will clear everything from the context and free the memory */
+    secp256k1_context_destroy(ctx);
+
+    /* It's best practice to try to clear secrets from memory after using them.
+     * This is done because some bugs can allow an attacker to leak memory, for
+     * example through "out of bounds" array access (see Heartbleed), Or the OS
+     * swapping them to disk. Hence, we overwrite the secret key buffer with zeros.
+     *
+     * TODO: Prevent these writes from being optimized out, as any good compiler
+     * will remove any writes that aren't used. */
+    memset(seckey, 0, sizeof(seckey));
+
+    return 0;
+}
diff --git a/include/secp256k1.h b/include/secp256k1.h
index 57114b8f26cfca..86ab7e556dde41 100644
--- a/include/secp256k1.h
+++ b/include/secp256k1.h
@@ -169,6 +169,17 @@ typedef int (*secp256k1_nonce_function)(
 #  define SECP256K1_ARG_NONNULL(_x)
 # endif
 
+/** Attribute for marking functions, types, and variables as deprecated */
+#if !defined(SECP256K1_BUILD) && defined(__has_attribute)
+# if __has_attribute(__deprecated__)
+#  define SECP256K1_DEPRECATED(_msg) __attribute__ ((__deprecated__(_msg)))
+# else
+#  define SECP256K1_DEPRECATED(_msg)
+# endif
+#else
+# define SECP256K1_DEPRECATED(_msg)
+#endif
+
 /** All flags' lower 8 bits indicate what they're for. Do not use directly. */
 #define SECP256K1_FLAGS_TYPE_MASK ((1 << 8) - 1)
 #define SECP256K1_FLAGS_TYPE_CONTEXT (1 << 0)
@@ -641,7 +652,8 @@ SECP256K1_API SECP256K1_WARN_UNUSED_RESULT int secp256k1_ec_seckey_negate(
 SECP256K1_API SECP256K1_WARN_UNUSED_RESULT int secp256k1_ec_privkey_negate(
     const secp256k1_context* ctx,
     unsigned char *seckey
-) SECP256K1_ARG_NONNULL(1) SECP256K1_ARG_NONNULL(2);
+) SECP256K1_ARG_NONNULL(1) SECP256K1_ARG_NONNULL(2)
+  SECP256K1_DEPRECATED("Use secp256k1_ec_seckey_negate instead");
 
 /** Negates a public key in place.
  *
@@ -681,7 +693,8 @@ SECP256K1_API SECP256K1_WARN_UNUSED_RESULT int secp256k1_ec_privkey_tweak_add(
     const secp256k1_context* ctx,
     unsigned char *seckey,
     const unsigned char *tweak32
-) SECP256K1_ARG_NONNULL(1) SECP256K1_ARG_NONNULL(2) SECP256K1_ARG_NONNULL(3);
+) SECP256K1_ARG_NONNULL(1) SECP256K1_ARG_NONNULL(2) SECP256K1_ARG_NONNULL(3)
+  SECP256K1_DEPRECATED("Use secp256k1_ec_seckey_tweak_add instead");
 
 /** Tweak a public key by adding tweak times the generator to it.
  *
@@ -727,7 +740,8 @@ SECP256K1_API SECP256K1_WARN_UNUSED_RESULT int secp256k1_ec_privkey_tweak_mul(
     const secp256k1_context* ctx,
     unsigned char *seckey,
     const unsigned char *tweak32
-) SECP256K1_ARG_NONNULL(1) SECP256K1_ARG_NONNULL(2) SECP256K1_ARG_NONNULL(3);
+) SECP256K1_ARG_NONNULL(1) SECP256K1_ARG_NONNULL(2) SECP256K1_ARG_NONNULL(3)
+  SECP256K1_DEPRECATED("Use secp256k1_ec_seckey_tweak_mul instead");
 
 /** Tweak a public key by multiplying it by a tweak value.
  *
@@ -800,7 +814,7 @@ SECP256K1_API SECP256K1_WARN_UNUSED_RESULT int secp256k1_ec_pubkey_combine(
  *  implementations optimized for a specific tag can precompute the SHA256 state
  *  after hashing the tag hashes.
  *
- *  Returns 0 if the arguments are invalid and 1 otherwise.
+ *  Returns: 1 always.
  *  Args:    ctx: pointer to a context object
  *  Out:  hash32: pointer to a 32-byte array to store the resulting hash
  *  In:      tag: pointer to an array containing the tag
diff --git a/include/secp256k1_extrakeys.h b/include/secp256k1_extrakeys.h
index a64d561b60bff8..09cbeaaa8007e5 100644
--- a/include/secp256k1_extrakeys.h
+++ b/include/secp256k1_extrakeys.h
@@ -81,8 +81,7 @@ SECP256K1_API int secp256k1_xonly_pubkey_cmp(
 
 /** Converts a secp256k1_pubkey into a secp256k1_xonly_pubkey.
  *
- *  Returns: 1 if the public key was successfully converted
- *           0 otherwise
+ *  Returns: 1 always.
  *
  *  Args:         ctx: pointer to a context object.
  *  Out: xonly_pubkey: pointer to an x-only public key object for placing the converted public key.
@@ -172,7 +171,7 @@ SECP256K1_API SECP256K1_WARN_UNUSED_RESULT int secp256k1_keypair_create(
 
 /** Get the secret key from a keypair.
  *
- *  Returns: 0 if the arguments are invalid. 1 otherwise.
+ *  Returns: 1 always.
  *  Args:   ctx: pointer to a context object.
  *  Out: seckey: pointer to a 32-byte buffer for the secret key.
  *  In: keypair: pointer to a keypair.
@@ -185,7 +184,7 @@ SECP256K1_API SECP256K1_WARN_UNUSED_RESULT int secp256k1_keypair_sec(
 
 /** Get the public key from a keypair.
  *
- *  Returns: 0 if the arguments are invalid. 1 otherwise.
+ *  Returns: 1 always.
  *  Args:    ctx: pointer to a context object.
  *  Out: pubkey: pointer to a pubkey object. If 1 is returned, it is set to
  *               the keypair public key. If not, it's set to an invalid value.
@@ -202,7 +201,7 @@ SECP256K1_API SECP256K1_WARN_UNUSED_RESULT int secp256k1_keypair_pub(
  *  This is the same as calling secp256k1_keypair_pub and then
  *  secp256k1_xonly_pubkey_from_pubkey.
  *
- *  Returns: 0 if the arguments are invalid. 1 otherwise.
+ *  Returns: 1 always.
  *  Args:   ctx: pointer to a context object.
  *  Out: pubkey: pointer to an xonly_pubkey object. If 1 is returned, it is set
  *               to the keypair public key after converting it to an
diff --git a/include/secp256k1_schnorrsig.h b/include/secp256k1_schnorrsig.h
index e971ddc2aa8f2c..5fedcb07b05191 100644
--- a/include/secp256k1_schnorrsig.h
+++ b/include/secp256k1_schnorrsig.h
@@ -116,7 +116,7 @@ typedef struct {
  *                BIP-340 "Default Signing" for a full explanation of this
  *                argument and for guidance if randomness is expensive.
  */
-SECP256K1_API int secp256k1_schnorrsig_sign(
+SECP256K1_API int secp256k1_schnorrsig_sign32(
     const secp256k1_context* ctx,
     unsigned char *sig64,
     const unsigned char *msg32,
@@ -124,6 +124,17 @@ SECP256K1_API int secp256k1_schnorrsig_sign(
     const unsigned char *aux_rand32
 ) SECP256K1_ARG_NONNULL(1) SECP256K1_ARG_NONNULL(2) SECP256K1_ARG_NONNULL(3) SECP256K1_ARG_NONNULL(4);
 
+/** Same as secp256k1_schnorrsig_sign32, but DEPRECATED. Will be removed in
+ *  future versions. */
+SECP256K1_API int secp256k1_schnorrsig_sign(
+    const secp256k1_context* ctx,
+    unsigned char *sig64,
+    const unsigned char *msg32,
+    const secp256k1_keypair *keypair,
+    const unsigned char *aux_rand32
+) SECP256K1_ARG_NONNULL(1) SECP256K1_ARG_NONNULL(2) SECP256K1_ARG_NONNULL(3) SECP256K1_ARG_NONNULL(4)
+  SECP256K1_DEPRECATED("Use secp256k1_schnorrsig_sign32 instead");
+
 /** Create a Schnorr signature with a more flexible API.
  *
  *  Same arguments as secp256k1_schnorrsig_sign except that it allows signing
diff --git a/sage/group_prover.sage b/sage/group_prover.sage
index b200bfeae3d1c6..9305c215d59288 100644
--- a/sage/group_prover.sage
+++ b/sage/group_prover.sage
@@ -164,6 +164,9 @@ class constraints:
   def negate(self):
     return constraints(zero=self.nonzero, nonzero=self.zero)
 
+  def map(self, fun):
+    return constraints(zero={fun(k): v for k, v in self.zero.items()}, nonzero={fun(k): v for k, v in self.nonzero.items()})
+
   def __add__(self, other):
     zero = self.zero.copy()
     zero.update(other.zero)
@@ -177,6 +180,30 @@ class constraints:
   def __repr__(self):
     return "%s" % self
 
+def normalize_factor(p):
+  """Normalizes the sign of primitive polynomials (as returned by factor())
+
+  This function ensures that the polynomial has a positive leading coefficient.
+
+  This is necessary because recent sage versions (starting with v9.3 or v9.4,
+  we don't know) are inconsistent about the placement of the minus sign in
+  polynomial factorizations:
+  ```
+  sage: R.<ax,bx,ay,by,Az,Bz,Ai,Bi> = PolynomialRing(QQ,8,order='invlex')
+  sage: R((-2 * (bx - ax)) ^ 1).factor()
+  (-2) * (bx - ax)
+  sage: R((-2 * (bx - ax)) ^ 2).factor()
+  (4) * (-bx + ax)^2
+  sage: R((-2 * (bx - ax)) ^ 3).factor()
+  (8) * (-bx + ax)^3
+  ```
+  """
+  # Assert p is not 0 and that its non-zero coeffients are coprime.
+  # (We could just work with the primitive part p/p.content() but we want to be
+  # aware if factor() does not return a primitive part in future sage versions.)
+  assert p.content() == 1
+  # Ensure that the first non-zero coefficient is positive.
+  return p if p.lc() > 0 else -p
 
 def conflicts(R, con):
   """Check whether any of the passed non-zero assumptions is implied by the zero assumptions"""
@@ -204,10 +231,10 @@ def get_nonzero_set(R, assume):
   nonzero = set()
   for nz in map(numerator, assume.nonzero):
     for (f,n) in nz.factor():
-      nonzero.add(f)
+      nonzero.add(normalize_factor(f))
     rnz = zero.reduce(nz)
     for (f,n) in rnz.factor():
-      nonzero.add(f)
+      nonzero.add(normalize_factor(f))
   return nonzero
 
 
@@ -222,27 +249,27 @@ def prove_nonzero(R, exprs, assume):
       return (False, [exprs[expr]])
   allexprs = reduce(lambda a,b: numerator(a)*numerator(b), exprs, 1)
   for (f, n) in allexprs.factor():
-    if f not in nonzero:
+    if normalize_factor(f) not in nonzero:
       ok = False
   if ok:
     return (True, None)
   ok = True
-  for (f, n) in zero.reduce(numerator(allexprs)).factor():
-    if f not in nonzero:
+  for (f, n) in zero.reduce(allexprs).factor():
+    if normalize_factor(f) not in nonzero:
       ok = False
   if ok:
     return (True, None)
   ok = True
   for expr in exprs:
     for (f,n) in numerator(expr).factor():
-      if f not in nonzero:
+      if normalize_factor(f) not in nonzero:
         ok = False
   if ok:
     return (True, None)
   ok = True
   for expr in exprs:
     for (f,n) in zero.reduce(numerator(expr)).factor():
-      if f not in nonzero:
+      if normalize_factor(f) not in nonzero:
         expl.add(exprs[expr])
   if expl:
     return (False, list(expl))
@@ -254,7 +281,7 @@ def prove_zero(R, exprs, assume):
   """Check whether all of the passed expressions are provably zero, given assumptions"""
   r, e = prove_nonzero(R, dict(map(lambda x: (fastfrac(R, x.bot, 1), exprs[x]), exprs)), assume)
   if not r:
-    return (False, map(lambda x: "Possibly zero denominator: %s" % x, e))
+    return (False, list(map(lambda x: "Possibly zero denominator: %s" % x, e)))
   zero = R.ideal(list(map(numerator, assume.zero)))
   nonzero = prod(x for x in assume.nonzero)
   expl = []
@@ -279,8 +306,8 @@ def describe_extra(R, assume, assumeExtra):
     if base not in zero:
       add = []
       for (f, n) in numerator(base).factor():
-        if f not in nonzero:
-          add += ["%s" % f]
+        if normalize_factor(f) not in nonzero:
+          add += ["%s" % normalize_factor(f)]
       if add:
         ret.add((" * ".join(add)) + " = 0 [%s]" % assumeExtra.zero[base])
   # Iterate over the extra nonzero expressions
@@ -288,8 +315,8 @@ def describe_extra(R, assume, assumeExtra):
     nzr = zeroextra.reduce(numerator(nz))
     if nzr not in zeroextra:
       for (f,n) in nzr.factor():
-        if zeroextra.reduce(f) not in nonzero:
-          ret.add("%s != 0" % zeroextra.reduce(f))
+        if normalize_factor(zeroextra.reduce(f)) not in nonzero:
+          ret.add("%s != 0" % normalize_factor(zeroextra.reduce(f)))
   return ", ".join(x for x in ret)
 
 
@@ -299,22 +326,21 @@ def check_symbolic(R, assumeLaw, assumeAssert, assumeBranch, require):
 
   if conflicts(R, assume):
     # This formula does not apply
-    return None
+    return (True, None)
 
   describe = describe_extra(R, assumeLaw + assumeBranch, assumeAssert)
+  if describe != "":
+    describe = " (assuming " + describe + ")"
 
   ok, msg = prove_zero(R, require.zero, assume)
   if not ok:
-    return "FAIL, %s fails (assuming %s)" % (str(msg), describe)
+    return (False, "FAIL, %s fails%s" % (str(msg), describe))
 
   res, expl = prove_nonzero(R, require.nonzero, assume)
   if not res:
-    return "FAIL, %s fails (assuming %s)" % (str(expl), describe)
+    return (False, "FAIL, %s fails%s" % (str(expl), describe))
 
-  if describe != "":
-    return "OK (assuming %s)" % describe
-  else:
-    return "OK"
+  return (True, "OK%s" % describe)
 
 
 def concrete_verify(c):
diff --git a/sage/prove_group_implementations.sage b/sage/prove_group_implementations.sage
index a97e732f7fa3ab..96ce33506a4622 100644
--- a/sage/prove_group_implementations.sage
+++ b/sage/prove_group_implementations.sage
@@ -8,25 +8,20 @@ load("weierstrass_prover.sage")
 def formula_secp256k1_gej_double_var(a):
   """libsecp256k1's secp256k1_gej_double_var, used by various addition functions"""
   rz = a.Z * a.Y
-  rz = rz * 2
-  t1 = a.X^2
-  t1 = t1 * 3
-  t2 = t1^2
-  t3 = a.Y^2
-  t3 = t3 * 2
-  t4 = t3^2
-  t4 = t4 * 2
-  t3 = t3 * a.X
-  rx = t3
-  rx = rx * 4
-  rx = -rx
-  rx = rx + t2
-  t2 = -t2
-  t3 = t3 * 6
-  t3 = t3 + t2
-  ry = t1 * t3
-  t2 = -t4
-  ry = ry + t2
+  s = a.Y^2
+  l = a.X^2
+  l = l * 3
+  l = l / 2
+  t = -s
+  t = t * a.X
+  rx = l^2
+  rx = rx + t
+  rx = rx + t
+  s = s^2
+  t = t + rx
+  ry = t * l
+  ry = ry + s
+  ry = -ry
   return jacobianpoint(rx, ry, rz)
 
 def formula_secp256k1_gej_add_var(branch, a, b):
@@ -197,7 +192,8 @@ def formula_secp256k1_gej_add_ge(branch, a, b):
     rr_alt = rr
     m_alt = m
   n = m_alt^2
-  q = n * t
+  q = -t
+  q = q * n
   n = n^2
   if degenerate:
     n = m
@@ -210,8 +206,6 @@ def formula_secp256k1_gej_add_ge(branch, a, b):
     zeroes.update({rz : 'r.z=0'})
   else:
     nonzeroes.update({rz : 'r.z!=0'})
-  rz = rz * 2
-  q = -q
   t = t + q
   rx = t
   t = t * 2
@@ -219,8 +213,7 @@ def formula_secp256k1_gej_add_ge(branch, a, b):
   t = t * rr_alt
   t = t + n
   ry = -t
-  rx = rx * 4
-  ry = ry * 4
+  ry = ry / 2
   if a_infinity:
     rx = b.X
     ry = b.Y
@@ -292,15 +285,18 @@ def formula_secp256k1_gej_add_ge_old(branch, a, b):
   return (constraints(zero={b.Z - 1 : 'b.z=1', b.Infinity : 'b_finite'}), constraints(zero=zero, nonzero=nonzero), jacobianpoint(rx, ry, rz))
 
 if __name__ == "__main__":
-  check_symbolic_jacobian_weierstrass("secp256k1_gej_add_var", 0, 7, 5, formula_secp256k1_gej_add_var)
-  check_symbolic_jacobian_weierstrass("secp256k1_gej_add_ge_var", 0, 7, 5, formula_secp256k1_gej_add_ge_var)
-  check_symbolic_jacobian_weierstrass("secp256k1_gej_add_zinv_var", 0, 7, 5, formula_secp256k1_gej_add_zinv_var)
-  check_symbolic_jacobian_weierstrass("secp256k1_gej_add_ge", 0, 7, 16, formula_secp256k1_gej_add_ge)
-  check_symbolic_jacobian_weierstrass("secp256k1_gej_add_ge_old [should fail]", 0, 7, 4, formula_secp256k1_gej_add_ge_old)
+  success = True
+  success = success & check_symbolic_jacobian_weierstrass("secp256k1_gej_add_var", 0, 7, 5, formula_secp256k1_gej_add_var)
+  success = success & check_symbolic_jacobian_weierstrass("secp256k1_gej_add_ge_var", 0, 7, 5, formula_secp256k1_gej_add_ge_var)
+  success = success & check_symbolic_jacobian_weierstrass("secp256k1_gej_add_zinv_var", 0, 7, 5, formula_secp256k1_gej_add_zinv_var)
+  success = success & check_symbolic_jacobian_weierstrass("secp256k1_gej_add_ge", 0, 7, 16, formula_secp256k1_gej_add_ge)
+  success = success & (not check_symbolic_jacobian_weierstrass("secp256k1_gej_add_ge_old [should fail]", 0, 7, 4, formula_secp256k1_gej_add_ge_old))
 
   if len(sys.argv) >= 2 and sys.argv[1] == "--exhaustive":
-    check_exhaustive_jacobian_weierstrass("secp256k1_gej_add_var", 0, 7, 5, formula_secp256k1_gej_add_var, 43)
-    check_exhaustive_jacobian_weierstrass("secp256k1_gej_add_ge_var", 0, 7, 5, formula_secp256k1_gej_add_ge_var, 43)
-    check_exhaustive_jacobian_weierstrass("secp256k1_gej_add_zinv_var", 0, 7, 5, formula_secp256k1_gej_add_zinv_var, 43)
-    check_exhaustive_jacobian_weierstrass("secp256k1_gej_add_ge", 0, 7, 16, formula_secp256k1_gej_add_ge, 43)
-    check_exhaustive_jacobian_weierstrass("secp256k1_gej_add_ge_old [should fail]", 0, 7, 4, formula_secp256k1_gej_add_ge_old, 43)
+    success = success & check_exhaustive_jacobian_weierstrass("secp256k1_gej_add_var", 0, 7, 5, formula_secp256k1_gej_add_var, 43)
+    success = success & check_exhaustive_jacobian_weierstrass("secp256k1_gej_add_ge_var", 0, 7, 5, formula_secp256k1_gej_add_ge_var, 43)
+    success = success & check_exhaustive_jacobian_weierstrass("secp256k1_gej_add_zinv_var", 0, 7, 5, formula_secp256k1_gej_add_zinv_var, 43)
+    success = success & check_exhaustive_jacobian_weierstrass("secp256k1_gej_add_ge", 0, 7, 16, formula_secp256k1_gej_add_ge, 43)
+    success = success & (not check_exhaustive_jacobian_weierstrass("secp256k1_gej_add_ge_old [should fail]", 0, 7, 4, formula_secp256k1_gej_add_ge_old, 43))
+
+  sys.exit(int(not success))
diff --git a/sage/weierstrass_prover.sage b/sage/weierstrass_prover.sage
index b770c6dafe2f02..be9cfd4c76d875 100644
--- a/sage/weierstrass_prover.sage
+++ b/sage/weierstrass_prover.sage
@@ -184,6 +184,7 @@ def check_exhaustive_jacobian_weierstrass(name, A, B, branches, formula, p):
       if r:
         points.append(point)
 
+  ret = True
   for za in range(1, p):
     for zb in range(1, p):
       for pa in points:
@@ -211,8 +212,11 @@ def check_exhaustive_jacobian_weierstrass(name, A, B, branches, formula, p):
                         match = True
                       r, e = concrete_verify(require)
                       if not r:
+                        ret = False
                         print("  failure in branch %i for (%s,%s,%s,%s) + (%s,%s,%s,%s) = (%s,%s,%s,%s): %s" % (branch, pA.X, pA.Y, pA.Z, pA.Infinity, pB.X, pB.Y, pB.Z, pB.Infinity, pC.X, pC.Y, pC.Z, pC.Infinity, e))
+
   print()
+  return ret
 
 
 def check_symbolic_function(R, assumeAssert, assumeBranch, f, A, B, pa, pb, pA, pB, pC):
@@ -244,15 +248,21 @@ def check_symbolic_jacobian_weierstrass(name, A, B, branches, formula):
 
   print("Formula " + name + ":")
   count = 0
+  ret = True
   for branch in range(branches):
     assumeFormula, assumeBranch, pC = formula(branch, pA, pB)
+    assumeBranch = assumeBranch.map(lift)
+    assumeFormula = assumeFormula.map(lift)
     pC.X = lift(pC.X)
     pC.Y = lift(pC.Y)
     pC.Z = lift(pC.Z)
     pC.Infinity = lift(pC.Infinity)
 
     for key in laws_jacobian_weierstrass:
-      res[key].append((check_symbolic_function(R, assumeFormula, assumeBranch, laws_jacobian_weierstrass[key], A, B, pa, pb, pA, pB, pC), branch))
+      success, msg = check_symbolic_function(R, assumeFormula, assumeBranch, laws_jacobian_weierstrass[key], A, B, pa, pb, pA, pB, pC)
+      if not success:
+        ret = False
+      res[key].append((msg, branch))
 
   for key in res:
     print("  %s:" % key)
@@ -262,3 +272,4 @@ def check_symbolic_jacobian_weierstrass(name, A, B, branches, formula):
         print("    branch %i: %s" % (x[1], x[0]))
 
   print()
+  return ret
diff --git a/src/bench_internal.c b/src/bench_internal.c
index aed82161273d9e..3c145f306c5918 100644
--- a/src/bench_internal.c
+++ b/src/bench_internal.c
@@ -140,6 +140,15 @@ void bench_scalar_inverse_var(void* arg, int iters) {
     CHECK(j <= iters);
 }
 
+void bench_field_half(void* arg, int iters) {
+    int i;
+    bench_inv *data = (bench_inv*)arg;
+
+    for (i = 0; i < iters; i++) {
+        secp256k1_fe_half(&data->fe[0]);
+    }
+}
+
 void bench_field_normalize(void* arg, int iters) {
     int i;
     bench_inv *data = (bench_inv*)arg;
@@ -354,6 +363,7 @@ int main(int argc, char **argv) {
     if (d || have_flag(argc, argv, "scalar") || have_flag(argc, argv, "inverse")) run_benchmark("scalar_inverse", bench_scalar_inverse, bench_setup, NULL, &data, 10, iters);
     if (d || have_flag(argc, argv, "scalar") || have_flag(argc, argv, "inverse")) run_benchmark("scalar_inverse_var", bench_scalar_inverse_var, bench_setup, NULL, &data, 10, iters);
 
+    if (d || have_flag(argc, argv, "field") || have_flag(argc, argv, "half")) run_benchmark("field_half", bench_field_half, bench_setup, NULL, &data, 10, iters*100);
     if (d || have_flag(argc, argv, "field") || have_flag(argc, argv, "normalize")) run_benchmark("field_normalize", bench_field_normalize, bench_setup, NULL, &data, 10, iters*100);
     if (d || have_flag(argc, argv, "field") || have_flag(argc, argv, "normalize")) run_benchmark("field_normalize_weak", bench_field_normalize_weak, bench_setup, NULL, &data, 10, iters*100);
     if (d || have_flag(argc, argv, "field") || have_flag(argc, argv, "sqr")) run_benchmark("field_sqr", bench_field_sqr, bench_setup, NULL, &data, 10, iters*10);
diff --git a/src/ecmult_compute_table.h b/src/ecmult_compute_table.h
new file mode 100644
index 00000000000000..665f87ff3d5b5d
--- /dev/null
+++ b/src/ecmult_compute_table.h
@@ -0,0 +1,16 @@
+/*****************************************************************************************************
+ * Copyright (c) 2013, 2014, 2017, 2021 Pieter Wuille, Andrew Poelstra, Jonas Nick, Russell O'Connor *
+ * Distributed under the MIT software license, see the accompanying                                  *
+ * file COPYING or https://www.opensource.org/licenses/mit-license.php.                              *
+ *****************************************************************************************************/
+
+#ifndef SECP256K1_ECMULT_COMPUTE_TABLE_H
+#define SECP256K1_ECMULT_COMPUTE_TABLE_H
+
+/* Construct table of all odd multiples of gen in range 1..(2**(window_g-1)-1). */
+static void secp256k1_ecmult_compute_table(secp256k1_ge_storage* table, int window_g, const secp256k1_gej* gen);
+
+/* Like secp256k1_ecmult_compute_table, but one for both gen and gen*2^128. */
+static void secp256k1_ecmult_compute_two_tables(secp256k1_ge_storage* table, secp256k1_ge_storage* table_128, int window_g, const secp256k1_ge* gen);
+
+#endif /* SECP256K1_ECMULT_COMPUTE_TABLE_H */
diff --git a/src/ecmult_compute_table_impl.h b/src/ecmult_compute_table_impl.h
new file mode 100644
index 00000000000000..69d59ce595604e
--- /dev/null
+++ b/src/ecmult_compute_table_impl.h
@@ -0,0 +1,49 @@
+/*****************************************************************************************************
+ * Copyright (c) 2013, 2014, 2017, 2021 Pieter Wuille, Andrew Poelstra, Jonas Nick, Russell O'Connor *
+ * Distributed under the MIT software license, see the accompanying                                  *
+ * file COPYING or https://www.opensource.org/licenses/mit-license.php.                              *
+ *****************************************************************************************************/
+
+#ifndef SECP256K1_ECMULT_COMPUTE_TABLE_IMPL_H
+#define SECP256K1_ECMULT_COMPUTE_TABLE_IMPL_H
+
+#include "ecmult_compute_table.h"
+#include "group_impl.h"
+#include "field_impl.h"
+#include "ecmult.h"
+#include "util.h"
+
+static void secp256k1_ecmult_compute_table(secp256k1_ge_storage* table, int window_g, const secp256k1_gej* gen) {
+    secp256k1_gej gj;
+    secp256k1_ge ge, dgen;
+    int j;
+
+    gj = *gen;
+    secp256k1_ge_set_gej_var(&ge, &gj);
+    secp256k1_ge_to_storage(&table[0], &ge);
+
+    secp256k1_gej_double_var(&gj, gen, NULL);
+    secp256k1_ge_set_gej_var(&dgen, &gj);
+
+    for (j = 1; j < ECMULT_TABLE_SIZE(window_g); ++j) {
+        secp256k1_gej_set_ge(&gj, &ge);
+        secp256k1_gej_add_ge_var(&gj, &gj, &dgen, NULL);
+        secp256k1_ge_set_gej_var(&ge, &gj);
+        secp256k1_ge_to_storage(&table[j], &ge);
+    }
+}
+
+/* Like secp256k1_ecmult_compute_table, but one for both gen and gen*2^128. */
+static void secp256k1_ecmult_compute_two_tables(secp256k1_ge_storage* table, secp256k1_ge_storage* table_128, int window_g, const secp256k1_ge* gen) {
+    secp256k1_gej gj;
+    int i;
+
+    secp256k1_gej_set_ge(&gj, gen);
+    secp256k1_ecmult_compute_table(table, window_g, &gj);
+    for (i = 0; i < 128; ++i) {
+        secp256k1_gej_double_var(&gj, &gj, NULL);
+    }
+    secp256k1_ecmult_compute_table(table_128, window_g, &gj);
+}
+
+#endif /* SECP256K1_ECMULT_COMPUTE_TABLE_IMPL_H */
diff --git a/src/ecmult_const_impl.h b/src/ecmult_const_impl.h
index 30b151ff9abdd5..12dbcc6c5b6924 100644
--- a/src/ecmult_const_impl.h
+++ b/src/ecmult_const_impl.h
@@ -12,6 +12,19 @@
 #include "ecmult_const.h"
 #include "ecmult_impl.h"
 
+/** Fill a table 'pre' with precomputed odd multiples of a.
+ *
+ *  The resulting point set is brought to a single constant Z denominator, stores the X and Y
+ *  coordinates as ge_storage points in pre, and stores the global Z in globalz.
+ *  It only operates on tables sized for WINDOW_A wnaf multiples.
+ */
+static void secp256k1_ecmult_odd_multiples_table_globalz_windowa(secp256k1_ge *pre, secp256k1_fe *globalz, const secp256k1_gej *a) {
+    secp256k1_fe zr[ECMULT_TABLE_SIZE(WINDOW_A)];
+
+    secp256k1_ecmult_odd_multiples_table(ECMULT_TABLE_SIZE(WINDOW_A), pre, zr, globalz, a);
+    secp256k1_ge_table_set_globalz(ECMULT_TABLE_SIZE(WINDOW_A), pre, zr);
+}
+
 /* This is like `ECMULT_TABLE_GET_GE` but is constant time */
 #define ECMULT_CONST_TABLE_GET_GE(r,pre,n,w) do { \
     int m = 0; \
@@ -40,7 +53,6 @@
     secp256k1_fe_cmov(&(r)->y, &neg_y, (n) != abs_n); \
 } while(0)
 
-
 /** Convert a number to WNAF notation.
  *  The number becomes represented by sum(2^{wi} * wnaf[i], i=0..WNAF_SIZE(w)+1) - return_val.
  *  It has the following guarantees:
@@ -56,7 +68,7 @@
  */
 static int secp256k1_wnaf_const(int *wnaf, const secp256k1_scalar *scalar, int w, int size) {
     int global_sign;
-    int skew = 0;
+    int skew;
     int word = 0;
 
     /* 1 2 3 */
@@ -64,9 +76,7 @@ static int secp256k1_wnaf_const(int *wnaf, const secp256k1_scalar *scalar, int w
     int u;
 
     int flip;
-    int bit;
-    secp256k1_scalar s;
-    int not_neg_one;
+    secp256k1_scalar s = *scalar;
 
     VERIFY_CHECK(w > 0);
     VERIFY_CHECK(size > 0);
@@ -74,33 +84,19 @@ static int secp256k1_wnaf_const(int *wnaf, const secp256k1_scalar *scalar, int w
     /* Note that we cannot handle even numbers by negating them to be odd, as is
      * done in other implementations, since if our scalars were specified to have
      * width < 256 for performance reasons, their negations would have width 256
-     * and we'd lose any performance benefit. Instead, we use a technique from
-     * Section 4.2 of the Okeya/Tagaki paper, which is to add either 1 (for even)
-     * or 2 (for odd) to the number we are encoding, returning a skew value indicating
+     * and we'd lose any performance benefit. Instead, we use a variation of a
+     * technique from Section 4.2 of the Okeya/Tagaki paper, which is to add 1 to the
+     * number we are encoding when it is even, returning a skew value indicating
      * this, and having the caller compensate after doing the multiplication.
      *
      * In fact, we _do_ want to negate numbers to minimize their bit-lengths (and in
      * particular, to ensure that the outputs from the endomorphism-split fit into
-     * 128 bits). If we negate, the parity of our number flips, inverting which of
-     * {1, 2} we want to add to the scalar when ensuring that it's odd. Further
-     * complicating things, -1 interacts badly with `secp256k1_scalar_cadd_bit` and
-     * we need to special-case it in this logic. */
-    flip = secp256k1_scalar_is_high(scalar);
-    /* We add 1 to even numbers, 2 to odd ones, noting that negation flips parity */
-    bit = flip ^ !secp256k1_scalar_is_even(scalar);
-    /* We check for negative one, since adding 2 to it will cause an overflow */
-    secp256k1_scalar_negate(&s, scalar);
-    not_neg_one = !secp256k1_scalar_is_one(&s);
-    s = *scalar;
-    secp256k1_scalar_cadd_bit(&s, bit, not_neg_one);
-    /* If we had negative one, flip == 1, s.d[0] == 0, bit == 1, so caller expects
-     * that we added two to it and flipped it. In fact for -1 these operations are
-     * identical. We only flipped, but since skewing is required (in the sense that
-     * the skew must be 1 or 2, never zero) and flipping is not, we need to change
-     * our flags to claim that we only skewed. */
+     * 128 bits). If we negate, the parity of our number flips, affecting whether
+     * we want to add to the scalar to ensure that it's odd. */
+    flip = secp256k1_scalar_is_high(&s);
+    skew = flip ^ secp256k1_scalar_is_even(&s);
+    secp256k1_scalar_cadd_bit(&s, 0, skew);
     global_sign = secp256k1_scalar_cond_negate(&s, flip);
-    global_sign *= not_neg_one * 2 - 1;
-    skew = 1 << bit;
 
     /* 4 */
     u_last = secp256k1_scalar_shr_int(&s, w);
@@ -214,42 +210,22 @@ static void secp256k1_ecmult_const(secp256k1_gej *r, const secp256k1_ge *a, cons
         }
     }
 
-    secp256k1_fe_mul(&r->z, &r->z, &Z);
-
     {
         /* Correct for wNAF skew */
-        secp256k1_ge correction = *a;
-        secp256k1_ge_storage correction_1_stor;
-        secp256k1_ge_storage correction_lam_stor;
-        secp256k1_ge_storage a2_stor;
         secp256k1_gej tmpj;
-        secp256k1_gej_set_ge(&tmpj, &correction);
-        secp256k1_gej_double_var(&tmpj, &tmpj, NULL);
-        secp256k1_ge_set_gej(&correction, &tmpj);
-        secp256k1_ge_to_storage(&correction_1_stor, a);
-        if (size > 128) {
-            secp256k1_ge_to_storage(&correction_lam_stor, a);
-        }
-        secp256k1_ge_to_storage(&a2_stor, &correction);
 
-        /* For odd numbers this is 2a (so replace it), for even ones a (so no-op) */
-        secp256k1_ge_storage_cmov(&correction_1_stor, &a2_stor, skew_1 == 2);
-        if (size > 128) {
-            secp256k1_ge_storage_cmov(&correction_lam_stor, &a2_stor, skew_lam == 2);
-        }
-
-        /* Apply the correction */
-        secp256k1_ge_from_storage(&correction, &correction_1_stor);
-        secp256k1_ge_neg(&correction, &correction);
-        secp256k1_gej_add_ge(r, r, &correction);
+        secp256k1_ge_neg(&tmpa, &pre_a[0]);
+        secp256k1_gej_add_ge(&tmpj, r, &tmpa);
+        secp256k1_gej_cmov(r, &tmpj, skew_1);
 
         if (size > 128) {
-            secp256k1_ge_from_storage(&correction, &correction_lam_stor);
-            secp256k1_ge_neg(&correction, &correction);
-            secp256k1_ge_mul_lambda(&correction, &correction);
-            secp256k1_gej_add_ge(r, r, &correction);
+            secp256k1_ge_neg(&tmpa, &pre_a_lam[0]);
+            secp256k1_gej_add_ge(&tmpj, r, &tmpa);
+            secp256k1_gej_cmov(r, &tmpj, skew_lam);
         }
     }
+
+    secp256k1_fe_mul(&r->z, &r->z, &Z);
 }
 
 #endif /* SECP256K1_ECMULT_CONST_IMPL_H */
diff --git a/src/ecmult_gen_prec.h b/src/ecmult_gen_compute_table.h
similarity index 60%
rename from src/ecmult_gen_prec.h
rename to src/ecmult_gen_compute_table.h
index 0cfcde9b7994a3..e577158d923492 100644
--- a/src/ecmult_gen_prec.h
+++ b/src/ecmult_gen_compute_table.h
@@ -4,11 +4,11 @@
  * file COPYING or https://www.opensource.org/licenses/mit-license.php.*
  ***********************************************************************/
 
-#ifndef SECP256K1_ECMULT_GEN_PREC_H
-#define SECP256K1_ECMULT_GEN_PREC_H
+#ifndef SECP256K1_ECMULT_GEN_COMPUTE_TABLE_H
+#define SECP256K1_ECMULT_GEN_COMPUTE_TABLE_H
 
 #include "ecmult_gen.h"
 
-static void secp256k1_ecmult_gen_create_prec_table(secp256k1_ge_storage* table, const secp256k1_ge* gen, int bits);
+static void secp256k1_ecmult_gen_compute_table(secp256k1_ge_storage* table, const secp256k1_ge* gen, int bits);
 
-#endif /* SECP256K1_ECMULT_GEN_PREC_H */
+#endif /* SECP256K1_ECMULT_GEN_COMPUTE_TABLE_H */
diff --git a/src/ecmult_gen_prec_impl.h b/src/ecmult_gen_compute_table_impl.h
similarity index 90%
rename from src/ecmult_gen_prec_impl.h
rename to src/ecmult_gen_compute_table_impl.h
index bac76c8b13bb85..ff6a2992dc997d 100644
--- a/src/ecmult_gen_prec_impl.h
+++ b/src/ecmult_gen_compute_table_impl.h
@@ -4,16 +4,16 @@
  * file COPYING or https://www.opensource.org/licenses/mit-license.php.*
  ***********************************************************************/
 
-#ifndef SECP256K1_ECMULT_GEN_PREC_IMPL_H
-#define SECP256K1_ECMULT_GEN_PREC_IMPL_H
+#ifndef SECP256K1_ECMULT_GEN_COMPUTE_TABLE_IMPL_H
+#define SECP256K1_ECMULT_GEN_COMPUTE_TABLE_IMPL_H
 
-#include "ecmult_gen_prec.h"
+#include "ecmult_gen_compute_table.h"
 #include "group_impl.h"
 #include "field_impl.h"
 #include "ecmult_gen.h"
 #include "util.h"
 
-static void secp256k1_ecmult_gen_create_prec_table(secp256k1_ge_storage* table, const secp256k1_ge* gen, int bits) {
+static void secp256k1_ecmult_gen_compute_table(secp256k1_ge_storage* table, const secp256k1_ge* gen, int bits) {
     int g = ECMULT_GEN_PREC_G(bits);
     int n = ECMULT_GEN_PREC_N(bits);
 
@@ -78,4 +78,4 @@ static void secp256k1_ecmult_gen_create_prec_table(secp256k1_ge_storage* table,
     free(prec);
 }
 
-#endif /* SECP256K1_ECMULT_GEN_PREC_IMPL_H */
+#endif /* SECP256K1_ECMULT_GEN_COMPUTE_TABLE_IMPL_H */
diff --git a/src/ecmult_gen_impl.h b/src/ecmult_gen_impl.h
index 6a6ab9a4b517b7..2c8a503acc4831 100644
--- a/src/ecmult_gen_impl.h
+++ b/src/ecmult_gen_impl.h
@@ -12,7 +12,7 @@
 #include "group.h"
 #include "ecmult_gen.h"
 #include "hash_impl.h"
-#include "ecmult_gen_static_prec_table.h"
+#include "precomputed_ecmult_gen.h"
 
 static void secp256k1_ecmult_gen_context_build(secp256k1_ecmult_gen_context *ctx) {
     secp256k1_ecmult_gen_blind(ctx, NULL);
diff --git a/src/ecmult_impl.h b/src/ecmult_impl.h
index 5bd4d4d23dc1aa..bbc820c77c748e 100644
--- a/src/ecmult_impl.h
+++ b/src/ecmult_impl.h
@@ -14,7 +14,7 @@
 #include "group.h"
 #include "scalar.h"
 #include "ecmult.h"
-#include "ecmult_static_pre_g.h"
+#include "precomputed_ecmult.h"
 
 #if defined(EXHAUSTIVE_TEST_ORDER)
 /* We need to lower these values for exhaustive tests because
@@ -47,7 +47,7 @@
 
 /* The number of objects allocated on the scratch space for ecmult_multi algorithms */
 #define PIPPENGER_SCRATCH_OBJECTS 6
-#define STRAUSS_SCRATCH_OBJECTS 7
+#define STRAUSS_SCRATCH_OBJECTS 5
 
 #define PIPPENGER_MAX_BUCKET_WINDOW 12
 
@@ -56,14 +56,23 @@
 
 #define ECMULT_MAX_POINTS_PER_BATCH 5000000
 
-/** Fill a table 'prej' with precomputed odd multiples of a. Prej will contain
- *  the values [1*a,3*a,...,(2*n-1)*a], so it space for n values. zr[0] will
- *  contain prej[0].z / a.z. The other zr[i] values = prej[i].z / prej[i-1].z.
- *  Prej's Z values are undefined, except for the last value.
+/** Fill a table 'pre_a' with precomputed odd multiples of a.
+ *  pre_a will contain [1*a,3*a,...,(2*n-1)*a], so it needs space for n group elements.
+ *  zr needs space for n field elements.
+ *
+ *  Although pre_a is an array of _ge rather than _gej, it actually represents elements
+ *  in Jacobian coordinates with their z coordinates omitted. The omitted z-coordinates
+ *  can be recovered using z and zr. Using the notation z(b) to represent the omitted
+ *  z coordinate of b:
+ *  - z(pre_a[n-1]) = 'z'
+ *  - z(pre_a[i-1]) = z(pre_a[i]) / zr[i] for n > i > 0
+ *
+ *  Lastly the zr[0] value, which isn't used above, is set so that:
+ *  - a.z = z(pre_a[0]) / zr[0]
  */
-static void secp256k1_ecmult_odd_multiples_table(int n, secp256k1_gej *prej, secp256k1_fe *zr, const secp256k1_gej *a) {
-    secp256k1_gej d;
-    secp256k1_ge a_ge, d_ge;
+static void secp256k1_ecmult_odd_multiples_table(int n, secp256k1_ge *pre_a, secp256k1_fe *zr, secp256k1_fe *z, const secp256k1_gej *a) {
+    secp256k1_gej d, ai;
+    secp256k1_ge d_ge;
     int i;
 
     VERIFY_CHECK(!a->infinity);
@@ -71,75 +80,74 @@ static void secp256k1_ecmult_odd_multiples_table(int n, secp256k1_gej *prej, sec
     secp256k1_gej_double_var(&d, a, NULL);
 
     /*
-     * Perform the additions on an isomorphism where 'd' is affine: drop the z coordinate
-     * of 'd', and scale the 1P starting value's x/y coordinates without changing its z.
+     * Perform the additions using an isomorphic curve Y^2 = X^3 + 7*C^6 where C := d.z.
+     * The isomorphism, phi, maps a secp256k1 point (x, y) to the point (x*C^2, y*C^3) on the other curve.
+     * In Jacobian coordinates phi maps (x, y, z) to (x*C^2, y*C^3, z) or, equivalently to (x, y, z/C).
+     *
+     *     phi(x, y, z) = (x*C^2, y*C^3, z) = (x, y, z/C)
+     *   d_ge := phi(d) = (d.x, d.y, 1)
+     *     ai := phi(a) = (a.x*C^2, a.y*C^3, a.z)
+     *
+     * The group addition functions work correctly on these isomorphic curves.
+     * In particular phi(d) is easy to represent in affine coordinates under this isomorphism.
+     * This lets us use the faster secp256k1_gej_add_ge_var group addition function that we wouldn't be able to use otherwise.
      */
-    d_ge.x = d.x;
-    d_ge.y = d.y;
-    d_ge.infinity = 0;
-
-    secp256k1_ge_set_gej_zinv(&a_ge, a, &d.z);
-    prej[0].x = a_ge.x;
-    prej[0].y = a_ge.y;
-    prej[0].z = a->z;
-    prej[0].infinity = 0;
+    secp256k1_ge_set_xy(&d_ge, &d.x, &d.y);
+    secp256k1_ge_set_gej_zinv(&pre_a[0], a, &d.z);
+    secp256k1_gej_set_ge(&ai, &pre_a[0]);
+    ai.z = a->z;
 
+    /* pre_a[0] is the point (a.x*C^2, a.y*C^3, a.z*C) which is equvalent to a.
+     * Set zr[0] to C, which is the ratio between the omitted z(pre_a[0]) value and a.z.
+     */
     zr[0] = d.z;
+
     for (i = 1; i < n; i++) {
-        secp256k1_gej_add_ge_var(&prej[i], &prej[i-1], &d_ge, &zr[i]);
+        secp256k1_gej_add_ge_var(&ai, &ai, &d_ge, &zr[i]);
+        secp256k1_ge_set_xy(&pre_a[i], &ai.x, &ai.y);
     }
 
-    /*
-     * Each point in 'prej' has a z coordinate too small by a factor of 'd.z'. Only
-     * the final point's z coordinate is actually used though, so just update that.
+    /* Multiply the last z-coordinate by C to undo the isomorphism.
+     * Since the z-coordinates of the pre_a values are implied by the zr array of z-coordinate ratios,
+     * undoing the isomorphism here undoes the isomorphism for all pre_a values.
      */
-    secp256k1_fe_mul(&prej[n-1].z, &prej[n-1].z, &d.z);
+    secp256k1_fe_mul(z, &ai.z, &d.z);
 }
 
-/** Fill a table 'pre' with precomputed odd multiples of a.
- *
- *  The resulting point set is brought to a single constant Z denominator, stores the X and Y
- *  coordinates as ge_storage points in pre, and stores the global Z in rz.
- *  It only operates on tables sized for WINDOW_A wnaf multiples.
- *
- *  To compute a*P + b*G, we compute a table for P using this function,
- *  and use the precomputed table in <ecmult_static_pre_g.h> for G.
- */
-static void secp256k1_ecmult_odd_multiples_table_globalz_windowa(secp256k1_ge *pre, secp256k1_fe *globalz, const secp256k1_gej *a) {
-    secp256k1_gej prej[ECMULT_TABLE_SIZE(WINDOW_A)];
-    secp256k1_fe zr[ECMULT_TABLE_SIZE(WINDOW_A)];
+#define SECP256K1_ECMULT_TABLE_VERIFY(n,w) \
+    VERIFY_CHECK(((n) & 1) == 1); \
+    VERIFY_CHECK((n) >= -((1 << ((w)-1)) - 1)); \
+    VERIFY_CHECK((n) <=  ((1 << ((w)-1)) - 1));
 
-    /* Compute the odd multiples in Jacobian form. */
-    secp256k1_ecmult_odd_multiples_table(ECMULT_TABLE_SIZE(WINDOW_A), prej, zr, a);
-    /* Bring them to the same Z denominator. */
-    secp256k1_ge_globalz_set_table_gej(ECMULT_TABLE_SIZE(WINDOW_A), pre, globalz, prej, zr);
+SECP256K1_INLINE static void secp256k1_ecmult_table_get_ge(secp256k1_ge *r, const secp256k1_ge *pre, int n, int w) {
+    SECP256K1_ECMULT_TABLE_VERIFY(n,w)
+    if (n > 0) {
+        *r = pre[(n-1)/2];
+    } else {
+        *r = pre[(-n-1)/2];
+        secp256k1_fe_negate(&(r->y), &(r->y), 1);
+    }
 }
 
-/** The following two macro retrieves a particular odd multiple from a table
- *  of precomputed multiples. */
-#define ECMULT_TABLE_GET_GE(r,pre,n,w) do { \
-    VERIFY_CHECK(((n) & 1) == 1); \
-    VERIFY_CHECK((n) >= -((1 << ((w)-1)) - 1)); \
-    VERIFY_CHECK((n) <=  ((1 << ((w)-1)) - 1)); \
-    if ((n) > 0) { \
-        *(r) = (pre)[((n)-1)/2]; \
-    } else { \
-        *(r) = (pre)[(-(n)-1)/2]; \
-        secp256k1_fe_negate(&((r)->y), &((r)->y), 1); \
-    } \
-} while(0)
-
-#define ECMULT_TABLE_GET_GE_STORAGE(r,pre,n,w) do { \
-    VERIFY_CHECK(((n) & 1) == 1); \
-    VERIFY_CHECK((n) >= -((1 << ((w)-1)) - 1)); \
-    VERIFY_CHECK((n) <=  ((1 << ((w)-1)) - 1)); \
-    if ((n) > 0) { \
-        secp256k1_ge_from_storage((r), &(pre)[((n)-1)/2]); \
-    } else { \
-        secp256k1_ge_from_storage((r), &(pre)[(-(n)-1)/2]); \
-        secp256k1_fe_negate(&((r)->y), &((r)->y), 1); \
-    } \
-} while(0)
+SECP256K1_INLINE static void secp256k1_ecmult_table_get_ge_lambda(secp256k1_ge *r, const secp256k1_ge *pre, const secp256k1_fe *x, int n, int w) {
+    SECP256K1_ECMULT_TABLE_VERIFY(n,w)
+    if (n > 0) {
+        secp256k1_ge_set_xy(r, &x[(n-1)/2], &pre[(n-1)/2].y);
+    } else {
+        secp256k1_ge_set_xy(r, &x[(-n-1)/2], &pre[(-n-1)/2].y);
+        secp256k1_fe_negate(&(r->y), &(r->y), 1);
+    }
+}
+
+SECP256K1_INLINE static void secp256k1_ecmult_table_get_ge_storage(secp256k1_ge *r, const secp256k1_ge_storage *pre, int n, int w) {
+    SECP256K1_ECMULT_TABLE_VERIFY(n,w)
+    if (n > 0) {
+        secp256k1_ge_from_storage(r, &pre[(n-1)/2]);
+    } else {
+        secp256k1_ge_from_storage(r, &pre[(-n-1)/2]);
+        secp256k1_fe_negate(&(r->y), &(r->y), 1);
+    }
+}
 
 /** Convert a number to WNAF notation. The number becomes represented by sum(2^i * wnaf[i], i=0..bits),
  *  with the following guarantees:
@@ -201,19 +209,16 @@ static int secp256k1_ecmult_wnaf(int *wnaf, int len, const secp256k1_scalar *a,
 }
 
 struct secp256k1_strauss_point_state {
-    secp256k1_scalar na_1, na_lam;
     int wnaf_na_1[129];
     int wnaf_na_lam[129];
     int bits_na_1;
     int bits_na_lam;
-    size_t input_pos;
 };
 
 struct secp256k1_strauss_state {
-    secp256k1_gej* prej;
-    secp256k1_fe* zr;
+    /* aux is used to hold z-ratios, and then used to hold pre_a[i].x * BETA values. */
+    secp256k1_fe* aux;
     secp256k1_ge* pre_a;
-    secp256k1_ge* pre_a_lam;
     struct secp256k1_strauss_point_state* ps;
 };
 
@@ -231,17 +236,19 @@ static void secp256k1_ecmult_strauss_wnaf(const struct secp256k1_strauss_state *
     size_t np;
     size_t no = 0;
 
+    secp256k1_fe_set_int(&Z, 1);
     for (np = 0; np < num; ++np) {
+        secp256k1_gej tmp;
+        secp256k1_scalar na_1, na_lam;
         if (secp256k1_scalar_is_zero(&na[np]) || secp256k1_gej_is_infinity(&a[np])) {
             continue;
         }
-        state->ps[no].input_pos = np;
         /* split na into na_1 and na_lam (where na = na_1 + na_lam*lambda, and na_1 and na_lam are ~128 bit) */
-        secp256k1_scalar_split_lambda(&state->ps[no].na_1, &state->ps[no].na_lam, &na[np]);
+        secp256k1_scalar_split_lambda(&na_1, &na_lam, &na[np]);
 
         /* build wnaf representation for na_1 and na_lam. */
-        state->ps[no].bits_na_1   = secp256k1_ecmult_wnaf(state->ps[no].wnaf_na_1,   129, &state->ps[no].na_1,   WINDOW_A);
-        state->ps[no].bits_na_lam = secp256k1_ecmult_wnaf(state->ps[no].wnaf_na_lam, 129, &state->ps[no].na_lam, WINDOW_A);
+        state->ps[no].bits_na_1   = secp256k1_ecmult_wnaf(state->ps[no].wnaf_na_1,   129, &na_1,   WINDOW_A);
+        state->ps[no].bits_na_lam = secp256k1_ecmult_wnaf(state->ps[no].wnaf_na_lam, 129, &na_lam, WINDOW_A);
         VERIFY_CHECK(state->ps[no].bits_na_1 <= 129);
         VERIFY_CHECK(state->ps[no].bits_na_lam <= 129);
         if (state->ps[no].bits_na_1 > bits) {
@@ -250,40 +257,36 @@ static void secp256k1_ecmult_strauss_wnaf(const struct secp256k1_strauss_state *
         if (state->ps[no].bits_na_lam > bits) {
             bits = state->ps[no].bits_na_lam;
         }
-        ++no;
-    }
 
-    /* Calculate odd multiples of a.
-     * All multiples are brought to the same Z 'denominator', which is stored
-     * in Z. Due to secp256k1' isomorphism we can do all operations pretending
-     * that the Z coordinate was 1, use affine addition formulae, and correct
-     * the Z coordinate of the result once at the end.
-     * The exception is the precomputed G table points, which are actually
-     * affine. Compared to the base used for other points, they have a Z ratio
-     * of 1/Z, so we can use secp256k1_gej_add_zinv_var, which uses the same
-     * isomorphism to efficiently add with a known Z inverse.
-     */
-    if (no > 0) {
-        /* Compute the odd multiples in Jacobian form. */
-        secp256k1_ecmult_odd_multiples_table(ECMULT_TABLE_SIZE(WINDOW_A), state->prej, state->zr, &a[state->ps[0].input_pos]);
-        for (np = 1; np < no; ++np) {
-            secp256k1_gej tmp = a[state->ps[np].input_pos];
+        /* Calculate odd multiples of a.
+         * All multiples are brought to the same Z 'denominator', which is stored
+         * in Z. Due to secp256k1' isomorphism we can do all operations pretending
+         * that the Z coordinate was 1, use affine addition formulae, and correct
+         * the Z coordinate of the result once at the end.
+         * The exception is the precomputed G table points, which are actually
+         * affine. Compared to the base used for other points, they have a Z ratio
+         * of 1/Z, so we can use secp256k1_gej_add_zinv_var, which uses the same
+         * isomorphism to efficiently add with a known Z inverse.
+         */
+        tmp = a[np];
+        if (no) {
 #ifdef VERIFY
-            secp256k1_fe_normalize_var(&(state->prej[(np - 1) * ECMULT_TABLE_SIZE(WINDOW_A) + ECMULT_TABLE_SIZE(WINDOW_A) - 1].z));
+            secp256k1_fe_normalize_var(&Z);
 #endif
-            secp256k1_gej_rescale(&tmp, &(state->prej[(np - 1) * ECMULT_TABLE_SIZE(WINDOW_A) + ECMULT_TABLE_SIZE(WINDOW_A) - 1].z));
-            secp256k1_ecmult_odd_multiples_table(ECMULT_TABLE_SIZE(WINDOW_A), state->prej + np * ECMULT_TABLE_SIZE(WINDOW_A), state->zr + np * ECMULT_TABLE_SIZE(WINDOW_A), &tmp);
-            secp256k1_fe_mul(state->zr + np * ECMULT_TABLE_SIZE(WINDOW_A), state->zr + np * ECMULT_TABLE_SIZE(WINDOW_A), &(a[state->ps[np].input_pos].z));
+            secp256k1_gej_rescale(&tmp, &Z);
         }
-        /* Bring them to the same Z denominator. */
-        secp256k1_ge_globalz_set_table_gej(ECMULT_TABLE_SIZE(WINDOW_A) * no, state->pre_a, &Z, state->prej, state->zr);
-    } else {
-        secp256k1_fe_set_int(&Z, 1);
+        secp256k1_ecmult_odd_multiples_table(ECMULT_TABLE_SIZE(WINDOW_A), state->pre_a + no * ECMULT_TABLE_SIZE(WINDOW_A), state->aux + no * ECMULT_TABLE_SIZE(WINDOW_A), &Z, &tmp);
+        if (no) secp256k1_fe_mul(state->aux + no * ECMULT_TABLE_SIZE(WINDOW_A), state->aux + no * ECMULT_TABLE_SIZE(WINDOW_A), &(a[np].z));
+
+        ++no;
     }
 
+    /* Bring them to the same Z denominator. */
+    secp256k1_ge_table_set_globalz(ECMULT_TABLE_SIZE(WINDOW_A) * no, state->pre_a, state->aux);
+
     for (np = 0; np < no; ++np) {
         for (i = 0; i < ECMULT_TABLE_SIZE(WINDOW_A); i++) {
-            secp256k1_ge_mul_lambda(&state->pre_a_lam[np * ECMULT_TABLE_SIZE(WINDOW_A) + i], &state->pre_a[np * ECMULT_TABLE_SIZE(WINDOW_A) + i]);
+            secp256k1_fe_mul(&state->aux[np * ECMULT_TABLE_SIZE(WINDOW_A) + i], &state->pre_a[np * ECMULT_TABLE_SIZE(WINDOW_A) + i].x, &secp256k1_const_beta);
         }
     }
 
@@ -309,20 +312,20 @@ static void secp256k1_ecmult_strauss_wnaf(const struct secp256k1_strauss_state *
         secp256k1_gej_double_var(r, r, NULL);
         for (np = 0; np < no; ++np) {
             if (i < state->ps[np].bits_na_1 && (n = state->ps[np].wnaf_na_1[i])) {
-                ECMULT_TABLE_GET_GE(&tmpa, state->pre_a + np * ECMULT_TABLE_SIZE(WINDOW_A), n, WINDOW_A);
+                secp256k1_ecmult_table_get_ge(&tmpa, state->pre_a + np * ECMULT_TABLE_SIZE(WINDOW_A), n, WINDOW_A);
                 secp256k1_gej_add_ge_var(r, r, &tmpa, NULL);
             }
             if (i < state->ps[np].bits_na_lam && (n = state->ps[np].wnaf_na_lam[i])) {
-                ECMULT_TABLE_GET_GE(&tmpa, state->pre_a_lam + np * ECMULT_TABLE_SIZE(WINDOW_A), n, WINDOW_A);
+                secp256k1_ecmult_table_get_ge_lambda(&tmpa, state->pre_a + np * ECMULT_TABLE_SIZE(WINDOW_A), state->aux + np * ECMULT_TABLE_SIZE(WINDOW_A), n, WINDOW_A);
                 secp256k1_gej_add_ge_var(r, r, &tmpa, NULL);
             }
         }
         if (i < bits_ng_1 && (n = wnaf_ng_1[i])) {
-            ECMULT_TABLE_GET_GE_STORAGE(&tmpa, secp256k1_pre_g, n, WINDOW_G);
+            secp256k1_ecmult_table_get_ge_storage(&tmpa, secp256k1_pre_g, n, WINDOW_G);
             secp256k1_gej_add_zinv_var(r, r, &tmpa, &Z);
         }
         if (i < bits_ng_128 && (n = wnaf_ng_128[i])) {
-            ECMULT_TABLE_GET_GE_STORAGE(&tmpa, secp256k1_pre_g_128, n, WINDOW_G);
+            secp256k1_ecmult_table_get_ge_storage(&tmpa, secp256k1_pre_g_128, n, WINDOW_G);
             secp256k1_gej_add_zinv_var(r, r, &tmpa, &Z);
         }
     }
@@ -333,23 +336,19 @@ static void secp256k1_ecmult_strauss_wnaf(const struct secp256k1_strauss_state *
 }
 
 static void secp256k1_ecmult(secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_scalar *na, const secp256k1_scalar *ng) {
-    secp256k1_gej prej[ECMULT_TABLE_SIZE(WINDOW_A)];
-    secp256k1_fe zr[ECMULT_TABLE_SIZE(WINDOW_A)];
+    secp256k1_fe aux[ECMULT_TABLE_SIZE(WINDOW_A)];
     secp256k1_ge pre_a[ECMULT_TABLE_SIZE(WINDOW_A)];
     struct secp256k1_strauss_point_state ps[1];
-    secp256k1_ge pre_a_lam[ECMULT_TABLE_SIZE(WINDOW_A)];
     struct secp256k1_strauss_state state;
 
-    state.prej = prej;
-    state.zr = zr;
+    state.aux = aux;
     state.pre_a = pre_a;
-    state.pre_a_lam = pre_a_lam;
     state.ps = ps;
     secp256k1_ecmult_strauss_wnaf(&state, r, 1, a, na, ng);
 }
 
 static size_t secp256k1_strauss_scratch_size(size_t n_points) {
-    static const size_t point_size = (2 * sizeof(secp256k1_ge) + sizeof(secp256k1_gej) + sizeof(secp256k1_fe)) * ECMULT_TABLE_SIZE(WINDOW_A) + sizeof(struct secp256k1_strauss_point_state) + sizeof(secp256k1_gej) + sizeof(secp256k1_scalar);
+    static const size_t point_size = (sizeof(secp256k1_ge) + sizeof(secp256k1_fe)) * ECMULT_TABLE_SIZE(WINDOW_A) + sizeof(struct secp256k1_strauss_point_state) + sizeof(secp256k1_gej) + sizeof(secp256k1_scalar);
     return n_points*point_size;
 }
 
@@ -370,13 +369,11 @@ static int secp256k1_ecmult_strauss_batch(const secp256k1_callback* error_callba
      * constant and strauss_scratch_size accordingly. */
     points = (secp256k1_gej*)secp256k1_scratch_alloc(error_callback, scratch, n_points * sizeof(secp256k1_gej));
     scalars = (secp256k1_scalar*)secp256k1_scratch_alloc(error_callback, scratch, n_points * sizeof(secp256k1_scalar));
-    state.prej = (secp256k1_gej*)secp256k1_scratch_alloc(error_callback, scratch, n_points * ECMULT_TABLE_SIZE(WINDOW_A) * sizeof(secp256k1_gej));
-    state.zr = (secp256k1_fe*)secp256k1_scratch_alloc(error_callback, scratch, n_points * ECMULT_TABLE_SIZE(WINDOW_A) * sizeof(secp256k1_fe));
+    state.aux = (secp256k1_fe*)secp256k1_scratch_alloc(error_callback, scratch, n_points * ECMULT_TABLE_SIZE(WINDOW_A) * sizeof(secp256k1_fe));
     state.pre_a = (secp256k1_ge*)secp256k1_scratch_alloc(error_callback, scratch, n_points * ECMULT_TABLE_SIZE(WINDOW_A) * sizeof(secp256k1_ge));
-    state.pre_a_lam = (secp256k1_ge*)secp256k1_scratch_alloc(error_callback, scratch, n_points * ECMULT_TABLE_SIZE(WINDOW_A) * sizeof(secp256k1_ge));
     state.ps = (struct secp256k1_strauss_point_state*)secp256k1_scratch_alloc(error_callback, scratch, n_points * sizeof(struct secp256k1_strauss_point_state));
 
-    if (points == NULL || scalars == NULL || state.prej == NULL || state.zr == NULL || state.pre_a == NULL || state.pre_a_lam == NULL || state.ps == NULL) {
+    if (points == NULL || scalars == NULL || state.aux == NULL || state.pre_a == NULL || state.ps == NULL) {
         secp256k1_scratch_apply_checkpoint(error_callback, scratch, scratch_checkpoint);
         return 0;
     }
diff --git a/src/field.h b/src/field.h
index 55679a2fc130aa..2584a494eeb091 100644
--- a/src/field.h
+++ b/src/field.h
@@ -32,6 +32,12 @@
 #error "Please select wide multiplication implementation"
 #endif
 
+static const secp256k1_fe secp256k1_fe_one = SECP256K1_FE_CONST(0, 0, 0, 0, 0, 0, 0, 1);
+static const secp256k1_fe secp256k1_const_beta = SECP256K1_FE_CONST(
+    0x7ae96a2bul, 0x657c0710ul, 0x6e64479eul, 0xac3434e9ul,
+    0x9cf04975ul, 0x12f58995ul, 0xc1396c28ul, 0x719501eeul
+);
+
 /** Normalize a field element. This brings the field element to a canonical representation, reduces
  *  its magnitude to 1, and reduces it modulo field size `p`.
  */
@@ -124,4 +130,13 @@ static void secp256k1_fe_storage_cmov(secp256k1_fe_storage *r, const secp256k1_f
 /** If flag is true, set *r equal to *a; otherwise leave it. Constant-time.  Both *r and *a must be initialized.*/
 static void secp256k1_fe_cmov(secp256k1_fe *r, const secp256k1_fe *a, int flag);
 
+/** Halves the value of a field element modulo the field prime. Constant-time.
+ *  For an input magnitude 'm', the output magnitude is set to 'floor(m/2) + 1'.
+ *  The output is not guaranteed to be normalized, regardless of the input. */
+static void secp256k1_fe_half(secp256k1_fe *r);
+
+/** Sets each limb of 'r' to its upper bound at magnitude 'm'. The output will also have its
+ *  magnitude set to 'm' and is normalized if (and only if) 'm' is zero. */
+static void secp256k1_fe_get_bounds(secp256k1_fe *r, int m);
+
 #endif /* SECP256K1_FIELD_H */
diff --git a/src/field_10x26_impl.h b/src/field_10x26_impl.h
index 4363e727e76e8d..21742bf6eb6c4b 100644
--- a/src/field_10x26_impl.h
+++ b/src/field_10x26_impl.h
@@ -11,6 +11,15 @@
 #include "field.h"
 #include "modinv32_impl.h"
 
+/** See the comment at the top of field_5x52_impl.h for more details.
+ *
+ *  Here, we represent field elements as 10 uint32_t's in base 2^26, least significant first,
+ *  where limbs can contain >26 bits.
+ *  A magnitude M means:
+ *  - 2*M*(2^22-1) is the max (inclusive) of the most significant limb
+ *  - 2*M*(2^26-1) is the max (inclusive) of the remaining limbs
+ */
+
 #ifdef VERIFY
 static void secp256k1_fe_verify(const secp256k1_fe *a) {
     const uint32_t *d = a->n;
@@ -40,6 +49,26 @@ static void secp256k1_fe_verify(const secp256k1_fe *a) {
 }
 #endif
 
+static void secp256k1_fe_get_bounds(secp256k1_fe *r, int m) {
+    VERIFY_CHECK(m >= 0);
+    VERIFY_CHECK(m <= 2048);
+    r->n[0] = 0x3FFFFFFUL * 2 * m;
+    r->n[1] = 0x3FFFFFFUL * 2 * m;
+    r->n[2] = 0x3FFFFFFUL * 2 * m;
+    r->n[3] = 0x3FFFFFFUL * 2 * m;
+    r->n[4] = 0x3FFFFFFUL * 2 * m;
+    r->n[5] = 0x3FFFFFFUL * 2 * m;
+    r->n[6] = 0x3FFFFFFUL * 2 * m;
+    r->n[7] = 0x3FFFFFFUL * 2 * m;
+    r->n[8] = 0x3FFFFFFUL * 2 * m;
+    r->n[9] = 0x03FFFFFUL * 2 * m;
+#ifdef VERIFY
+    r->magnitude = m;
+    r->normalized = (m == 0);
+    secp256k1_fe_verify(r);
+#endif
+}
+
 static void secp256k1_fe_normalize(secp256k1_fe *r) {
     uint32_t t0 = r->n[0], t1 = r->n[1], t2 = r->n[2], t3 = r->n[3], t4 = r->n[4],
              t5 = r->n[5], t6 = r->n[6], t7 = r->n[7], t8 = r->n[8], t9 = r->n[9];
@@ -391,6 +420,10 @@ SECP256K1_INLINE static void secp256k1_fe_negate(secp256k1_fe *r, const secp256k
 #ifdef VERIFY
     VERIFY_CHECK(a->magnitude <= m);
     secp256k1_fe_verify(a);
+    VERIFY_CHECK(0x3FFFC2FUL * 2 * (m + 1) >= 0x3FFFFFFUL * 2 * m);
+    VERIFY_CHECK(0x3FFFFBFUL * 2 * (m + 1) >= 0x3FFFFFFUL * 2 * m);
+    VERIFY_CHECK(0x3FFFFFFUL * 2 * (m + 1) >= 0x3FFFFFFUL * 2 * m);
+    VERIFY_CHECK(0x03FFFFFUL * 2 * (m + 1) >= 0x03FFFFFUL * 2 * m);
 #endif
     r->n[0] = 0x3FFFC2FUL * 2 * (m + 1) - a->n[0];
     r->n[1] = 0x3FFFFBFUL * 2 * (m + 1) - a->n[1];
@@ -1120,6 +1153,82 @@ static SECP256K1_INLINE void secp256k1_fe_cmov(secp256k1_fe *r, const secp256k1_
 #endif
 }
 
+static SECP256K1_INLINE void secp256k1_fe_half(secp256k1_fe *r) {
+    uint32_t t0 = r->n[0], t1 = r->n[1], t2 = r->n[2], t3 = r->n[3], t4 = r->n[4],
+             t5 = r->n[5], t6 = r->n[6], t7 = r->n[7], t8 = r->n[8], t9 = r->n[9];
+    uint32_t one = (uint32_t)1;
+    uint32_t mask = -(t0 & one) >> 6;
+
+#ifdef VERIFY
+    secp256k1_fe_verify(r);
+    VERIFY_CHECK(r->magnitude < 32);
+#endif
+
+    /* Bounds analysis (over the rationals).
+     *
+     * Let m = r->magnitude
+     *     C = 0x3FFFFFFUL * 2
+     *     D = 0x03FFFFFUL * 2
+     *
+     * Initial bounds: t0..t8 <= C * m
+     *                     t9 <= D * m
+     */
+
+    t0 += 0x3FFFC2FUL & mask;
+    t1 += 0x3FFFFBFUL & mask;
+    t2 += mask;
+    t3 += mask;
+    t4 += mask;
+    t5 += mask;
+    t6 += mask;
+    t7 += mask;
+    t8 += mask;
+    t9 += mask >> 4;
+
+    VERIFY_CHECK((t0 & one) == 0);
+
+    /* t0..t8: added <= C/2
+     *     t9: added <= D/2
+     *
+     * Current bounds: t0..t8 <= C * (m + 1/2)
+     *                     t9 <= D * (m + 1/2)
+     */
+
+    r->n[0] = (t0 >> 1) + ((t1 & one) << 25);
+    r->n[1] = (t1 >> 1) + ((t2 & one) << 25);
+    r->n[2] = (t2 >> 1) + ((t3 & one) << 25);
+    r->n[3] = (t3 >> 1) + ((t4 & one) << 25);
+    r->n[4] = (t4 >> 1) + ((t5 & one) << 25);
+    r->n[5] = (t5 >> 1) + ((t6 & one) << 25);
+    r->n[6] = (t6 >> 1) + ((t7 & one) << 25);
+    r->n[7] = (t7 >> 1) + ((t8 & one) << 25);
+    r->n[8] = (t8 >> 1) + ((t9 & one) << 25);
+    r->n[9] = (t9 >> 1);
+
+    /* t0..t8: shifted right and added <= C/4 + 1/2
+     *     t9: shifted right
+     *
+     * Current bounds: t0..t8 <= C * (m/2 + 1/2)
+     *                     t9 <= D * (m/2 + 1/4)
+     */
+
+#ifdef VERIFY
+    /* Therefore the output magnitude (M) has to be set such that:
+     *     t0..t8: C * M >= C * (m/2 + 1/2)
+     *         t9: D * M >= D * (m/2 + 1/4)
+     *
+     * It suffices for all limbs that, for any input magnitude m:
+     *     M >= m/2 + 1/2
+     *
+     * and since we want the smallest such integer value for M:
+     *     M == floor(m/2) + 1
+     */
+    r->magnitude = (r->magnitude >> 1) + 1;
+    r->normalized = 0;
+    secp256k1_fe_verify(r);
+#endif
+}
+
 static SECP256K1_INLINE void secp256k1_fe_storage_cmov(secp256k1_fe_storage *r, const secp256k1_fe_storage *a, int flag) {
     uint32_t mask0, mask1;
     VG_CHECK_VERIFY(r->n, sizeof(r->n));
diff --git a/src/field_5x52_impl.h b/src/field_5x52_impl.h
index b56bdd13534c43..6bd202f5871305 100644
--- a/src/field_5x52_impl.h
+++ b/src/field_5x52_impl.h
@@ -22,11 +22,18 @@
 #endif
 
 /** Implements arithmetic modulo FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFC2F,
- *  represented as 5 uint64_t's in base 2^52. The values are allowed to contain >52 each. In particular,
- *  each FieldElem has a 'magnitude' associated with it. Internally, a magnitude M means each element
- *  is at most M*(2^53-1), except the most significant one, which is limited to M*(2^49-1). All operations
- *  accept any input with magnitude at most M, and have different rules for propagating magnitude to their
- *  output.
+ *  represented as 5 uint64_t's in base 2^52, least significant first. Note that the limbs are allowed to
+ *  contain >52 bits each.
+ *
+ *  Each field element has a 'magnitude' associated with it. Internally, a magnitude M means:
+ *  - 2*M*(2^48-1) is the max (inclusive) of the most significant limb
+ *  - 2*M*(2^52-1) is the max (inclusive) of the remaining limbs
+ *
+ *  Operations have different rules for propagating magnitude to their outputs. If an operation takes a
+ *  magnitude M as a parameter, that means the magnitude of input field elements can be at most M (inclusive).
+ *
+ *  Each field element also has a 'normalized' flag. A field element is normalized if its magnitude is either
+ *  0 or 1, and its value is already reduced modulo the order of the field.
  */
 
 #ifdef VERIFY
@@ -51,6 +58,21 @@ static void secp256k1_fe_verify(const secp256k1_fe *a) {
 }
 #endif
 
+static void secp256k1_fe_get_bounds(secp256k1_fe *r, int m) {
+    VERIFY_CHECK(m >= 0);
+    VERIFY_CHECK(m <= 2048);
+    r->n[0] = 0xFFFFFFFFFFFFFULL * 2 * m;
+    r->n[1] = 0xFFFFFFFFFFFFFULL * 2 * m;
+    r->n[2] = 0xFFFFFFFFFFFFFULL * 2 * m;
+    r->n[3] = 0xFFFFFFFFFFFFFULL * 2 * m;
+    r->n[4] = 0x0FFFFFFFFFFFFULL * 2 * m;
+#ifdef VERIFY
+    r->magnitude = m;
+    r->normalized = (m == 0);
+    secp256k1_fe_verify(r);
+#endif
+}
+
 static void secp256k1_fe_normalize(secp256k1_fe *r) {
     uint64_t t0 = r->n[0], t1 = r->n[1], t2 = r->n[2], t3 = r->n[3], t4 = r->n[4];
 
@@ -377,6 +399,9 @@ SECP256K1_INLINE static void secp256k1_fe_negate(secp256k1_fe *r, const secp256k
 #ifdef VERIFY
     VERIFY_CHECK(a->magnitude <= m);
     secp256k1_fe_verify(a);
+    VERIFY_CHECK(0xFFFFEFFFFFC2FULL * 2 * (m + 1) >= 0xFFFFFFFFFFFFFULL * 2 * m);
+    VERIFY_CHECK(0xFFFFFFFFFFFFFULL * 2 * (m + 1) >= 0xFFFFFFFFFFFFFULL * 2 * m);
+    VERIFY_CHECK(0x0FFFFFFFFFFFFULL * 2 * (m + 1) >= 0x0FFFFFFFFFFFFULL * 2 * m);
 #endif
     r->n[0] = 0xFFFFEFFFFFC2FULL * 2 * (m + 1) - a->n[0];
     r->n[1] = 0xFFFFFFFFFFFFFULL * 2 * (m + 1) - a->n[1];
@@ -467,6 +492,71 @@ static SECP256K1_INLINE void secp256k1_fe_cmov(secp256k1_fe *r, const secp256k1_
 #endif
 }
 
+static SECP256K1_INLINE void secp256k1_fe_half(secp256k1_fe *r) {
+    uint64_t t0 = r->n[0], t1 = r->n[1], t2 = r->n[2], t3 = r->n[3], t4 = r->n[4];
+    uint64_t one = (uint64_t)1;
+    uint64_t mask = -(t0 & one) >> 12;
+
+#ifdef VERIFY
+    secp256k1_fe_verify(r);
+    VERIFY_CHECK(r->magnitude < 32);
+#endif
+
+    /* Bounds analysis (over the rationals).
+     *
+     * Let m = r->magnitude
+     *     C = 0xFFFFFFFFFFFFFULL * 2
+     *     D = 0x0FFFFFFFFFFFFULL * 2
+     *
+     * Initial bounds: t0..t3 <= C * m
+     *                     t4 <= D * m
+     */
+
+    t0 += 0xFFFFEFFFFFC2FULL & mask;
+    t1 += mask;
+    t2 += mask;
+    t3 += mask;
+    t4 += mask >> 4;
+
+    VERIFY_CHECK((t0 & one) == 0);
+
+    /* t0..t3: added <= C/2
+     *     t4: added <= D/2
+     *
+     * Current bounds: t0..t3 <= C * (m + 1/2)
+     *                     t4 <= D * (m + 1/2)
+     */
+
+    r->n[0] = (t0 >> 1) + ((t1 & one) << 51);
+    r->n[1] = (t1 >> 1) + ((t2 & one) << 51);
+    r->n[2] = (t2 >> 1) + ((t3 & one) << 51);
+    r->n[3] = (t3 >> 1) + ((t4 & one) << 51);
+    r->n[4] = (t4 >> 1);
+
+    /* t0..t3: shifted right and added <= C/4 + 1/2
+     *     t4: shifted right
+     *
+     * Current bounds: t0..t3 <= C * (m/2 + 1/2)
+     *                     t4 <= D * (m/2 + 1/4)
+     */
+
+#ifdef VERIFY
+    /* Therefore the output magnitude (M) has to be set such that:
+     *     t0..t3: C * M >= C * (m/2 + 1/2)
+     *         t4: D * M >= D * (m/2 + 1/4)
+     *
+     * It suffices for all limbs that, for any input magnitude m:
+     *     M >= m/2 + 1/2
+     *
+     * and since we want the smallest such integer value for M:
+     *     M == floor(m/2) + 1
+     */
+    r->magnitude = (r->magnitude >> 1) + 1;
+    r->normalized = 0;
+    secp256k1_fe_verify(r);
+#endif
+}
+
 static SECP256K1_INLINE void secp256k1_fe_storage_cmov(secp256k1_fe_storage *r, const secp256k1_fe_storage *a, int flag) {
     uint64_t mask0, mask1;
     VG_CHECK_VERIFY(r->n, sizeof(r->n));
diff --git a/src/field_impl.h b/src/field_impl.h
index 374284a1f4ce97..0a4a04d9ac2a99 100644
--- a/src/field_impl.h
+++ b/src/field_impl.h
@@ -135,6 +135,4 @@ static int secp256k1_fe_sqrt(secp256k1_fe *r, const secp256k1_fe *a) {
     return secp256k1_fe_equal(&t1, a);
 }
 
-static const secp256k1_fe secp256k1_fe_one = SECP256K1_FE_CONST(0, 0, 0, 0, 0, 0, 0, 1);
-
 #endif /* SECP256K1_FIELD_IMPL_H */
diff --git a/src/gen_ecmult_static_pre_g.c b/src/gen_ecmult_static_pre_g.c
deleted file mode 100644
index ba1d1f17d7d96b..00000000000000
--- a/src/gen_ecmult_static_pre_g.c
+++ /dev/null
@@ -1,131 +0,0 @@
-/*****************************************************************************************************
- * Copyright (c) 2013, 2014, 2017, 2021 Pieter Wuille, Andrew Poelstra, Jonas Nick, Russell O'Connor *
- * Distributed under the MIT software license, see the accompanying                                  *
- * file COPYING or https://www.opensource.org/licenses/mit-license.php.                              *
- *****************************************************************************************************/
-
-#include <inttypes.h>
-#include <stdio.h>
-
-/* Autotools creates libsecp256k1-config.h, of which ECMULT_WINDOW_SIZE is needed.
-   ifndef guard so downstream users can define their own if they do not use autotools. */
-#if !defined(ECMULT_WINDOW_SIZE)
-#include "libsecp256k1-config.h"
-#endif
-
-#include "../include/secp256k1.h"
-#include "assumptions.h"
-#include "util.h"
-#include "field_impl.h"
-#include "group_impl.h"
-#include "ecmult.h"
-
-void print_table(FILE *fp, const char *name, int window_g, const secp256k1_gej *gen, int with_conditionals) {
-    static secp256k1_gej gj;
-    static secp256k1_ge ge, dgen;
-    static secp256k1_ge_storage ges;
-    int j;
-    int i;
-
-    gj = *gen;
-    secp256k1_ge_set_gej_var(&ge, &gj);
-    secp256k1_ge_to_storage(&ges, &ge);
-
-    fprintf(fp, "static const secp256k1_ge_storage %s[ECMULT_TABLE_SIZE(WINDOW_G)] = {\n", name);
-    fprintf(fp, " S(%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32
-                  ",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32")\n",
-                SECP256K1_GE_STORAGE_CONST_GET(ges));
-
-    secp256k1_gej_double_var(&gj, gen, NULL);
-    secp256k1_ge_set_gej_var(&dgen, &gj);
-
-    j = 1;
-    for(i = 3; i <= window_g; ++i) {
-        if (with_conditionals) {
-            fprintf(fp, "#if ECMULT_TABLE_SIZE(WINDOW_G) > %ld\n", ECMULT_TABLE_SIZE(i-1));
-        }
-        for(;j < ECMULT_TABLE_SIZE(i); ++j) {
-            secp256k1_gej_set_ge(&gj, &ge);
-            secp256k1_gej_add_ge_var(&gj, &gj, &dgen, NULL);
-            secp256k1_ge_set_gej_var(&ge, &gj);
-            secp256k1_ge_to_storage(&ges, &ge);
-
-            fprintf(fp, ",S(%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32
-                          ",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32")\n",
-                        SECP256K1_GE_STORAGE_CONST_GET(ges));
-        }
-        if (with_conditionals) {
-            fprintf(fp, "#endif\n");
-        }
-    }
-    fprintf(fp, "};\n");
-}
-
-void print_two_tables(FILE *fp, int window_g, const secp256k1_ge *g, int with_conditionals) {
-    secp256k1_gej gj;
-    int i;
-
-    secp256k1_gej_set_ge(&gj, g);
-    print_table(fp, "secp256k1_pre_g", window_g, &gj, with_conditionals);
-    for (i = 0; i < 128; ++i) {
-        secp256k1_gej_double_var(&gj, &gj, NULL);
-    }
-    print_table(fp, "secp256k1_pre_g_128", window_g, &gj, with_conditionals);
-}
-
-int main(void) {
-    const secp256k1_ge g = SECP256K1_G;
-    const secp256k1_ge g_13 = SECP256K1_G_ORDER_13;
-    const secp256k1_ge g_199 = SECP256K1_G_ORDER_199;
-    const int window_g_13 = 4;
-    const int window_g_199 = 8;
-    FILE* fp;
-
-    fp = fopen("src/ecmult_static_pre_g.h","w");
-    if (fp == NULL) {
-        fprintf(stderr, "Could not open src/ecmult_static_pre_g.h for writing!\n");
-        return -1;
-    }
-
-    fprintf(fp, "/* This file was automatically generated by gen_ecmult_static_pre_g. */\n");
-    fprintf(fp, "/* This file contains an array secp256k1_pre_g with odd multiples of the base point G and\n");
-    fprintf(fp, " * an array secp256k1_pre_g_128 with odd multiples of 2^128*G for accelerating the computation of a*P + b*G.\n");
-    fprintf(fp, " */\n");
-    fprintf(fp, "#ifndef SECP256K1_ECMULT_STATIC_PRE_G_H\n");
-    fprintf(fp, "#define SECP256K1_ECMULT_STATIC_PRE_G_H\n");
-    fprintf(fp, "#include \"group.h\"\n");
-    fprintf(fp, "#ifdef S\n");
-    fprintf(fp, "   #error macro identifier S already in use.\n");
-    fprintf(fp, "#endif\n");
-    fprintf(fp, "#define S(a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p) "
-                "SECP256K1_GE_STORAGE_CONST(0x##a##u,0x##b##u,0x##c##u,0x##d##u,0x##e##u,0x##f##u,0x##g##u,"
-                "0x##h##u,0x##i##u,0x##j##u,0x##k##u,0x##l##u,0x##m##u,0x##n##u,0x##o##u,0x##p##u)\n");
-    fprintf(fp, "#if ECMULT_TABLE_SIZE(ECMULT_WINDOW_SIZE) > %ld\n", ECMULT_TABLE_SIZE(ECMULT_WINDOW_SIZE));
-    fprintf(fp, "   #error configuration mismatch, invalid ECMULT_WINDOW_SIZE. Try deleting ecmult_static_pre_g.h before the build.\n");
-    fprintf(fp, "#endif\n");
-    fprintf(fp, "#if defined(EXHAUSTIVE_TEST_ORDER)\n");
-    fprintf(fp, "#if EXHAUSTIVE_TEST_ORDER == 13\n");
-    fprintf(fp, "#define WINDOW_G %d\n", window_g_13);
-
-    print_two_tables(fp, window_g_13, &g_13, 0);
-
-    fprintf(fp, "#elif EXHAUSTIVE_TEST_ORDER == 199\n");
-    fprintf(fp, "#define WINDOW_G %d\n", window_g_199);
-
-    print_two_tables(fp, window_g_199, &g_199, 0);
-
-    fprintf(fp, "#else\n");
-    fprintf(fp, "   #error No known generator for the specified exhaustive test group order.\n");
-    fprintf(fp, "#endif\n");
-    fprintf(fp, "#else /* !defined(EXHAUSTIVE_TEST_ORDER) */\n");
-    fprintf(fp, "#define WINDOW_G ECMULT_WINDOW_SIZE\n");
-
-    print_two_tables(fp, ECMULT_WINDOW_SIZE, &g, 1);
-
-    fprintf(fp, "#endif\n");
-    fprintf(fp, "#undef S\n");
-    fprintf(fp, "#endif\n");
-    fclose(fp);
-
-    return 0;
-}
diff --git a/src/group.h b/src/group.h
index b9cd334dae26c7..bb7dae1cf781e1 100644
--- a/src/group.h
+++ b/src/group.h
@@ -9,7 +9,10 @@
 
 #include "field.h"
 
-/** A group element of the secp256k1 curve, in affine coordinates. */
+/** A group element in affine coordinates on the secp256k1 curve,
+ *  or occasionally on an isomorphic curve of the form y^2 = x^3 + 7*t^6.
+ *  Note: For exhaustive test mode, secp256k1 is replaced by a small subgroup of a different curve.
+ */
 typedef struct {
     secp256k1_fe x;
     secp256k1_fe y;
@@ -19,7 +22,9 @@ typedef struct {
 #define SECP256K1_GE_CONST(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p) {SECP256K1_FE_CONST((a),(b),(c),(d),(e),(f),(g),(h)), SECP256K1_FE_CONST((i),(j),(k),(l),(m),(n),(o),(p)), 0}
 #define SECP256K1_GE_CONST_INFINITY {SECP256K1_FE_CONST(0, 0, 0, 0, 0, 0, 0, 0), SECP256K1_FE_CONST(0, 0, 0, 0, 0, 0, 0, 0), 1}
 
-/** A group element of the secp256k1 curve, in jacobian coordinates. */
+/** A group element of the secp256k1 curve, in jacobian coordinates.
+ *  Note: For exhastive test mode, sepc256k1 is replaced by a small subgroup of a different curve.
+ */
 typedef struct {
     secp256k1_fe x; /* actual X: x/z^2 */
     secp256k1_fe y; /* actual Y: y/z^3 */
@@ -64,12 +69,24 @@ static void secp256k1_ge_set_gej_var(secp256k1_ge *r, secp256k1_gej *a);
 /** Set a batch of group elements equal to the inputs given in jacobian coordinates */
 static void secp256k1_ge_set_all_gej_var(secp256k1_ge *r, const secp256k1_gej *a, size_t len);
 
-/** Bring a batch inputs given in jacobian coordinates (with known z-ratios) to
- *  the same global z "denominator". zr must contain the known z-ratios such
- *  that mul(a[i].z, zr[i+1]) == a[i+1].z. zr[0] is ignored. The x and y
- *  coordinates of the result are stored in r, the common z coordinate is
- *  stored in globalz. */
-static void secp256k1_ge_globalz_set_table_gej(size_t len, secp256k1_ge *r, secp256k1_fe *globalz, const secp256k1_gej *a, const secp256k1_fe *zr);
+/** Bring a batch of inputs to the same global z "denominator", based on ratios between
+ *  (omitted) z coordinates of adjacent elements.
+ *
+ *  Although the elements a[i] are _ge rather than _gej, they actually represent elements
+ *  in Jacobian coordinates with their z coordinates omitted.
+ *
+ *  Using the notation z(b) to represent the omitted z coordinate of b, the array zr of
+ *  z coordinate ratios must satisfy zr[i] == z(a[i]) / z(a[i-1]) for 0 < 'i' < len.
+ *  The zr[0] value is unused.
+ *
+ *  This function adjusts the coordinates of 'a' in place so that for all 'i', z(a[i]) == z(a[len-1]).
+ *  In other words, the initial value of z(a[len-1]) becomes the global z "denominator". Only the
+ *  a[i].x and a[i].y coordinates are explicitly modified; the adjustment of the omitted z coordinate is
+ *  implicit.
+ *
+ *  The coordinates of the final element a[len-1] are not changed.
+ */
+static void secp256k1_ge_table_set_globalz(size_t len, secp256k1_ge *a, const secp256k1_fe *zr);
 
 /** Set a group element (affine) equal to the point at infinity. */
 static void secp256k1_ge_set_infinity(secp256k1_ge *r);
@@ -124,6 +141,9 @@ static void secp256k1_ge_to_storage(secp256k1_ge_storage *r, const secp256k1_ge
 /** Convert a group element back from the storage type. */
 static void secp256k1_ge_from_storage(secp256k1_ge *r, const secp256k1_ge_storage *a);
 
+/** If flag is true, set *r equal to *a; otherwise leave it. Constant-time.  Both *r and *a must be initialized.*/
+static void secp256k1_gej_cmov(secp256k1_gej *r, const secp256k1_gej *a, int flag);
+
 /** If flag is true, set *r equal to *a; otherwise leave it. Constant-time.  Both *r and *a must be initialized.*/
 static void secp256k1_ge_storage_cmov(secp256k1_ge_storage *r, const secp256k1_ge_storage *a, int flag);
 
diff --git a/src/group_impl.h b/src/group_impl.h
index bce9fbdad56424..b19b02a01fafbf 100644
--- a/src/group_impl.h
+++ b/src/group_impl.h
@@ -161,27 +161,26 @@ static void secp256k1_ge_set_all_gej_var(secp256k1_ge *r, const secp256k1_gej *a
     }
 }
 
-static void secp256k1_ge_globalz_set_table_gej(size_t len, secp256k1_ge *r, secp256k1_fe *globalz, const secp256k1_gej *a, const secp256k1_fe *zr) {
+static void secp256k1_ge_table_set_globalz(size_t len, secp256k1_ge *a, const secp256k1_fe *zr) {
     size_t i = len - 1;
     secp256k1_fe zs;
 
     if (len > 0) {
-        /* The z of the final point gives us the "global Z" for the table. */
-        r[i].x = a[i].x;
-        r[i].y = a[i].y;
         /* Ensure all y values are in weak normal form for fast negation of points */
-        secp256k1_fe_normalize_weak(&r[i].y);
-        *globalz = a[i].z;
-        r[i].infinity = 0;
+        secp256k1_fe_normalize_weak(&a[i].y);
         zs = zr[i];
 
         /* Work our way backwards, using the z-ratios to scale the x/y values. */
         while (i > 0) {
+            secp256k1_gej tmpa;
             if (i != len - 1) {
                 secp256k1_fe_mul(&zs, &zs, &zr[i]);
             }
             i--;
-            secp256k1_ge_set_gej_zinv(&r[i], &a[i], &zs);
+            tmpa.x = a[i].x;
+            tmpa.y = a[i].y;
+            tmpa.infinity = 0;
+            secp256k1_ge_set_gej_zinv(&a[i], &tmpa, &zs);
         }
     }
 }
@@ -272,37 +271,35 @@ static int secp256k1_ge_is_valid_var(const secp256k1_ge *a) {
 }
 
 static SECP256K1_INLINE void secp256k1_gej_double(secp256k1_gej *r, const secp256k1_gej *a) {
-    /* Operations: 3 mul, 4 sqr, 0 normalize, 12 mul_int/add/negate.
-     *
-     * Note that there is an implementation described at
-     *     https://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling-dbl-2009-l
-     * which trades a multiply for a square, but in practice this is actually slower,
-     * mainly because it requires more normalizations.
-     */
-    secp256k1_fe t1,t2,t3,t4;
+    /* Operations: 3 mul, 4 sqr, 8 add/half/mul_int/negate */
+    secp256k1_fe l, s, t;
 
     r->infinity = a->infinity;
 
-    secp256k1_fe_mul(&r->z, &a->z, &a->y);
-    secp256k1_fe_mul_int(&r->z, 2);       /* Z' = 2*Y*Z (2) */
-    secp256k1_fe_sqr(&t1, &a->x);
-    secp256k1_fe_mul_int(&t1, 3);         /* T1 = 3*X^2 (3) */
-    secp256k1_fe_sqr(&t2, &t1);           /* T2 = 9*X^4 (1) */
-    secp256k1_fe_sqr(&t3, &a->y);
-    secp256k1_fe_mul_int(&t3, 2);         /* T3 = 2*Y^2 (2) */
-    secp256k1_fe_sqr(&t4, &t3);
-    secp256k1_fe_mul_int(&t4, 2);         /* T4 = 8*Y^4 (2) */
-    secp256k1_fe_mul(&t3, &t3, &a->x);    /* T3 = 2*X*Y^2 (1) */
-    r->x = t3;
-    secp256k1_fe_mul_int(&r->x, 4);       /* X' = 8*X*Y^2 (4) */
-    secp256k1_fe_negate(&r->x, &r->x, 4); /* X' = -8*X*Y^2 (5) */
-    secp256k1_fe_add(&r->x, &t2);         /* X' = 9*X^4 - 8*X*Y^2 (6) */
-    secp256k1_fe_negate(&t2, &t2, 1);     /* T2 = -9*X^4 (2) */
-    secp256k1_fe_mul_int(&t3, 6);         /* T3 = 12*X*Y^2 (6) */
-    secp256k1_fe_add(&t3, &t2);           /* T3 = 12*X*Y^2 - 9*X^4 (8) */
-    secp256k1_fe_mul(&r->y, &t1, &t3);    /* Y' = 36*X^3*Y^2 - 27*X^6 (1) */
-    secp256k1_fe_negate(&t2, &t4, 2);     /* T2 = -8*Y^4 (3) */
-    secp256k1_fe_add(&r->y, &t2);         /* Y' = 36*X^3*Y^2 - 27*X^6 - 8*Y^4 (4) */
+    /* Formula used:
+     * L = (3/2) * X1^2
+     * S = Y1^2
+     * T = -X1*S
+     * X3 = L^2 + 2*T
+     * Y3 = -(L*(X3 + T) + S^2)
+     * Z3 = Y1*Z1
+     */
+
+    secp256k1_fe_mul(&r->z, &a->z, &a->y); /* Z3 = Y1*Z1 (1) */
+    secp256k1_fe_sqr(&s, &a->y);           /* S = Y1^2 (1) */
+    secp256k1_fe_sqr(&l, &a->x);           /* L = X1^2 (1) */
+    secp256k1_fe_mul_int(&l, 3);           /* L = 3*X1^2 (3) */
+    secp256k1_fe_half(&l);                 /* L = 3/2*X1^2 (2) */
+    secp256k1_fe_negate(&t, &s, 1);        /* T = -S (2) */
+    secp256k1_fe_mul(&t, &t, &a->x);       /* T = -X1*S (1) */
+    secp256k1_fe_sqr(&r->x, &l);           /* X3 = L^2 (1) */
+    secp256k1_fe_add(&r->x, &t);           /* X3 = L^2 + T (2) */
+    secp256k1_fe_add(&r->x, &t);           /* X3 = L^2 + 2*T (3) */
+    secp256k1_fe_sqr(&s, &s);              /* S' = S^2 (1) */
+    secp256k1_fe_add(&t, &r->x);           /* T' = X3 + T (4) */
+    secp256k1_fe_mul(&r->y, &t, &l);       /* Y3 = L*(X3 + T) (1) */
+    secp256k1_fe_add(&r->y, &s);           /* Y3 = L*(X3 + T) + S^2 (2) */
+    secp256k1_fe_negate(&r->y, &r->y, 2);  /* Y3 = -(L*(X3 + T) + S^2) (3) */
 }
 
 static void secp256k1_gej_double_var(secp256k1_gej *r, const secp256k1_gej *a, secp256k1_fe *rzr) {
@@ -327,7 +324,6 @@ static void secp256k1_gej_double_var(secp256k1_gej *r, const secp256k1_gej *a, s
     if (rzr != NULL) {
         *rzr = a->y;
         secp256k1_fe_normalize_weak(rzr);
-        secp256k1_fe_mul_int(rzr, 2);
     }
 
     secp256k1_gej_double(r, a);
@@ -493,8 +489,7 @@ static void secp256k1_gej_add_zinv_var(secp256k1_gej *r, const secp256k1_gej *a,
 
 
 static void secp256k1_gej_add_ge(secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_ge *b) {
-    /* Operations: 7 mul, 5 sqr, 4 normalize, 21 mul_int/add/negate/cmov */
-    static const secp256k1_fe fe_1 = SECP256K1_FE_CONST(0, 0, 0, 0, 0, 0, 0, 1);
+    /* Operations: 7 mul, 5 sqr, 24 add/cmov/half/mul_int/negate/normalize_weak/normalizes_to_zero */
     secp256k1_fe zz, u1, u2, s1, s2, t, tt, m, n, q, rr;
     secp256k1_fe m_alt, rr_alt;
     int infinity, degenerate;
@@ -515,11 +510,11 @@ static void secp256k1_gej_add_ge(secp256k1_gej *r, const secp256k1_gej *a, const
      *    Z = Z1*Z2
      *    T = U1+U2
      *    M = S1+S2
-     *    Q = T*M^2
+     *    Q = -T*M^2
      *    R = T^2-U1*U2
-     *    X3 = 4*(R^2-Q)
-     *    Y3 = 4*(R*(3*Q-2*R^2)-M^4)
-     *    Z3 = 2*M*Z
+     *    X3 = R^2+Q
+     *    Y3 = -(R*(2*X3+Q)+M^4)/2
+     *    Z3 = M*Z
      *  (Note that the paper uses xi = Xi / Zi and yi = Yi / Zi instead.)
      *
      *  This formula has the benefit of being the same for both addition
@@ -583,7 +578,8 @@ static void secp256k1_gej_add_ge(secp256k1_gej *r, const secp256k1_gej *a, const
      * and denominator of lambda; R and M represent the explicit
      * expressions x1^2 + x2^2 + x1x2 and y1 + y2. */
     secp256k1_fe_sqr(&n, &m_alt);                       /* n = Malt^2 (1) */
-    secp256k1_fe_mul(&q, &n, &t);                       /* q = Q = T*Malt^2 (1) */
+    secp256k1_fe_negate(&q, &t, 2);                     /* q = -T (3) */
+    secp256k1_fe_mul(&q, &q, &n);                       /* q = Q = -T*Malt^2 (1) */
     /* These two lines use the observation that either M == Malt or M == 0,
      * so M^3 * Malt is either Malt^4 (which is computed by squaring), or
      * zero (which is "computed" by cmov). So the cost is one squaring
@@ -591,26 +587,21 @@ static void secp256k1_gej_add_ge(secp256k1_gej *r, const secp256k1_gej *a, const
     secp256k1_fe_sqr(&n, &n);
     secp256k1_fe_cmov(&n, &m, degenerate);              /* n = M^3 * Malt (2) */
     secp256k1_fe_sqr(&t, &rr_alt);                      /* t = Ralt^2 (1) */
-    secp256k1_fe_mul(&r->z, &a->z, &m_alt);             /* r->z = Malt*Z (1) */
+    secp256k1_fe_mul(&r->z, &a->z, &m_alt);             /* r->z = Z3 = Malt*Z (1) */
     infinity = secp256k1_fe_normalizes_to_zero(&r->z) & ~a->infinity;
-    secp256k1_fe_mul_int(&r->z, 2);                     /* r->z = Z3 = 2*Malt*Z (2) */
-    secp256k1_fe_negate(&q, &q, 1);                     /* q = -Q (2) */
-    secp256k1_fe_add(&t, &q);                           /* t = Ralt^2-Q (3) */
-    secp256k1_fe_normalize_weak(&t);
-    r->x = t;                                           /* r->x = Ralt^2-Q (1) */
-    secp256k1_fe_mul_int(&t, 2);                        /* t = 2*x3 (2) */
-    secp256k1_fe_add(&t, &q);                           /* t = 2*x3 - Q: (4) */
-    secp256k1_fe_mul(&t, &t, &rr_alt);                  /* t = Ralt*(2*x3 - Q) (1) */
-    secp256k1_fe_add(&t, &n);                           /* t = Ralt*(2*x3 - Q) + M^3*Malt (3) */
-    secp256k1_fe_negate(&r->y, &t, 3);                  /* r->y = Ralt*(Q - 2x3) - M^3*Malt (4) */
-    secp256k1_fe_normalize_weak(&r->y);
-    secp256k1_fe_mul_int(&r->x, 4);                     /* r->x = X3 = 4*(Ralt^2-Q) */
-    secp256k1_fe_mul_int(&r->y, 4);                     /* r->y = Y3 = 4*Ralt*(Q - 2x3) - 4*M^3*Malt (4) */
+    secp256k1_fe_add(&t, &q);                           /* t = Ralt^2 + Q (2) */
+    r->x = t;                                           /* r->x = X3 = Ralt^2 + Q (2) */
+    secp256k1_fe_mul_int(&t, 2);                        /* t = 2*X3 (4) */
+    secp256k1_fe_add(&t, &q);                           /* t = 2*X3 + Q (5) */
+    secp256k1_fe_mul(&t, &t, &rr_alt);                  /* t = Ralt*(2*X3 + Q) (1) */
+    secp256k1_fe_add(&t, &n);                           /* t = Ralt*(2*X3 + Q) + M^3*Malt (3) */
+    secp256k1_fe_negate(&r->y, &t, 3);                  /* r->y = -(Ralt*(2*X3 + Q) + M^3*Malt) (4) */
+    secp256k1_fe_half(&r->y);                           /* r->y = Y3 = -(Ralt*(2*X3 + Q) + M^3*Malt)/2 (3) */
 
     /** In case a->infinity == 1, replace r with (b->x, b->y, 1). */
     secp256k1_fe_cmov(&r->x, &b->x, a->infinity);
     secp256k1_fe_cmov(&r->y, &b->y, a->infinity);
-    secp256k1_fe_cmov(&r->z, &fe_1, a->infinity);
+    secp256k1_fe_cmov(&r->z, &secp256k1_fe_one, a->infinity);
     r->infinity = infinity;
 }
 
@@ -642,18 +633,22 @@ static void secp256k1_ge_from_storage(secp256k1_ge *r, const secp256k1_ge_storag
     r->infinity = 0;
 }
 
+static SECP256K1_INLINE void secp256k1_gej_cmov(secp256k1_gej *r, const secp256k1_gej *a, int flag) {
+    secp256k1_fe_cmov(&r->x, &a->x, flag);
+    secp256k1_fe_cmov(&r->y, &a->y, flag);
+    secp256k1_fe_cmov(&r->z, &a->z, flag);
+
+    r->infinity ^= (r->infinity ^ a->infinity) & flag;
+}
+
 static SECP256K1_INLINE void secp256k1_ge_storage_cmov(secp256k1_ge_storage *r, const secp256k1_ge_storage *a, int flag) {
     secp256k1_fe_storage_cmov(&r->x, &a->x, flag);
     secp256k1_fe_storage_cmov(&r->y, &a->y, flag);
 }
 
 static void secp256k1_ge_mul_lambda(secp256k1_ge *r, const secp256k1_ge *a) {
-    static const secp256k1_fe beta = SECP256K1_FE_CONST(
-        0x7ae96a2bul, 0x657c0710ul, 0x6e64479eul, 0xac3434e9ul,
-        0x9cf04975ul, 0x12f58995ul, 0xc1396c28ul, 0x719501eeul
-    );
     *r = *a;
-    secp256k1_fe_mul(&r->x, &r->x, &beta);
+    secp256k1_fe_mul(&r->x, &r->x, &secp256k1_const_beta);
 }
 
 static int secp256k1_ge_is_in_correct_subgroup(const secp256k1_ge* ge) {
diff --git a/src/hash.h b/src/hash.h
index 0947a096943a9d..4e0384cfbffcaa 100644
--- a/src/hash.h
+++ b/src/hash.h
@@ -12,8 +12,8 @@
 
 typedef struct {
     uint32_t s[8];
-    uint32_t buf[16]; /* In big endian */
-    size_t bytes;
+    unsigned char buf[64];
+    uint64_t bytes;
 } secp256k1_sha256;
 
 static void secp256k1_sha256_initialize(secp256k1_sha256 *hash);
diff --git a/src/hash_impl.h b/src/hash_impl.h
index f8cd3a1634113e..0991fe78383b40 100644
--- a/src/hash_impl.h
+++ b/src/hash_impl.h
@@ -28,12 +28,6 @@
     (h) = t1 + t2; \
 } while(0)
 
-#if defined(SECP256K1_BIG_ENDIAN)
-#define BE32(x) (x)
-#elif defined(SECP256K1_LITTLE_ENDIAN)
-#define BE32(p) ((((p) & 0xFF) << 24) | (((p) & 0xFF00) << 8) | (((p) & 0xFF0000) >> 8) | (((p) & 0xFF000000) >> 24))
-#endif
-
 static void secp256k1_sha256_initialize(secp256k1_sha256 *hash) {
     hash->s[0] = 0x6a09e667ul;
     hash->s[1] = 0xbb67ae85ul;
@@ -47,26 +41,26 @@ static void secp256k1_sha256_initialize(secp256k1_sha256 *hash) {
 }
 
 /** Perform one SHA-256 transformation, processing 16 big endian 32-bit words. */
-static void secp256k1_sha256_transform(uint32_t* s, const uint32_t* chunk) {
+static void secp256k1_sha256_transform(uint32_t* s, const unsigned char* buf) {
     uint32_t a = s[0], b = s[1], c = s[2], d = s[3], e = s[4], f = s[5], g = s[6], h = s[7];
     uint32_t w0, w1, w2, w3, w4, w5, w6, w7, w8, w9, w10, w11, w12, w13, w14, w15;
 
-    Round(a, b, c, d, e, f, g, h, 0x428a2f98, w0 = BE32(chunk[0]));
-    Round(h, a, b, c, d, e, f, g, 0x71374491, w1 = BE32(chunk[1]));
-    Round(g, h, a, b, c, d, e, f, 0xb5c0fbcf, w2 = BE32(chunk[2]));
-    Round(f, g, h, a, b, c, d, e, 0xe9b5dba5, w3 = BE32(chunk[3]));
-    Round(e, f, g, h, a, b, c, d, 0x3956c25b, w4 = BE32(chunk[4]));
-    Round(d, e, f, g, h, a, b, c, 0x59f111f1, w5 = BE32(chunk[5]));
-    Round(c, d, e, f, g, h, a, b, 0x923f82a4, w6 = BE32(chunk[6]));
-    Round(b, c, d, e, f, g, h, a, 0xab1c5ed5, w7 = BE32(chunk[7]));
-    Round(a, b, c, d, e, f, g, h, 0xd807aa98, w8 = BE32(chunk[8]));
-    Round(h, a, b, c, d, e, f, g, 0x12835b01, w9 = BE32(chunk[9]));
-    Round(g, h, a, b, c, d, e, f, 0x243185be, w10 = BE32(chunk[10]));
-    Round(f, g, h, a, b, c, d, e, 0x550c7dc3, w11 = BE32(chunk[11]));
-    Round(e, f, g, h, a, b, c, d, 0x72be5d74, w12 = BE32(chunk[12]));
-    Round(d, e, f, g, h, a, b, c, 0x80deb1fe, w13 = BE32(chunk[13]));
-    Round(c, d, e, f, g, h, a, b, 0x9bdc06a7, w14 = BE32(chunk[14]));
-    Round(b, c, d, e, f, g, h, a, 0xc19bf174, w15 = BE32(chunk[15]));
+    Round(a, b, c, d, e, f, g, h, 0x428a2f98,  w0 = secp256k1_read_be32(&buf[0]));
+    Round(h, a, b, c, d, e, f, g, 0x71374491,  w1 = secp256k1_read_be32(&buf[4]));
+    Round(g, h, a, b, c, d, e, f, 0xb5c0fbcf,  w2 = secp256k1_read_be32(&buf[8]));
+    Round(f, g, h, a, b, c, d, e, 0xe9b5dba5,  w3 = secp256k1_read_be32(&buf[12]));
+    Round(e, f, g, h, a, b, c, d, 0x3956c25b,  w4 = secp256k1_read_be32(&buf[16]));
+    Round(d, e, f, g, h, a, b, c, 0x59f111f1,  w5 = secp256k1_read_be32(&buf[20]));
+    Round(c, d, e, f, g, h, a, b, 0x923f82a4,  w6 = secp256k1_read_be32(&buf[24]));
+    Round(b, c, d, e, f, g, h, a, 0xab1c5ed5,  w7 = secp256k1_read_be32(&buf[28]));
+    Round(a, b, c, d, e, f, g, h, 0xd807aa98,  w8 = secp256k1_read_be32(&buf[32]));
+    Round(h, a, b, c, d, e, f, g, 0x12835b01,  w9 = secp256k1_read_be32(&buf[36]));
+    Round(g, h, a, b, c, d, e, f, 0x243185be, w10 = secp256k1_read_be32(&buf[40]));
+    Round(f, g, h, a, b, c, d, e, 0x550c7dc3, w11 = secp256k1_read_be32(&buf[44]));
+    Round(e, f, g, h, a, b, c, d, 0x72be5d74, w12 = secp256k1_read_be32(&buf[48]));
+    Round(d, e, f, g, h, a, b, c, 0x80deb1fe, w13 = secp256k1_read_be32(&buf[52]));
+    Round(c, d, e, f, g, h, a, b, 0x9bdc06a7, w14 = secp256k1_read_be32(&buf[56]));
+    Round(b, c, d, e, f, g, h, a, 0xc19bf174, w15 = secp256k1_read_be32(&buf[60]));
 
     Round(a, b, c, d, e, f, g, h, 0xe49b69c1, w0 += sigma1(w14) + w9 + sigma0(w1));
     Round(h, a, b, c, d, e, f, g, 0xefbe4786, w1 += sigma1(w15) + w10 + sigma0(w2));
@@ -136,7 +130,7 @@ static void secp256k1_sha256_write(secp256k1_sha256 *hash, const unsigned char *
     while (len >= 64 - bufsize) {
         /* Fill the buffer, and process it. */
         size_t chunk_len = 64 - bufsize;
-        memcpy(((unsigned char*)hash->buf) + bufsize, data, chunk_len);
+        memcpy(hash->buf + bufsize, data, chunk_len);
         data += chunk_len;
         len -= chunk_len;
         secp256k1_sha256_transform(hash->s, hash->buf);
@@ -149,19 +143,19 @@ static void secp256k1_sha256_write(secp256k1_sha256 *hash, const unsigned char *
 }
 
 static void secp256k1_sha256_finalize(secp256k1_sha256 *hash, unsigned char *out32) {
-    static const unsigned char pad[64] = {0x80, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0};
-    uint32_t sizedesc[2];
-    uint32_t out[8];
-    int i = 0;
-    sizedesc[0] = BE32(hash->bytes >> 29);
-    sizedesc[1] = BE32(hash->bytes << 3);
+    static const unsigned char pad[64] = {0x80};
+    unsigned char sizedesc[8];
+    int i;
+    /* The maximum message size of SHA256 is 2^64-1 bits. */
+    VERIFY_CHECK(hash->bytes < ((uint64_t)1 << 61));
+    secp256k1_write_be32(&sizedesc[0], hash->bytes >> 29);
+    secp256k1_write_be32(&sizedesc[4], hash->bytes << 3);
     secp256k1_sha256_write(hash, pad, 1 + ((119 - (hash->bytes % 64)) % 64));
-    secp256k1_sha256_write(hash, (const unsigned char*)sizedesc, 8);
+    secp256k1_sha256_write(hash, sizedesc, 8);
     for (i = 0; i < 8; i++) {
-        out[i] = BE32(hash->s[i]);
+        secp256k1_write_be32(&out32[4*i], hash->s[i]);
         hash->s[i] = 0;
     }
-    memcpy(out32, (const unsigned char*)out, 32);
 }
 
 /* Initializes a sha256 struct and writes the 64 byte string
@@ -285,7 +279,6 @@ static void secp256k1_rfc6979_hmac_sha256_finalize(secp256k1_rfc6979_hmac_sha256
     rng->retry = 0;
 }
 
-#undef BE32
 #undef Round
 #undef sigma1
 #undef sigma0
diff --git a/src/modules/ecdh/tests_impl.h b/src/modules/ecdh/tests_impl.h
index be07447a4b9957..10b7075c384f2c 100644
--- a/src/modules/ecdh/tests_impl.h
+++ b/src/modules/ecdh/tests_impl.h
@@ -60,7 +60,7 @@ void test_ecdh_generator_basepoint(void) {
 
     s_one[31] = 1;
     /* Check against pubkey creation when the basepoint is the generator */
-    for (i = 0; i < 100; ++i) {
+    for (i = 0; i < 2 * count; ++i) {
         secp256k1_sha256 sha;
         unsigned char s_b32[32];
         unsigned char output_ecdh[65];
@@ -123,10 +123,43 @@ void test_bad_scalar(void) {
     CHECK(secp256k1_ecdh(ctx, output, &point, s_overflow, ecdh_hash_function_test_fail, NULL) == 0);
 }
 
+/** Test that ECDH(sG, 1/s) == ECDH((1/s)G, s) == ECDH(G, 1) for a few random s. */
+void test_result_basepoint(void) {
+    secp256k1_pubkey point;
+    secp256k1_scalar rand;
+    unsigned char s[32];
+    unsigned char s_inv[32];
+    unsigned char out[32];
+    unsigned char out_inv[32];
+    unsigned char out_base[32];
+    int i;
+
+    unsigned char s_one[32] = { 0 };
+    s_one[31] = 1;
+    CHECK(secp256k1_ec_pubkey_create(ctx, &point, s_one) == 1);
+    CHECK(secp256k1_ecdh(ctx, out_base, &point, s_one, NULL, NULL) == 1);
+
+    for (i = 0; i < 2 * count; i++) {
+        random_scalar_order(&rand);
+        secp256k1_scalar_get_b32(s, &rand);
+        secp256k1_scalar_inverse(&rand, &rand);
+        secp256k1_scalar_get_b32(s_inv, &rand);
+
+        CHECK(secp256k1_ec_pubkey_create(ctx, &point, s) == 1);
+        CHECK(secp256k1_ecdh(ctx, out, &point, s_inv, NULL, NULL) == 1);
+        CHECK(secp256k1_memcmp_var(out, out_base, 32) == 0);
+
+        CHECK(secp256k1_ec_pubkey_create(ctx, &point, s_inv) == 1);
+        CHECK(secp256k1_ecdh(ctx, out_inv, &point, s, NULL, NULL) == 1);
+        CHECK(secp256k1_memcmp_var(out_inv, out_base, 32) == 0);
+    }
+}
+
 void run_ecdh_tests(void) {
     test_ecdh_api();
     test_ecdh_generator_basepoint();
     test_bad_scalar();
+    test_result_basepoint();
 }
 
 #endif /* SECP256K1_MODULE_ECDH_TESTS_H */
diff --git a/src/modules/schnorrsig/main_impl.h b/src/modules/schnorrsig/main_impl.h
index 94e3ee414d6d1a..cd651591c4e162 100644
--- a/src/modules/schnorrsig/main_impl.h
+++ b/src/modules/schnorrsig/main_impl.h
@@ -192,11 +192,15 @@ static int secp256k1_schnorrsig_sign_internal(const secp256k1_context* ctx, unsi
     return ret;
 }
 
-int secp256k1_schnorrsig_sign(const secp256k1_context* ctx, unsigned char *sig64, const unsigned char *msg32, const secp256k1_keypair *keypair, const unsigned char *aux_rand32) {
+int secp256k1_schnorrsig_sign32(const secp256k1_context* ctx, unsigned char *sig64, const unsigned char *msg32, const secp256k1_keypair *keypair, const unsigned char *aux_rand32) {
     /* We cast away const from the passed aux_rand32 argument since we know the default nonce function does not modify it. */
     return secp256k1_schnorrsig_sign_internal(ctx, sig64, msg32, 32, keypair, secp256k1_nonce_function_bip340, (unsigned char*)aux_rand32);
 }
 
+int secp256k1_schnorrsig_sign(const secp256k1_context* ctx, unsigned char *sig64, const unsigned char *msg32, const secp256k1_keypair *keypair, const unsigned char *aux_rand32) {
+    return secp256k1_schnorrsig_sign32(ctx, sig64, msg32, keypair, aux_rand32);
+}
+
 int secp256k1_schnorrsig_sign_custom(const secp256k1_context* ctx, unsigned char *sig64, const unsigned char *msg, size_t msglen, const secp256k1_keypair *keypair, secp256k1_schnorrsig_extraparams *extraparams) {
     secp256k1_nonce_function_hardened noncefp = NULL;
     void *ndata = NULL;
diff --git a/src/modules/schnorrsig/tests_impl.h b/src/modules/schnorrsig/tests_impl.h
index 2efec8a2b9dc49..25840b8fa701c0 100644
--- a/src/modules/schnorrsig/tests_impl.h
+++ b/src/modules/schnorrsig/tests_impl.h
@@ -87,7 +87,7 @@ void run_nonce_function_bip340_tests(void) {
     CHECK(nonce_function_bip340(nonce, msg, msglen, key, pk, NULL, 0, NULL) == 0);
     CHECK(nonce_function_bip340(nonce, msg, msglen, key, pk, algo, algolen, NULL) == 1);
     /* Other algo is fine */
-    secp256k1_rfc6979_hmac_sha256_generate(&secp256k1_test_rng, algo, algolen);
+    secp256k1_testrand_bytes_test(algo, algolen);
     CHECK(nonce_function_bip340(nonce, msg, msglen, key, pk, algo, algolen, NULL) == 1);
 
     for (i = 0; i < count; i++) {
@@ -160,21 +160,21 @@ void test_schnorrsig_api(void) {
 
     /** main test body **/
     ecount = 0;
-    CHECK(secp256k1_schnorrsig_sign(none, sig, msg, &keypairs[0], NULL) == 1);
+    CHECK(secp256k1_schnorrsig_sign32(none, sig, msg, &keypairs[0], NULL) == 1);
     CHECK(ecount == 0);
-    CHECK(secp256k1_schnorrsig_sign(vrfy, sig, msg, &keypairs[0], NULL) == 1);
+    CHECK(secp256k1_schnorrsig_sign32(vrfy, sig, msg, &keypairs[0], NULL) == 1);
     CHECK(ecount == 0);
-    CHECK(secp256k1_schnorrsig_sign(sign, sig, msg, &keypairs[0], NULL) == 1);
+    CHECK(secp256k1_schnorrsig_sign32(sign, sig, msg, &keypairs[0], NULL) == 1);
     CHECK(ecount == 0);
-    CHECK(secp256k1_schnorrsig_sign(sign, NULL, msg, &keypairs[0], NULL) == 0);
+    CHECK(secp256k1_schnorrsig_sign32(sign, NULL, msg, &keypairs[0], NULL) == 0);
     CHECK(ecount == 1);
-    CHECK(secp256k1_schnorrsig_sign(sign, sig, NULL, &keypairs[0], NULL) == 0);
+    CHECK(secp256k1_schnorrsig_sign32(sign, sig, NULL, &keypairs[0], NULL) == 0);
     CHECK(ecount == 2);
-    CHECK(secp256k1_schnorrsig_sign(sign, sig, msg, NULL, NULL) == 0);
+    CHECK(secp256k1_schnorrsig_sign32(sign, sig, msg, NULL, NULL) == 0);
     CHECK(ecount == 3);
-    CHECK(secp256k1_schnorrsig_sign(sign, sig, msg, &invalid_keypair, NULL) == 0);
+    CHECK(secp256k1_schnorrsig_sign32(sign, sig, msg, &invalid_keypair, NULL) == 0);
     CHECK(ecount == 4);
-    CHECK(secp256k1_schnorrsig_sign(sttc, sig, msg, &keypairs[0], NULL) == 0);
+    CHECK(secp256k1_schnorrsig_sign32(sttc, sig, msg, &keypairs[0], NULL) == 0);
     CHECK(ecount == 5);
 
     ecount = 0;
@@ -202,7 +202,7 @@ void test_schnorrsig_api(void) {
     CHECK(ecount == 6);
 
     ecount = 0;
-    CHECK(secp256k1_schnorrsig_sign(sign, sig, msg, &keypairs[0], NULL) == 1);
+    CHECK(secp256k1_schnorrsig_sign32(sign, sig, msg, &keypairs[0], NULL) == 1);
     CHECK(secp256k1_schnorrsig_verify(none, sig, msg, sizeof(msg), &pk[0]) == 1);
     CHECK(ecount == 0);
     CHECK(secp256k1_schnorrsig_verify(sign, sig, msg, sizeof(msg), &pk[0]) == 1);
@@ -247,7 +247,7 @@ void test_schnorrsig_bip_vectors_check_signing(const unsigned char *sk, const un
     secp256k1_xonly_pubkey pk, pk_expected;
 
     CHECK(secp256k1_keypair_create(ctx, &keypair, sk));
-    CHECK(secp256k1_schnorrsig_sign(ctx, sig, msg32, &keypair, aux_rand));
+    CHECK(secp256k1_schnorrsig_sign32(ctx, sig, msg32, &keypair, aux_rand));
     CHECK(secp256k1_memcmp_var(sig, expected_sig, 64) == 0);
 
     CHECK(secp256k1_xonly_pubkey_parse(ctx, &pk_expected, pk_serialized));
@@ -740,8 +740,11 @@ void test_schnorrsig_sign(void) {
     secp256k1_testrand256(aux_rand);
     CHECK(secp256k1_keypair_create(ctx, &keypair, sk));
     CHECK(secp256k1_keypair_xonly_pub(ctx, &pk, NULL, &keypair));
-    CHECK(secp256k1_schnorrsig_sign(ctx, sig, msg, &keypair, NULL) == 1);
+    CHECK(secp256k1_schnorrsig_sign32(ctx, sig, msg, &keypair, NULL) == 1);
     CHECK(secp256k1_schnorrsig_verify(ctx, sig, msg, sizeof(msg), &pk));
+    /* Check that deprecated alias gives the same result */
+    CHECK(secp256k1_schnorrsig_sign(ctx, sig2, msg, &keypair, NULL) == 1);
+    CHECK(secp256k1_memcmp_var(sig, sig2, sizeof(sig)) == 0);
 
     /* Test different nonce functions */
     CHECK(secp256k1_schnorrsig_sign_custom(ctx, sig, msg, sizeof(msg), &keypair, &extraparams) == 1);
@@ -764,7 +767,7 @@ void test_schnorrsig_sign(void) {
     extraparams.noncefp = NULL;
     extraparams.ndata = aux_rand;
     CHECK(secp256k1_schnorrsig_sign_custom(ctx, sig, msg, sizeof(msg), &keypair, &extraparams) == 1);
-    CHECK(secp256k1_schnorrsig_sign(ctx, sig2, msg, &keypair, extraparams.ndata) == 1);
+    CHECK(secp256k1_schnorrsig_sign32(ctx, sig2, msg, &keypair, extraparams.ndata) == 1);
     CHECK(secp256k1_memcmp_var(sig, sig2, sizeof(sig)) == 0);
 }
 
@@ -787,7 +790,7 @@ void test_schnorrsig_sign_verify(void) {
 
     for (i = 0; i < N_SIGS; i++) {
         secp256k1_testrand256(msg[i]);
-        CHECK(secp256k1_schnorrsig_sign(ctx, sig[i], msg[i], &keypair, NULL));
+        CHECK(secp256k1_schnorrsig_sign32(ctx, sig[i], msg[i], &keypair, NULL));
         CHECK(secp256k1_schnorrsig_verify(ctx, sig[i], msg[i], sizeof(msg[i]), &pk));
     }
 
@@ -795,18 +798,18 @@ void test_schnorrsig_sign_verify(void) {
         /* Flip a few bits in the signature and in the message and check that
          * verify and verify_batch (TODO) fail */
         size_t sig_idx = secp256k1_testrand_int(N_SIGS);
-        size_t byte_idx = secp256k1_testrand_int(32);
+        size_t byte_idx = secp256k1_testrand_bits(5);
         unsigned char xorbyte = secp256k1_testrand_int(254)+1;
         sig[sig_idx][byte_idx] ^= xorbyte;
         CHECK(!secp256k1_schnorrsig_verify(ctx, sig[sig_idx], msg[sig_idx], sizeof(msg[sig_idx]), &pk));
         sig[sig_idx][byte_idx] ^= xorbyte;
 
-        byte_idx = secp256k1_testrand_int(32);
+        byte_idx = secp256k1_testrand_bits(5);
         sig[sig_idx][32+byte_idx] ^= xorbyte;
         CHECK(!secp256k1_schnorrsig_verify(ctx, sig[sig_idx], msg[sig_idx], sizeof(msg[sig_idx]), &pk));
         sig[sig_idx][32+byte_idx] ^= xorbyte;
 
-        byte_idx = secp256k1_testrand_int(32);
+        byte_idx = secp256k1_testrand_bits(5);
         msg[sig_idx][byte_idx] ^= xorbyte;
         CHECK(!secp256k1_schnorrsig_verify(ctx, sig[sig_idx], msg[sig_idx], sizeof(msg[sig_idx]), &pk));
         msg[sig_idx][byte_idx] ^= xorbyte;
@@ -816,13 +819,13 @@ void test_schnorrsig_sign_verify(void) {
     }
 
     /* Test overflowing s */
-    CHECK(secp256k1_schnorrsig_sign(ctx, sig[0], msg[0], &keypair, NULL));
+    CHECK(secp256k1_schnorrsig_sign32(ctx, sig[0], msg[0], &keypair, NULL));
     CHECK(secp256k1_schnorrsig_verify(ctx, sig[0], msg[0], sizeof(msg[0]), &pk));
     memset(&sig[0][32], 0xFF, 32);
     CHECK(!secp256k1_schnorrsig_verify(ctx, sig[0], msg[0], sizeof(msg[0]), &pk));
 
     /* Test negative s */
-    CHECK(secp256k1_schnorrsig_sign(ctx, sig[0], msg[0], &keypair, NULL));
+    CHECK(secp256k1_schnorrsig_sign32(ctx, sig[0], msg[0], &keypair, NULL));
     CHECK(secp256k1_schnorrsig_verify(ctx, sig[0], msg[0], sizeof(msg[0]), &pk));
     secp256k1_scalar_set_b32(&s, &sig[0][32], NULL);
     secp256k1_scalar_negate(&s, &s);
@@ -873,7 +876,7 @@ void test_schnorrsig_taproot(void) {
 
     /* Key spend */
     secp256k1_testrand256(msg);
-    CHECK(secp256k1_schnorrsig_sign(ctx, sig, msg, &keypair, NULL) == 1);
+    CHECK(secp256k1_schnorrsig_sign32(ctx, sig, msg, &keypair, NULL) == 1);
     /* Verify key spend */
     CHECK(secp256k1_xonly_pubkey_parse(ctx, &output_pk, output_pk_bytes) == 1);
     CHECK(secp256k1_schnorrsig_verify(ctx, sig, msg, sizeof(msg), &output_pk) == 1);
diff --git a/src/precompute_ecmult.c b/src/precompute_ecmult.c
new file mode 100644
index 00000000000000..5ccbcb3c5757ca
--- /dev/null
+++ b/src/precompute_ecmult.c
@@ -0,0 +1,96 @@
+/*****************************************************************************************************
+ * Copyright (c) 2013, 2014, 2017, 2021 Pieter Wuille, Andrew Poelstra, Jonas Nick, Russell O'Connor *
+ * Distributed under the MIT software license, see the accompanying                                  *
+ * file COPYING or https://www.opensource.org/licenses/mit-license.php.                              *
+ *****************************************************************************************************/
+
+#include <inttypes.h>
+#include <stdio.h>
+
+/* Autotools creates libsecp256k1-config.h, of which ECMULT_WINDOW_SIZE is needed.
+   ifndef guard so downstream users can define their own if they do not use autotools. */
+#if !defined(ECMULT_WINDOW_SIZE)
+#include "libsecp256k1-config.h"
+#endif
+
+#include "../include/secp256k1.h"
+#include "assumptions.h"
+#include "util.h"
+#include "field_impl.h"
+#include "group_impl.h"
+#include "ecmult.h"
+#include "ecmult_compute_table_impl.h"
+
+static void print_table(FILE *fp, const char *name, int window_g, const secp256k1_ge_storage* table) {
+    int j;
+    int i;
+
+    fprintf(fp, "const secp256k1_ge_storage %s[ECMULT_TABLE_SIZE(WINDOW_G)] = {\n", name);
+    fprintf(fp, " S(%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32
+                  ",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32")\n",
+                SECP256K1_GE_STORAGE_CONST_GET(table[0]));
+
+    j = 1;
+    for(i = 3; i <= window_g; ++i) {
+        fprintf(fp, "#if WINDOW_G > %d\n", i-1);
+        for(;j < ECMULT_TABLE_SIZE(i); ++j) {
+            fprintf(fp, ",S(%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32
+                          ",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32",%"PRIx32")\n",
+                        SECP256K1_GE_STORAGE_CONST_GET(table[j]));
+        }
+        fprintf(fp, "#endif\n");
+    }
+    fprintf(fp, "};\n");
+}
+
+static void print_two_tables(FILE *fp, int window_g) {
+    secp256k1_ge_storage* table = malloc(ECMULT_TABLE_SIZE(window_g) * sizeof(secp256k1_ge_storage));
+    secp256k1_ge_storage* table_128 = malloc(ECMULT_TABLE_SIZE(window_g) * sizeof(secp256k1_ge_storage));
+
+    secp256k1_ecmult_compute_two_tables(table, table_128, window_g, &secp256k1_ge_const_g);
+
+    print_table(fp, "secp256k1_pre_g", window_g, table);
+    print_table(fp, "secp256k1_pre_g_128", window_g, table_128);
+
+    free(table);
+    free(table_128);
+}
+
+int main(void) {
+    /* Always compute all tables for window sizes up to 15. */
+    int window_g = (ECMULT_WINDOW_SIZE < 15) ? 15 : ECMULT_WINDOW_SIZE;
+    FILE* fp;
+
+    fp = fopen("src/precomputed_ecmult.c","w");
+    if (fp == NULL) {
+        fprintf(stderr, "Could not open src/precomputed_ecmult.h for writing!\n");
+        return -1;
+    }
+
+    fprintf(fp, "/* This file was automatically generated by precompute_ecmult. */\n");
+    fprintf(fp, "/* This file contains an array secp256k1_pre_g with odd multiples of the base point G and\n");
+    fprintf(fp, " * an array secp256k1_pre_g_128 with odd multiples of 2^128*G for accelerating the computation of a*P + b*G.\n");
+    fprintf(fp, " */\n");
+    fprintf(fp, "#if defined HAVE_CONFIG_H\n");
+    fprintf(fp, "#    include \"libsecp256k1-config.h\"\n");
+    fprintf(fp, "#endif\n");
+    fprintf(fp, "#include \"../include/secp256k1.h\"\n");
+    fprintf(fp, "#include \"group.h\"\n");
+    fprintf(fp, "#include \"ecmult.h\"\n");
+    fprintf(fp, "#include \"precomputed_ecmult.h\"\n");
+    fprintf(fp, "#define S(a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p) SECP256K1_GE_STORAGE_CONST(0x##a##u,0x##b##u,0x##c##u,0x##d##u,0x##e##u,0x##f##u,0x##g##u,0x##h##u,0x##i##u,0x##j##u,0x##k##u,0x##l##u,0x##m##u,0x##n##u,0x##o##u,0x##p##u)\n");
+    fprintf(fp, "#if ECMULT_WINDOW_SIZE > %d\n", window_g);
+    fprintf(fp, "   #error configuration mismatch, invalid ECMULT_WINDOW_SIZE. Try deleting precomputed_ecmult.c before the build.\n");
+    fprintf(fp, "#endif\n");
+    fprintf(fp, "#ifdef EXHAUSTIVE_TEST_ORDER\n");
+    fprintf(fp, "#    error Cannot compile precomputed_ecmult.c in exhaustive test mode\n");
+    fprintf(fp, "#endif /* EXHAUSTIVE_TEST_ORDER */\n");
+    fprintf(fp, "#define WINDOW_G ECMULT_WINDOW_SIZE\n");
+
+    print_two_tables(fp, window_g);
+
+    fprintf(fp, "#undef S\n");
+    fclose(fp);
+
+    return 0;
+}
diff --git a/src/gen_ecmult_gen_static_prec_table.c b/src/precompute_ecmult_gen.c
similarity index 61%
rename from src/gen_ecmult_gen_static_prec_table.c
rename to src/precompute_ecmult_gen.c
index 22923df3134b1c..7c6359c402f641 100644
--- a/src/gen_ecmult_gen_static_prec_table.c
+++ b/src/precompute_ecmult_gen.c
@@ -1,8 +1,8 @@
-/***********************************************************************
- * Copyright (c) 2013, 2014, 2015 Thomas Daede, Cory Fields            *
- * Distributed under the MIT software license, see the accompanying    *
- * file COPYING or https://www.opensource.org/licenses/mit-license.php.*
- ***********************************************************************/
+/*********************************************************************************
+ * Copyright (c) 2013, 2014, 2015, 2021 Thomas Daede, Cory Fields, Pieter Wuille *
+ * Distributed under the MIT software license, see the accompanying              *
+ * file COPYING or https://www.opensource.org/licenses/mit-license.php.          *
+ *********************************************************************************/
 
 #include <inttypes.h>
 #include <stdio.h>
@@ -12,10 +12,10 @@
 #include "util.h"
 #include "group.h"
 #include "ecmult_gen.h"
-#include "ecmult_gen_prec_impl.h"
+#include "ecmult_gen_compute_table_impl.h"
 
 int main(int argc, char **argv) {
-    const char outfile[] = "src/ecmult_gen_static_prec_table.h";
+    const char outfile[] = "src/precomputed_ecmult_gen.c";
     FILE* fp;
     int bits;
 
@@ -28,21 +28,20 @@ int main(int argc, char **argv) {
         return -1;
     }
 
-    fprintf(fp, "/* This file was automatically generated by gen_ecmult_gen_static_prec_table. */\n");
+    fprintf(fp, "/* This file was automatically generated by precompute_ecmult_gen. */\n");
     fprintf(fp, "/* See ecmult_gen_impl.h for details about the contents of this file. */\n");
-    fprintf(fp, "#ifndef SECP256K1_ECMULT_GEN_STATIC_PREC_TABLE_H\n");
-    fprintf(fp, "#define SECP256K1_ECMULT_GEN_STATIC_PREC_TABLE_H\n");
-
+    fprintf(fp, "#if defined HAVE_CONFIG_H\n");
+    fprintf(fp, "#    include \"libsecp256k1-config.h\"\n");
+    fprintf(fp, "#endif\n");
+    fprintf(fp, "#include \"../include/secp256k1.h\"\n");
     fprintf(fp, "#include \"group.h\"\n");
-
-    fprintf(fp, "#define S(a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p) "
-            "SECP256K1_GE_STORAGE_CONST(0x##a##u,0x##b##u,0x##c##u,0x##d##u,0x##e##u,0x##f##u,0x##g##u,"
-            "0x##h##u,0x##i##u,0x##j##u,0x##k##u,0x##l##u,0x##m##u,0x##n##u,0x##o##u,0x##p##u)\n");
-
+    fprintf(fp, "#include \"ecmult_gen.h\"\n");
+    fprintf(fp, "#include \"precomputed_ecmult_gen.h\"\n");
     fprintf(fp, "#ifdef EXHAUSTIVE_TEST_ORDER\n");
-    fprintf(fp, "static secp256k1_ge_storage secp256k1_ecmult_gen_prec_table[ECMULT_GEN_PREC_N(ECMULT_GEN_PREC_BITS)][ECMULT_GEN_PREC_G(ECMULT_GEN_PREC_BITS)];\n");
-    fprintf(fp, "#else\n");
-    fprintf(fp, "static const secp256k1_ge_storage secp256k1_ecmult_gen_prec_table[ECMULT_GEN_PREC_N(ECMULT_GEN_PREC_BITS)][ECMULT_GEN_PREC_G(ECMULT_GEN_PREC_BITS)] = {\n");
+    fprintf(fp, "#    error Cannot compile precomputed_ecmult_gen.c in exhaustive test mode\n");
+    fprintf(fp, "#endif /* EXHAUSTIVE_TEST_ORDER */\n");
+    fprintf(fp, "#define S(a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p) SECP256K1_GE_STORAGE_CONST(0x##a##u,0x##b##u,0x##c##u,0x##d##u,0x##e##u,0x##f##u,0x##g##u,0x##h##u,0x##i##u,0x##j##u,0x##k##u,0x##l##u,0x##m##u,0x##n##u,0x##o##u,0x##p##u)\n");
+    fprintf(fp, "const secp256k1_ge_storage secp256k1_ecmult_gen_prec_table[ECMULT_GEN_PREC_N(ECMULT_GEN_PREC_BITS)][ECMULT_GEN_PREC_G(ECMULT_GEN_PREC_BITS)] = {\n");
 
     for (bits = 2; bits <= 8; bits *= 2) {
         int g = ECMULT_GEN_PREC_G(bits);
@@ -50,7 +49,7 @@ int main(int argc, char **argv) {
         int inner, outer;
 
         secp256k1_ge_storage* table = checked_malloc(&default_error_callback, n * g * sizeof(secp256k1_ge_storage));
-        secp256k1_ecmult_gen_create_prec_table(table, &secp256k1_ge_const_g, bits);
+        secp256k1_ecmult_gen_compute_table(table, &secp256k1_ge_const_g, bits);
 
         fprintf(fp, "#if ECMULT_GEN_PREC_BITS == %d\n", bits);
         for(outer = 0; outer != n; outer++) {
@@ -74,9 +73,7 @@ int main(int argc, char **argv) {
     }
 
     fprintf(fp, "};\n");
-    fprintf(fp, "#endif /* EXHAUSTIVE_TEST_ORDER */\n");
-    fprintf(fp, "#undef SC\n");
-    fprintf(fp, "#endif /* SECP256K1_ECMULT_GEN_STATIC_PREC_TABLE_H */\n");
+    fprintf(fp, "#undef S\n");
     fclose(fp);
 
     return 0;
diff --git a/src/ecmult_static_pre_g.h b/src/precomputed_ecmult.c
similarity index 99%
rename from src/ecmult_static_pre_g.h
rename to src/precomputed_ecmult.c
index 9072fb268860c9..3e67f37b747dc3 100644
--- a/src/ecmult_static_pre_g.h
+++ b/src/precomputed_ecmult.c
@@ -1,187 +1,38 @@
-/* This file was automatically generated by gen_ecmult_static_pre_g. */
+/* This file was automatically generated by precompute_ecmult. */
 /* This file contains an array secp256k1_pre_g with odd multiples of the base point G and
  * an array secp256k1_pre_g_128 with odd multiples of 2^128*G for accelerating the computation of a*P + b*G.
  */
-#ifndef SECP256K1_ECMULT_STATIC_PRE_G_H
-#define SECP256K1_ECMULT_STATIC_PRE_G_H
-#include "group.h"
-#ifdef S
-   #error macro identifier S already in use.
+#if defined HAVE_CONFIG_H
+#    include "libsecp256k1-config.h"
 #endif
+#include "../include/secp256k1.h"
+#include "group.h"
+#include "ecmult.h"
+#include "precomputed_ecmult.h"
 #define S(a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p) SECP256K1_GE_STORAGE_CONST(0x##a##u,0x##b##u,0x##c##u,0x##d##u,0x##e##u,0x##f##u,0x##g##u,0x##h##u,0x##i##u,0x##j##u,0x##k##u,0x##l##u,0x##m##u,0x##n##u,0x##o##u,0x##p##u)
-#if ECMULT_TABLE_SIZE(ECMULT_WINDOW_SIZE) > 8192
-   #error configuration mismatch, invalid ECMULT_WINDOW_SIZE. Try deleting ecmult_static_pre_g.h before the build.
+#if ECMULT_WINDOW_SIZE > 15
+   #error configuration mismatch, invalid ECMULT_WINDOW_SIZE. Try deleting precomputed_ecmult.c before the build.
 #endif
-#if defined(EXHAUSTIVE_TEST_ORDER)
-#if EXHAUSTIVE_TEST_ORDER == 13
-#define WINDOW_G 4
-static const secp256k1_ge_storage secp256k1_pre_g[ECMULT_TABLE_SIZE(WINDOW_G)] = {
- S(c3459c3d,35326167,cd86cce8,7a2417f,5b8bd567,de8538ee,d507b0c,d128f5bb,8e467fec,cd30000a,6cc1184e,25d382c2,a2f4494e,2fbe9abc,8b64abac,d005fb24)
-,S(ae64a1bd,38872f22,f637b457,125cc859,e4c7a31b,cf553cf5,b96e7096,cc61cc10,8e467fec,cd30000a,6cc1184e,25d382c2,a2f4494e,2fbe9abc,8b64abac,d005fb24)
-,S(851695d4,9a83f8ef,919bb861,53cbcb16,630fb68a,ed0a766a,3ec693d6,8e6afa40,3c915051,a5fb60b4,fec49de6,e4385101,59f30035,b6502bc0,f5edba86,9753a96c)
-,S(0,0,0,0,0,0,0,1,c36eafae,5a049f4b,13b6219,1bc7aefe,a60cffca,49afd43f,a124578,68ac52c3)
-};
-static const secp256k1_ge_storage secp256k1_pre_g_128[ECMULT_TABLE_SIZE(WINDOW_G)] = {
- S(8e55c205,92466f75,3c417ec0,e600f626,bfac877c,52258a1c,3941145a,62753693,8e467fec,cd30000a,6cc1184e,25d382c2,a2f4494e,2fbe9abc,8b64abac,d005fb24)
-,S(c3459c3d,35326167,cd86cce8,7a2417f,5b8bd567,de8538ee,d507b0c,d128f5bb,8e467fec,cd30000a,6cc1184e,25d382c2,a2f4494e,2fbe9abc,8b64abac,d005fb24)
-,S(0,0,0,0,0,0,0,1,3c915051,a5fb60b4,fec49de6,e4385101,59f30035,b6502bc0,f5edba86,9753a96c)
-,S(7ae96a2b,657c0710,6e64479e,ac3434e9,9cf04975,12f58995,c1396c28,719501ee,c36eafae,5a049f4b,13b6219,1bc7aefe,a60cffca,49afd43f,a124578,68ac52c3)
-};
-#elif EXHAUSTIVE_TEST_ORDER == 199
-#define WINDOW_G 8
-static const secp256k1_ge_storage secp256k1_pre_g[ECMULT_TABLE_SIZE(WINDOW_G)] = {
- S(226e653f,c8df7744,9bacbf12,7d1dcbf9,87f05b2a,e7edbd28,1f564575,c48dcf18,a13872c2,e933bb17,5d9ffd5b,b5b6e10c,57fe3c00,baaaa15a,e003ec3e,9c269bae)
-,S(ff1755e,623c8369,f55edda4,2a5deef0,b32c57f4,80c5884f,d2a2dde1,b1c078c4,640db9f3,9dda2f51,ee3ef3db,775315aa,c06346f1,e31ff76f,83a24bb0,8fc93242)
-,S(64dd1439,5d19a544,a7a1e81b,b9d079b3,593e7022,6bcd444e,6dc8197a,1a6dc3e6,2c7f2dce,e421d852,d3bff68e,993c8bb4,c189d3be,bd4fa667,6a599f9f,8c639c50)
-,S(199d1eb1,5a28f0aa,7258651b,ff07ff13,1c988fa,dc41dc67,390c3172,54b98016,d670e3f1,6fc2382,46bf323f,127e7c14,576e3a64,5d5c41da,40f22b29,4fca876c)
-,S(3d303d12,f8eb67d0,ecaee514,bdd90e57,b58b6d6a,c896a26f,78c06103,52225b04,d282f481,b04bece2,60de6995,b58c4f0d,9c6121e0,d94f45da,f5da7f13,cfefef99)
-,S(76bb4d25,c7a005dc,49a5295f,bb92bf1e,5d0dd5f6,609c2008,54f37361,23a6f9f3,e2295c71,21478f52,d4d18aa,72cd82ae,1995d37e,6ef2e05a,4dea6ee2,6371fbe3)
-,S(8735fb32,6950ae71,31cd03f7,cc1511d8,65c87e84,748c9824,8ccc576e,5480ed8,39cfeb15,665f371f,535d56c1,317a8d62,29ea2827,943a98f1,cee7f685,86e47eec)
-,S(8521a020,a8e7cc5e,d9534d74,d4656581,33869bc,1940548d,6cc35ba4,e3c4323a,c63014ea,99a0c8e0,aca2a93e,ce85729d,d615d7d8,6bc5670e,31180979,791b7d43)
-,S(2f63a396,4d14ca73,2167e033,306863d8,f75605c7,4ea6c917,890dd50d,fc20f53b,e2295c71,21478f52,d4d18aa,72cd82ae,1995d37e,6ef2e05a,4dea6ee2,6371fbe3)
-,S(c479fee8,e2202b42,1c9346fb,4e59f720,95a70dea,2ade9641,58bad01c,2c37f,6897db9d,5a3d8948,ea97069e,de75661,b1ad74e4,2b1daa4f,533d88ba,67137731)
-,S(4d756b2,e61d9806,e02ec830,f1fd44f8,46438689,c528346,15a1addf,ac864bcd,c90704c9,8f72d8be,638a36dc,e8768909,fac7921c,dc54d314,89fbb3e,6da9018a)
-,S(baeab1a8,bca6e88,740deeb5,93887afe,295155ba,fa50f307,baf2de95,2841dca6,a88e0162,de086e20,587faa5c,bca8f572,318c17e9,5668819c,13c4d5d9,2d1c75dc)
-,S(47bbafcd,f612ca3e,703f7721,623bde14,926fc00d,a6bf9e9,41ddb2d5,587626e3,d160b834,5807531a,327a7269,3627dab6,f9770a12,14eb182d,fa73ad15,41b57ff0)
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-,S(eb5c4120,4917d176,6fea883b,e680dfd9,233d9654,97fac48b,76f5d0d4,7f6dbf73,97682462,a5c276b7,1568f961,f218a99e,4e528b1b,d4e255b0,acc27744,98ec84fe)
-,S(4097892c,5478baea,62e8da20,6d17fcd8,b8eed45f,24fb7648,fe742508,8da60a4e,10d4b884,5636de20,a009a491,83906957,9d88bc00,5ac1c7a7,3262898b,8bca16b)
-,S(9963e528,754cbba8,29622664,1aec11b0,3fba2739,40f61cbd,5d4c0478,5429f40,d670e3f1,6fc2382,46bf323f,127e7c14,576e3a64,5d5c41da,40f22b29,4fca876c)
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-,S(851695d4,9a83f8ef,919bb861,53cbcb16,630fb68a,ed0a766a,3ec693d6,8e6afa40,cb47a6a4,91313828,f036dbd7,64173a40,473d3969,5cc47758,511d0a81,badb02f1)
-,S(f31a7dfe,4d2e4a98,aaf3dd70,25441c7a,2da056b0,db6795e8,3a39d76f,b316bd92,17e98bbd,6e144db,959b7aea,cd766958,7be26db5,445ea19d,dab8bc90,1f527cb4)
-,S(2266b5e,d5cef55,81e36315,c5fc5c53,a1c73b12,c24c8de7,9afef2a0,7a50212c,7a60fc27,e4ab0ba6,ea71c036,52b83c5c,5ab5dac8,ea43dec2,84719f8c,6d7c3c81)
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-,S(896c7bc,5cfeaecc,37b111ea,2deee270,c7cb24eb,6eb1cb80,a84e87d3,89d997c1,66446d0d,b01a679b,59afb34a,2bbf8cfe,f0a44870,f0e074ce,78a51a36,9bbbf33c)
-,S(709c8403,ed2fd0f9,5fac44f0,dfa96ec2,e7b357e4,4ebb3dcf,6f688f20,39d3d67d,d160b834,5807531a,327a7269,3627dab6,f9770a12,14eb182d,fa73ad15,41b57ff0)
-,S(142f854d,ba0b87a2,9826f018,86ab51ff,b9f19214,c23c07c0,f008a375,a1effdb6,74bafd32,bdf2bdfd,c65a52c3,5ef4f4d,9332f7b2,a0512a9f,f821f858,484bb9c)
-,S(66abbf18,f58dd1c9,4b6072c,eb182abe,c31d3af9,6d91060b,d233f462,5716f289,d380d231,1bde27ad,2c400971,66c3744b,3e762c41,42b05998,95a6605f,739c5fdf)
-,S(7a1b727f,c05c6de2,79b2e027,108fae3e,87ca9249,eb44207f,286e1cbe,4728027,9de02272,226d60c0,97e5907c,407f9e07,d4bc6959,a2079419,139a9578,4e48cb7f)
-,S(3d98fe85,5d42f661,e35d8ce3,cfee5f17,57bc3de9,ecec02fa,11a2d90,378ac0e8,bdc6dc52,8a55814e,da6bcb58,e2a73b96,9d975c7,a3270483,b419b29b,5263f45)
-,S(34f8450b,18bffd3f,cbadcd08,5da5411b,ba2ef195,55753768,bf777307,3a2d4cb4,5771fe9d,21f791df,a78055a3,43570a8d,ce73e816,a9977e63,ec3b2a25,d2e38653)
-,S(76dac07e,bc240289,791e6a9a,56abdc7,bdf2fac1,e7a0100b,5a0ee09e,3455cc33,8b4502cd,420d4202,39a5ad3c,fa10b0b2,6ccd084d,5faed560,7de07a6,fb7b4093)
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-,S(2cdce338,ae1f5aa0,55c76cb5,acbd084e,3284bb5d,b8cf9b4a,141cc8ee,1aa61e59,17e98bbd,6e144db,959b7aea,cd766958,7be26db5,445ea19d,dab8bc90,1f527cb4)
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-,S(b084fee5,5da62ee0,ebdebf3b,245e0d4e,def46aac,bfdc9f43,261d3e2b,eed946a0,80f3b79c,ba5f8457,3f2aa473,1d10c1f,9b6543e1,b8cec5e0,e89cd6df,868cc2aa)
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-};
-static const secp256k1_ge_storage secp256k1_pre_g_128[ECMULT_TABLE_SIZE(WINDOW_G)] = {
- S(ec823227,1d0e875a,8e1cf7e6,f9a35a15,749be650,1cc885c4,f383060b,46bd3d33,aa140b83,97ca0dcd,a15dcd4c,7509bb87,9a34231e,d0d3a586,543bba01,ae7545b8)
-,S(66abbf18,f58dd1c9,4b6072c,eb182abe,c31d3af9,6d91060b,d233f462,5716f289,d380d231,1bde27ad,2c400971,66c3744b,3e762c41,42b05998,95a6605f,739c5fdf)
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-,S(875c93a1,8ac5a86e,9e85e4aa,3c5ad0ac,3d697192,6384b05b,568604b3,8fee8fb6,c8c7236a,68522b1a,3f5c650e,4673345d,844bcccc,1ca54b7e,ab3ad885,6e6ee7e4)
-,S(f3a864ac,edc7852f,f4dfae93,5f8588a6,96ff17bf,7233134e,6704ceb,16f3b74c,39cfeb15,665f371f,535d56c1,317a8d62,29ea2827,943a98f1,cee7f685,86e47eec)
-,S(3e8a6608,17f28243,f363d67e,b874856f,1fc9810a,9da6c3f3,a72dd23f,ec45513e,8a8d4c8f,37d2572,6ca66369,59c9eca3,82bd076e,867092a0,b428ad99,2c54a63b)
-,S(47bbafcd,f612ca3e,703f7721,623bde14,926fc00d,a6bf9e9,41ddb2d5,587626e3,d160b834,5807531a,327a7269,3627dab6,f9770a12,14eb182d,fa73ad15,41b57ff0)
-,S(709c8403,ed2fd0f9,5fac44f0,dfa96ec2,e7b357e4,4ebb3dcf,6f688f20,39d3d67d,2e9f47cb,a7f8ace5,cd858d96,c9d82549,688f5ed,eb14e7d2,58c52e9,be4a7c3f)
-,S(66fd0ada,7159c2df,fdd925e7,2eb70772,743e39e3,4c10d5cf,f58eac4e,a30e4157,8a8d4c8f,37d2572,6ca66369,59c9eca3,82bd076e,867092a0,b428ad99,2c54a63b)
-,S(60d5d771,4e1e7589,e0b1e68c,ed7f881,73fca809,eae35685,65334942,79962dc2,99bb92f2,4fe59864,a6504cb5,d4407301,f5bb78f,f1f8b31,875ae5c8,644408f3)
-,S(3d98fe85,5d42f661,e35d8ce3,cfee5f17,57bc3de9,ecec02fa,11a2d90,378ac0e8,bdc6dc52,8a55814e,da6bcb58,e2a73b96,9d975c7,a3270483,b419b29b,5263f45)
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-,S(bcea96e9,4ff1b11e,6e8a09d6,ec594b00,597e2b0a,37b0e581,50a58d59,33220419,80f3b79c,ba5f8457,3f2aa473,1d10c1f,9b6543e1,b8cec5e0,e89cd6df,868cc2aa)
-,S(ae3796a4,823f3eb4,ea4bd677,110dc3fb,45537c3c,4d10d2e8,e758a3be,4875db83,ef2b477b,a9c921df,5ff65b6e,7c6f96a8,627743ff,a53e3858,cd9d7673,f7435ac4)
-,S(5245e215,52c6f785,2cccef32,e9f54774,cfb6bb9b,c363ae72,71174df4,6d77fdcc,ca78f1f0,894a6670,19609169,78c251dc,64f969e,c25be4b4,202508d3,22d07f3b)
-,S(26e2e75b,aeb04a72,8b893f67,d27ebfc6,b8984a88,6fbd8eee,5719107d,5a413f0f,61b51308,ac8643f0,7c18eda7,ef9a4196,aa66221,1b809acc,2b2cb538,23010ef8)
-,S(e0089ec9,4b25ac6,ff44b5da,2dfedb37,9fdaedf1,6bc8cecd,b1a95fa0,32431c73,e8167442,f91ebb24,6a648515,328996a7,841d924a,bba15e62,2547436e,e0ad7f7b)
-,S(f8f16642,266d8dc6,abfdbbc6,f6c47711,a192c051,7f00a633,b78076a7,a7884e72,4f6d0e5,148ce7e1,772f84c,b08903b8,71d2beb5,d0934488,8ab1c75d,42232790)
-,S(23082df9,a86b80fc,5185ee3c,6493763b,14a6e237,baf686aa,f589b649,8573d04c,bdc6dc52,8a55814e,da6bcb58,e2a73b96,9d975c7,a3270483,b419b29b,5263f45)
-,S(76dac07e,bc240289,791e6a9a,56abdc7,bdf2fac1,e7a0100b,5a0ee09e,3455cc33,8b4502cd,420d4202,39a5ad3c,fa10b0b2,6ccd084d,5faed560,7de07a6,fb7b4093)
-,S(8735fb32,6950ae71,31cd03f7,cc1511d8,65c87e84,748c9824,8ccc576e,5480ed8,c63014ea,99a0c8e0,aca2a93e,ce85729d,d615d7d8,6bc5670e,31180979,791b7d43)
-,S(969360d2,54e2dba9,e79d0789,c339250d,c438330a,a66addf9,f27e2ee8,fc9036ac,66446d0d,b01a679b,59afb34a,2bbf8cfe,f0a44870,f0e074ce,78a51a36,9bbbf33c)
-,S(74f5ba33,89d075d3,eebaa54d,73e9f038,881b7329,5623e833,b5e87beb,29ba3246,74bafd32,bdf2bdfd,c65a52c3,5ef4f4d,9332f7b2,a0512a9f,f821f858,484bb9c)
-,S(a4465b18,b7f702e2,a8165184,834ce490,68fb0a51,6658734b,6229eda0,605911a2,c8c7236a,68522b1a,3f5c650e,4673345d,844bcccc,1ca54b7e,ab3ad885,6e6ee7e4)
-,S(eac612b0,84cb452a,35d49c60,c0d0c0bd,359eb909,68a80186,92a028f,ece4c331,23575727,9da3d12b,c0004d12,e06a9cda,8405c91e,b8118260,2793e62f,16c2a29e)
-,S(f31a7dfe,4d2e4a98,aaf3dd70,25441c7a,2da056b0,db6795e8,3a39d76f,b316bd92,e8167442,f91ebb24,6a648515,328996a7,841d924a,bba15e62,2547436e,e0ad7f7b)
-,S(470ce05d,53f77ff0,c5ab8770,a4c11a3e,23f7d4fe,f5c0ab36,1b940a8a,6b8cf7b9,57d6adb1,5f0b1daa,1fce91d7,ccf75d45,9480b056,6f81ff8a,85612173,1cd8f2b8)
-,S(d9fbce92,515652cf,3714f87b,e16e505,91a28eb4,1bf7053,2ab42ebd,be900212,2d7d0b7e,4fb4131d,9f21966a,4a73b0f2,639ede1f,26b0ba25,a2580eb,30100c96)
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-,S(7fe309bb,e3e8edd3,3f5f94b,4247256e,41e62e69,fd9f2063,d878c6d6,3d85675f,dca8a8d8,625c2ed4,3fffb2ed,1f956325,7bfa36e1,47ee7d9f,d86c19cf,e93d5991)
-,S(64eaaad1,9c7e9d01,5cc997ff,c2f23022,c59805d,48a48e1,970b8847,10211659,fb092f1a,eb73181e,f88d07b3,4f76fc47,8e2d414a,2f6cbb77,754e38a1,bddcd49f)
-,S(59e10f43,eb4b2fb0,94f2f66d,1404dd08,ab9c2442,50bd16e0,21feb78f,e0380d01,e2295c71,21478f52,d4d18aa,72cd82ae,1995d37e,6ef2e05a,4dea6ee2,6371fbe3)
-,S(c979de1c,1318f60f,a9929ff3,a6621b9,d15226e8,2883556e,36a5f77b,28e3b585,d7bd974a,b40a1bd3,a35a1ee9,16d32768,48e7bcfc,be70851d,8ba758b9,996d2099)
-,S(64dd1439,5d19a544,a7a1e81b,b9d079b3,593e7022,6bcd444e,6dc8197a,1a6dc3e6,d380d231,1bde27ad,2c400971,66c3744b,3e762c41,42b05998,95a6605f,739c5fdf)
-,S(d8a3fd95,b7ed8e8a,674da307,94911393,303917c2,60944d1f,76261274,fba45757,a13872c2,e933bb17,5d9ffd5b,b5b6e10c,57fe3c00,baaaa15a,e003ec3e,9c269bae)
-,S(acee4285,57fc32a1,d3d14199,30cadaf0,613a630a,7a7e8c14,d08d05df,19143c3a,ca78f1f0,894a6670,19609169,78c251dc,64f969e,c25be4b4,202508d3,22d07f3b)
-,S(5d771e5d,6dc6c87,5ede8bae,4b27a9d4,3c5f8da2,8e84f5c3,501299c8,db16484c,859f03d8,1b54f459,158e3fc9,ad47c3a3,a54a2537,15bc213d,7b8e6072,9283bfae)
-,S(e8d3f45a,b5be455f,dc3c2270,34100ca2,b8d203e1,35a9ed3d,5c8b703c,ef4d9b48,2d7d0b7e,4fb4131d,9f21966a,4a73b0f2,639ede1f,26b0ba25,a2580eb,30100c96)
-,S(9963e528,754cbba8,29622664,1aec11b0,3fba2739,40f61cbd,5d4c0478,5429f40,298f1c0e,f903dc7d,b940cdc0,ed8183eb,a891c59b,a2a3be25,bf0dd4d5,b03574c3)
-,S(cbdb65,553cd5d8,ff61cf33,e53fdd9a,cf0ee159,c21dc578,be5bac2b,7973c229,ca78f1f0,894a6670,19609169,78c251dc,64f969e,c25be4b4,202508d3,22d07f3b)
-,S(5029bff6,d4c80347,738230c8,cb252906,471b5bc1,3d26a533,304f5f0d,808f756c,97682462,a5c276b7,1568f961,f218a99e,4e528b1b,d4e255b0,acc27744,98ec84fe)
-,S(80c5672b,3f773975,b5829101,6fa9de9b,33863263,8a978db,934f04c1,3a909b85,55ebf47c,6835f232,5ea232b3,8af64478,65cbdce1,2f2c5a79,abc445fd,518ab677)
-,S(226e653f,c8df7744,9bacbf12,7d1dcbf9,87f05b2a,e7edbd28,1f564575,c48dcf18,5ec78d3d,16cc44e8,a26002a4,4a491ef3,a801c3ff,45555ea5,1ffc13c0,63d96081)
-,S(521e20fc,9c7c0514,47f31e74,5bb81662,dac66374,9b891a6f,d9681cb6,21e3155c,9e4aecf7,5379bc0f,83e71258,1065be69,f5599dde,e47f6533,d4d34ac6,dcfeed37)
-,S(5472b5d0,4ba1ca80,236183b1,46e596e5,4b20b83b,248988c8,bbe8460e,3666f262,a829524e,a0f4e255,e0316e28,3308a2ba,6b7f4fa9,907e0075,7a9ede8b,e3270977)
-,S(2ecd421e,47399e76,60d10143,1789a437,89b54d23,31d1cb78,49cabda2,bc5174f9,c8268dfc,2da263eb,ac7d81d8,6b8b504d,6d6bc20f,745eef75,bc9bb378,1193744b)
-,S(4097892c,5478baea,62e8da20,6d17fcd8,b8eed45f,24fb7648,fe742508,8da60a4e,10d4b884,5636de20,a009a491,83906957,9d88bc00,5ac1c7a7,3262898b,8bca16b)
-,S(eb5c4120,4917d176,6fea883b,e680dfd9,233d9654,97fac48b,76f5d0d4,7f6dbf73,6897db9d,5a3d8948,ea97069e,de75661,b1ad74e4,2b1daa4f,533d88ba,67137731)
-,S(92b866ad,a37a3f2f,bc607717,96b2c74f,57dde74c,da8e015f,792df531,7eb21fa6,55ebf47c,6835f232,5ea232b3,8af64478,65cbdce1,2f2c5a79,abc445fd,518ab677)
-,S(a78dab,f2f2ef7f,4d424752,aa1aeaf5,50bec241,bf9ad129,3b9fe680,32b6141b,9de02272,226d60c0,97e5907c,407f9e07,d4bc6959,a2079419,139a9578,4e48cb7f)
-,S(b084fee5,5da62ee0,ebdebf3b,245e0d4e,def46aac,bfdc9f43,261d3e2b,eed946a0,80f3b79c,ba5f8457,3f2aa473,1d10c1f,9b6543e1,b8cec5e0,e89cd6df,868cc2aa)
-,S(ff1755e,623c8369,f55edda4,2a5deef0,b32c57f4,80c5884f,d2a2dde1,b1c078c4,640db9f3,9dda2f51,ee3ef3db,775315aa,c06346f1,e31ff76f,83a24bb0,8fc93242)
-,S(2cdce338,ae1f5aa0,55c76cb5,acbd084e,3284bb5d,b8cf9b4a,141cc8ee,1aa61e59,e8167442,f91ebb24,6a648515,328996a7,841d924a,bba15e62,2547436e,e0ad7f7b)
-,S(3faf6744,2aebb675,c6b75911,e3d34f4c,a8f5b669,a4c438f2,2c163659,3e36c1ea,fc0b579e,cbaf2c74,7b5c792d,7b66abf2,dfb1a44d,2ed6a417,7e1fc6bd,a989b623)
-,S(34772cad,ad5888f2,53a810b7,5b175b8d,e3a454e4,26a1b5a5,c003f222,8e7b45c0,d380d231,1bde27ad,2c400971,66c3744b,3e762c41,42b05998,95a6605f,739c5fdf)
-,S(853cffd4,4cb0a29e,390ad886,455566cc,2776ab74,55210e57,9bf1fcc0,c8d767ed,621fdd8d,dd929f3f,681a6f83,bf8061f8,2b4396a6,5df86be6,ec656a86,b1b730b0)
-,S(1130e02f,29480660,b2cb4f68,81da3f2c,1bdaf64,8df3b6ce,1a333738,29e4165e,10d4b884,5636de20,a009a491,83906957,9d88bc00,5ac1c7a7,3262898b,8bca16b)
-,S(a0627644,ebc6a423,1f3e113b,eedbf9d8,21d9374a,af2e7c55,14ee7395,aa9992b7,7a60fc27,e4ab0ba6,ea71c036,52b83c5c,5ab5dac8,ea43dec2,84719f8c,6d7c3c81)
-,S(1e864d74,9e96a16a,c4299b88,226aaff9,9a45ab9c,203853ac,ea0378ef,5715ded6,6120ecff,80d4804d,6dcfe173,882249c1,a0d2dc3b,ceaee73e,74e49d8b,79445b15)
-};
-#else
-   #error No known generator for the specified exhaustive test group order.
-#endif
-#else /* !defined(EXHAUSTIVE_TEST_ORDER) */
+#ifdef EXHAUSTIVE_TEST_ORDER
+#    error Cannot compile precomputed_ecmult.c in exhaustive test mode
+#endif /* EXHAUSTIVE_TEST_ORDER */
 #define WINDOW_G ECMULT_WINDOW_SIZE
-static const secp256k1_ge_storage secp256k1_pre_g[ECMULT_TABLE_SIZE(WINDOW_G)] = {
+const secp256k1_ge_storage secp256k1_pre_g[ECMULT_TABLE_SIZE(WINDOW_G)] = {
  S(79be667e,f9dcbbac,55a06295,ce870b07,29bfcdb,2dce28d9,59f2815b,16f81798,483ada77,26a3c465,5da4fbfc,e1108a8,fd17b448,a6855419,9c47d08f,fb10d4b8)
-#if ECMULT_TABLE_SIZE(WINDOW_G) > 1
+#if WINDOW_G > 2
 ,S(f9308a01,9258c310,49344f85,f89d5229,b531c845,836f99b0,8601f113,bce036f9,388f7b0f,632de814,fe337e6,2a37f356,6500a999,34c2231b,6cb9fd75,84b8e672)
 #endif
-#if ECMULT_TABLE_SIZE(WINDOW_G) > 2
+#if WINDOW_G > 3
 ,S(2f8bde4d,1a072093,55b4a725,a5c5128,e88b84bd,dc619ab7,cba8d569,b240efe4,d8ac2226,36e5e3d6,d4dba9dd,a6c9c426,f788271b,ab0d6840,dca87d3a,a6ac62d6)
 ,S(5cbdf064,6e5db4ea,a398f365,f2ea7a0e,3d419b7e,330e39c,e92bdded,cac4f9bc,6aebca40,ba255960,a3178d6d,861a54db,a813d0b8,13fde7b5,a5082628,87264da)
 #endif
-#if ECMULT_TABLE_SIZE(WINDOW_G) > 4
+#if WINDOW_G > 4
 ,S(acd484e2,f0c7f653,9ad178a,9f559abd,e0979697,4c57e714,c35f110d,fc27ccbe,cc338921,b0a7d9fd,64380971,763b61e9,add888a4,375f8e0f,5cc262a,c64f9c37)
 ,S(774ae7f8,58a9411e,5ef4246b,70c65aac,5649980b,e5c17891,bbec1789,5da008cb,d984a032,eb6b5e19,243dd56,d7b7b365,372db1e2,dff9d6a8,301d74c9,c953c61b)
 ,S(f28773c2,d975288b,c7d1d205,c3748651,b075fbc6,610e58cd,deeddf8f,19405aa8,ab0902e,8d880a89,758212eb,65cdaf47,3a1a06da,521fa91f,29b5cb52,db03ed81)
 ,S(d7924d4f,7d43ea96,5a465ae3,95ff411,31e5946f,3c85f79e,44adbcf8,e27e080e,581e2872,a86c72a6,83842ec2,28cc6def,ea40af2b,d896d3a5,c504dc9f,f6a26b58)
 #endif
-#if ECMULT_TABLE_SIZE(WINDOW_G) > 8
+#if WINDOW_G > 5
 ,S(defdea4c,db677750,a420fee8,7eacf21,eb9898ae,79b97687,66e4faa0,4a2d4a34,4211ab06,94635168,e997b0ea,d2a93dae,ced1f4a0,4a95c0f6,cfb199f6,9e56eb77)
 ,S(2b4ea0a7,97a443d2,93ef5cff,444f4979,f06acfeb,d7e86d27,74756561,38385b6c,85e89bc0,37945d93,b343083b,5a1c8613,1a01f60c,50269763,b570c854,e5c09b7a)
 ,S(352bbf4a,4cdd1256,4f93fa33,2ce33330,1d9ad402,71f81071,81340aef,25be59d5,321eb407,5348f534,d59c1825,9dda3e1f,4a1b3b2e,71b1039c,67bd3d8b,cf81998c)
@@ -191,7 +42,7 @@ static const secp256k1_ge_storage secp256k1_pre_g[ECMULT_TABLE_SIZE(WINDOW_G)] =
 ,S(c44d12c7,65d812e,8acf28d7,cbb19f90,11ecd9e9,fdf281b0,e6a3b5e8,7d22e7db,2119a460,ce326cdc,76c45926,c982fdac,e106e86,1edf61c5,a039063f,e0e6482)
 ,S(6a245bf6,dc698504,c89a20cf,ded60853,152b6953,36c28063,b61c65cb,d269e6b4,e022cf42,c2bd4a70,8b3f5126,f16a24ad,8b33ba48,d0423b6e,fd5e6348,100d8a82)
 #endif
-#if ECMULT_TABLE_SIZE(WINDOW_G) > 16
+#if WINDOW_G > 6
 ,S(1697ffa6,fd9de627,c077e3d2,fe541084,ce13300b,bec1146,f95ae57f,d0bd6a5,b9c398f1,86806f5d,27561506,e4557433,a2cf1500,9e498ae7,adee9d63,d01b2396)
 ,S(605bdb01,9981718b,986d0f07,e834cb0d,9deb8360,ffb7f61d,f982345e,f27a7479,2972d2d,e4f8d206,81a78d93,ec96fe23,c26bfae8,4fb14db4,3b01e1e9,56b8c49)
 ,S(62d14dab,4150bf49,7402fdc4,5a215e10,dcb01c35,4959b10c,fe31c7e9,d87ff33d,80fc06bd,8cc5b010,98088a19,50eed0db,1aa1329,67ab4722,35f56424,83b25eaf)
@@ -209,7 +60,7 @@ static const secp256k1_ge_storage secp256k1_pre_g[ECMULT_TABLE_SIZE(WINDOW_G)] =
 ,S(754e3239,f325570c,dbbf4a87,deee8a66,b7f2b334,79d468fb,c1a50743,bf56cc18,673fb86,e5bda30f,b3cd0ed3,4ea49a0,23ee33d0,197a695d,c5d9809,3c536683)
 ,S(e3e6bd10,71a1e96a,ff57859c,82d570f0,33080066,1d1c952f,9fe26946,91d9b9e8,59c9e0bb,a394e76f,40c0aa58,379a3cb6,a5a22839,93e90c41,67002af4,920e37f5)
 #endif
-#if ECMULT_TABLE_SIZE(WINDOW_G) > 32
+#if WINDOW_G > 7
 ,S(186b483d,56a0338,26ae73d8,8f732985,c4ccb1f3,2ba35f4b,4cc47fdc,f04aa6eb,3b952d32,c67cf77e,2e17446e,204180ab,21fb8090,895138b4,a4a797f8,6e80888b)
 ,S(df9d70a6,b9876ce5,44c98561,f4be4f72,5442e6d2,b737d9c9,1a832172,4ce0963f,55eb2daf,d84d6ccd,5f862b78,5dc39d4a,b1572227,20ef9da2,17b8c45c,f2ba2417)
 ,S(5edd5cc2,3c51e87a,497ca815,d5dce0f8,ab52554f,849ed899,5de64c5f,34ce7143,efae9c8d,bc141306,61e8cec0,30c89ad0,c13c66c0,d17a2905,cdc706ab,7399a868)
@@ -243,7 +94,7 @@ static const secp256k1_ge_storage secp256k1_pre_g[ECMULT_TABLE_SIZE(WINDOW_G)] =
 ,S(c4191636,5abb2b5d,9192f5f,2dbeafec,208f020f,12570a18,4dbadc3e,58595997,4f14351,d0087efa,49d245b3,28984989,d5caf945,f34bfc0,ed16e96b,58fa9913)
 ,S(841d6063,a586fa47,5a724604,da03bc5b,92a2e0d2,e0a36acf,e4c73a55,14742881,73867f5,9c0659e8,1904f9a1,c7543698,e62562d6,744c169c,e7a36de0,1a8d6154)
 #endif
-#if ECMULT_TABLE_SIZE(WINDOW_G) > 64
+#if WINDOW_G > 8
 ,S(5e95bb39,9a6971d3,76026947,f89bde2f,282b3381,928be4d,ed112ac4,d70e20d5,39f23f36,6809085b,eebfc711,81313775,a99c9aed,7d8ba38b,161384c7,46012865)
 ,S(36e4641a,53948fd4,76c39f8a,99fd974e,5ec07564,b5315d8b,f99471bc,a0ef2f66,d2424b1b,1abe4eb8,164227b0,85c9aa94,56ea1349,3fd563e0,6fd51cf5,694c78fc)
 ,S(336581e,a7bfbbb2,90c191a2,f507a41c,f5643842,170e914f,aeab27c2,c579f726,ead12168,595fe1be,99252129,b6e56b33,91f7ab14,10cd1e0e,f3dcdcab,d2fda224)
@@ -309,7 +160,7 @@ static const secp256k1_ge_storage secp256k1_pre_g[ECMULT_TABLE_SIZE(WINDOW_G)] =
 ,S(809a20c6,7d64900f,fb698c4c,825f6d5f,2310fb04,51c86934,5b7319f6,45605721,9e994980,d9917e22,b76b0619,27fa0414,3d096ccc,54963e6a,5ebfa5f3,f8e286c1)
 ,S(1b38903a,43f7f114,ed4500b4,eac7083f,defece1c,f29c6352,8d563446,f972c180,4036edc9,31a60ae8,89353f77,fd53de4a,2708b26b,6f5da72a,d3394119,daf408f9)
 #endif
-#if ECMULT_TABLE_SIZE(WINDOW_G) > 128
+#if WINDOW_G > 9
 ,S(90a80db6,eb294b9e,ab0b4e8d,dfa3efe7,263458ce,2d07566d,f4e6c588,68feef23,753c8b9f,9754f18d,87f21145,d9e2936b,5ee050b2,7bbd9681,442c76e9,2fcf91e6)
 ,S(c2c80f84,4b705998,12d62546,f60340e,3e6f3605,4a14546e,6dc25d47,376bea9b,86ca160d,68f4d4e7,18b495b8,91d3b1b5,73b871a7,2b4cf61,23abd448,3aa79c64)
 ,S(9cf60674,4cf4b5f3,fdf989d3,f19fb265,2d00cfe1,d5fcd692,a323ce11,a28e7553,8147cbf7,b973fcc1,5b57b6a3,cfad6863,edd0f30e,3c45b85d,c300c513,c247759d)
@@ -439,7 +290,7 @@ static const secp256k1_ge_storage secp256k1_pre_g[ECMULT_TABLE_SIZE(WINDOW_G)] =
 ,S(367807c9,a3606b4e,1b8c2616,ad528030,1dfcf686,40eddf02,fc59317c,230e9a86,1f023f2f,a2bbece7,3dba14c,124095cb,fdc4f92f,281a14,8304a412,c16ecae6)
 ,S(8ec4fdc3,9891f6af,1374e06f,c44b815,1b82541,75fc4909,acba5941,201af62b,2dc6cae5,cac2d887,83dca0e5,3c798f8,fe067bcf,5fc29751,13756cf7,ef4e5f1b)
 #endif
-#if ECMULT_TABLE_SIZE(WINDOW_G) > 256
+#if WINDOW_G > 10
 ,S(5cc24a6d,4c5b24f9,14542f91,e5fa937f,ffa08551,51b8b842,8729b06a,9178a263,b1da8635,81531a1f,bfb38a4e,419fa1fe,ca8d55a8,3ddbcb98,da19d5cf,fb7da472)
 ,S(83905926,c03905c3,a9644a6c,da810dd2,92602a50,50c52a21,9134fc4d,e3599e9f,4293260f,e8af6792,a20b115e,aa837638,9094298b,21d9de16,cf20e0c5,7a46089a)
 ,S(944b097e,4721e9dd,f8204ac3,d3878fa,e8fa6c14,34ae4822,481b2985,6589b6c7,5fc47565,30e9b095,f8b79643,1e745b99,1525bd4c,4764e8e,e8af4b96,9bd6ddf6)
@@ -697,7 +548,7 @@ static const secp256k1_ge_storage secp256k1_pre_g[ECMULT_TABLE_SIZE(WINDOW_G)] =
 ,S(d9a0c689,95283291,972d6a72,897181b2,6b4ae317,4e98a676,f75bbfbf,81be876e,d36ae886,a0acbc9a,d1564d97,4d3f0309,e9039b93,bb350d92,2b7f8c7a,c6bfa042)
 ,S(c7a36324,6aeb7c8c,991b2aa7,10abdf5c,fff29912,30b3a69f,be2dd481,7c7c3e0a,1298fdd7,e448d2d,79986302,6b4b2d49,2b32148,d6d1e5f8,15f5b0c0,5c9e21f)
 #endif
-#if ECMULT_TABLE_SIZE(WINDOW_G) > 512
+#if WINDOW_G > 11
 ,S(635cd7a0,5064d3bc,66535a0,4dbf563a,640d2464,c7fe0ac4,8304214f,e4985a86,e40265,913e77f6,46735cfc,af9e2f30,e8d5f047,f3281a4c,e0453e27,e9e3ae1f)
 ,S(5da624f,38edea25,37f5b632,695b481a,eca54bb7,2169cadc,c5b91e10,989ee5f5,d3ea6dba,a1080300,fffaeeb2,43a5256e,48c28822,37b16505,a220d771,ca721779)
 ,S(aa4f64a2,b19775ec,92c1f687,1fe4b6d5,ce05278c,2976c80e,fe19284f,e87d4d5e,f39f117e,95fce0fb,6976e949,9583a66c,224ea028,8396518e,293793dc,3ed4f240)
@@ -1211,7 +1062,7 @@ static const secp256k1_ge_storage secp256k1_pre_g[ECMULT_TABLE_SIZE(WINDOW_G)] =
 ,S(dd879eba,f870ee3d,f6e3f03,60833e12,fbdf844c,d6a5b76c,19802485,6649bbc1,e7bb04c3,a99a5c11,dd7a324c,1d5696c8,b041a12a,c6a538f7,3094716c,9943c55a)
 ,S(7aa4afbe,29b06a1e,da9319d5,72572b13,1df68f74,1b5f8d8e,d00ccc04,80f8016e,15f79a9f,b77b8587,7f02e3c1,3202b972,423fd00d,937c0d2d,c9d94b17,53fe7130)
 #endif
-#if ECMULT_TABLE_SIZE(WINDOW_G) > 1024
+#if WINDOW_G > 12
 ,S(9145f3f5,876a3265,5d7fee64,f2d6e660,c34354e9,1571d68b,a19e4cc5,8c39f890,45e0a95d,cf369fd0,5bc9ba7f,da108d6f,37650c1d,5766b6c9,98ecda28,b285d8ff)
 ,S(5450b752,36fb010d,1ef37afa,2077f3dc,a5a7f6c8,91a21317,13df740f,4511ac9e,a7c7ed57,62bef86d,4de1084d,3ded2d7b,13ff563a,67109ef0,8b6f0180,fe6ba175)
 ,S(ba6c0d72,5e48de2c,cb8256d6,7417d075,bbf2766f,d13501b8,4c32cf88,c0666a0f,cd2a9132,7137299f,50eda669,3a599032,bdecd64b,91bcc640,5ed186f3,e525e442)
@@ -2237,7 +2088,7 @@ static const secp256k1_ge_storage secp256k1_pre_g[ECMULT_TABLE_SIZE(WINDOW_G)] =
 ,S(6d109d8c,2505808f,7b4052d6,b170ec80,c68373bb,e02f2b62,1cd29773,a96acb3e,c8dea0de,511f6e40,5141088e,cb023bff,200c870b,9cb952ad,c5038b28,eadc9a57)
 ,S(9c219898,6d202467,c055c734,73557505,6105b3e,2874dbf2,8f7f0c39,66e1af88,8637496c,8eddff12,f9dd4146,36565e0f,efddf5cd,52d48455,ec9fb2d,8dd0914a)
 #endif
-#if ECMULT_TABLE_SIZE(WINDOW_G) > 2048
+#if WINDOW_G > 13
 ,S(22ca5039,e660f60,1036dd75,2bb973b,dcd104b5,dcb8f2e7,4de8edc6,c292b03a,86fd4471,1ef6b81e,4416449a,708fa0c9,cb6731ba,c403e58e,da5e915f,646b11e8)
 ,S(cfc18f02,cc004640,f2116fdd,1f6ca202,2e39be25,df75e27c,80bde842,e6b6f938,d62b58df,4f79d0e5,97ba2923,f708cbe5,c09236a4,d9d01398,6451d684,6290df7d)
 ,S(3388bcc2,3425458e,991f46ca,6235c50a,57490556,becbaf25,9d3c14f9,9a80a087,7d4aa2f8,6f65783,1c22316,6958fbf3,6c3dca5,d4ac413b,3102e13e,a645c016)
@@ -4287,7 +4138,7 @@ static const secp256k1_ge_storage secp256k1_pre_g[ECMULT_TABLE_SIZE(WINDOW_G)] =
 ,S(d5acff71,e82a455c,3a1ce292,689e8686,f441ce1b,e644e79a,bd6d0efe,29270865,aac6d48f,1b46e970,a044971a,ad13f033,ca8cde96,958d870e,dc7d80,1d26d5e8)
 ,S(5cf51c1b,2210d85,9d765e17,32109514,8f03fc57,51004b6a,91f098e2,e2711596,1eeb19e0,610df459,2c31e58e,aa2a2148,17fe9ee,a3995838,f395bdcf,26d5c3b3)
 #endif
-#if ECMULT_TABLE_SIZE(WINDOW_G) > 4096
+#if WINDOW_G > 14
 ,S(21456873,b58e3687,52c75800,8d3bcae,9efaf1f5,c5727842,25e3d854,8fd421cb,a2f2d10,c85e8a0f,4a136ad0,5df991ce,7d3c5585,a263d5cf,da50a4f4,3db76cb0)
 ,S(c10de5c0,51ae2d73,28ac06ca,cca840b4,ed7ab204,21c6122f,1d68fe7f,7893d38d,ee30e086,a891484,2ad4041,f9ab9c57,cff1a315,f0642d31,b31c2914,faac99e0)
 ,S(be778032,f12c1b77,9bba3d9e,d290ca90,30ac7050,bdd77a2,7eac09be,eea65c0b,8657348b,a1e27a63,1dd2a54b,e2d5270f,4cca817a,219c5378,4d4f73ba,2c932e63)
@@ -8386,22 +8237,22 @@ static const secp256k1_ge_storage secp256k1_pre_g[ECMULT_TABLE_SIZE(WINDOW_G)] =
 ,S(1e70619c,381a6adc,e5d925e0,c9c74f97,3c02ff64,ff2662d7,34efc485,d2bce895,c923f771,f543ffed,42935c28,8474aaaf,80a46ad4,3c579ce0,bb5e663d,668b24b3)
 #endif
 };
-static const secp256k1_ge_storage secp256k1_pre_g_128[ECMULT_TABLE_SIZE(WINDOW_G)] = {
+const secp256k1_ge_storage secp256k1_pre_g_128[ECMULT_TABLE_SIZE(WINDOW_G)] = {
  S(8f68b9d2,f63b5f33,9239c1ad,981f162e,e88c5678,723ea335,1b7b444c,9ec4c0da,662a9f2d,ba063986,de1d90c2,b6be215d,bbea2cfe,95510bfd,f23cbf79,501fff82)
-#if ECMULT_TABLE_SIZE(WINDOW_G) > 1
+#if WINDOW_G > 2
 ,S(38381dbe,2e509f22,8ba93363,f2451f08,fd845cb3,51d954be,18e2b8ed,d23809fa,e4a32d0a,fb917dc,b09405a5,520eb1cc,3681fccb,32d8f24d,bd707518,331fed52)
 #endif
-#if ECMULT_TABLE_SIZE(WINDOW_G) > 2
+#if WINDOW_G > 3
 ,S(49262724,e4372ae6,f6921b82,aa4699a1,f186aea5,40122630,3ea42648,97c2a310,1337e773,bca7abf9,5a2cfa56,9714303b,6d163612,a75ff8ce,c41b681,5e27ded0)
 ,S(e306568c,1a240c90,d5e253b3,e477e2f8,4dcc1a56,ff06db8d,1384b079,cebd2d31,eac6fe3,78934260,888f2b10,7f7d0db6,ffbc8042,be373826,692b4083,92546e44)
 #endif
-#if ECMULT_TABLE_SIZE(WINDOW_G) > 4
+#if WINDOW_G > 4
 ,S(3b9e100e,2428cefc,271b0e76,23fbd633,74ebf8d9,aab41dd9,c530c39e,363136b0,fafb9815,2d16bb71,df1533eb,8f475b26,a2ae28a3,3ad31f81,953ec16f,6cdbbc8a)
 ,S(bb0aad49,712ac9a9,2b76ca80,f5dedef7,17ca0768,8107beee,9608f047,2f485d3f,ea699c53,c5835479,8ecd201f,7297da34,895a5afa,31670bff,e7939250,3ca2f975)
 ,S(79090ac8,e4eefcc0,d4e8eb19,7afe0113,e1e58b4d,b01123de,4aeed33a,36718dc9,eaab722b,91905b8f,13d816cb,cd9aaa56,dd36afb7,ba9008b,963322b1,1cfae7c5)
 ,S(e77c81ad,e9f97b55,1c03dbbc,e549ba66,8dd71de7,cd775ad2,a269694c,7f60c7d1,3acf1478,eef81321,c5fc3b32,3ea81543,631470f7,1c2986d3,4ec581f2,82d72449)
 #endif
-#if ECMULT_TABLE_SIZE(WINDOW_G) > 8
+#if WINDOW_G > 5
 ,S(de2b5ce9,dbce511c,f2d8878e,3ded87cc,3d633dae,a2d45341,501fb3a4,55ccf6b0,f10576f3,d3c3e0e1,bbf717e9,8b1a3744,65b8c45a,c66318bb,34829eb7,11100666)
 ,S(d07bddff,d491a2fe,1ea59fbd,7c121217,29659ca5,de46658b,26b1460b,13c03c56,b2ad4708,cd3c97dd,f9c40e2,a1de04d5,61d963ff,8cc2eea7,6be3f60c,2b405ce7)
 ,S(82403e7c,5d3016af,3765ec4c,396ce8e1,f8da45c,434b8257,10edab41,bb6a4d51,d09661b,e27cb767,4456badd,b3e84051,99ab6ccc,4ec67c1b,11e92ead,7b463b19)
@@ -8411,7 +8262,7 @@ static const secp256k1_ge_storage secp256k1_pre_g_128[ECMULT_TABLE_SIZE(WINDOW_G
 ,S(1f56f096,b18a7499,a153a5ae,acf8be05,8496dd23,da8e6c19,215628fb,c0567ed0,fef22b8a,3b52f490,83004436,b65cd69,c94189f4,1a93c0d5,1fc13cb4,379dff58)
 ,S(1d9f69b1,a4a47432,e386f525,234aa30,79e947cf,cf203297,4e0fc05b,638e213b,d898ec17,949c0761,b38500c3,a2b1da24,5438d5b3,d3f6f720,41f15d6e,e4d4ccbb)
 #endif
-#if ECMULT_TABLE_SIZE(WINDOW_G) > 16
+#if WINDOW_G > 6
 ,S(128c913,4d9dcb78,12fc4361,5c67ad0,55213354,dc8008b1,aeb5a9dc,fb629efd,fee3e54a,dd152610,d9725936,99d662,c160c8e4,ec6f76e4,5ff41818,be67c96)
 ,S(21ec012f,5a95b94d,244b8d51,9756075c,301f2854,8e2c51fc,49c0e3d9,d1a9685,2def2105,77af497f,4c7fef71,6949f28e,7418eda6,fd5fc162,d128de19,3cde08ae)
 ,S(688f5202,fb9d8bc0,9e480e89,4c7cfc74,761c3be7,7dafb11c,58422836,3e331cc5,96ba7d59,63b541a7,2ec7cabd,92403434,1a393eaf,89eebd94,62d9c218,c7302cd3)
@@ -8429,7 +8280,7 @@ static const secp256k1_ge_storage secp256k1_pre_g_128[ECMULT_TABLE_SIZE(WINDOW_G
 ,S(144e88f6,3e73abff,72cac11e,7ddccf79,19e744e6,278941ae,18d1b797,e098e4e5,63cdbf3,5df3c655,c58197f,ea54633d,158705cf,7dc2eb3b,4e09f83c,3021837c)
 ,S(9436e3dc,489ecd8e,2d16a739,c9c73e3d,60e5bc93,68157039,75b8efbd,5c3a9081,1460531f,50cb6ebf,d1aa7806,ea84e7f7,8e8d76b2,b3a66d5e,3a0bf60,39a7e59c)
 #endif
-#if ECMULT_TABLE_SIZE(WINDOW_G) > 32
+#if WINDOW_G > 7
 ,S(9d3c2561,7a56d10b,46d9b01a,1710d193,e840e005,df669e76,1936c275,20890db9,6bbdc0bc,4c4ae9bc,c2dfee9b,82da9b94,1f89ffcd,e8af2aca,4467ce3,78521ea3)
 ,S(29f98e50,f51b7f8b,e18c6ae0,b453c4f2,d0aca5a8,b0e61d2d,dda8506,3fdb76c8,daf3bcdc,ae8e031c,73eb8b21,14058063,58a6ec30,ad379186,df80e3c7,f0e5d28f)
 ,S(d67d30c2,c71daa36,1805e31,1dd6046e,17a89752,94d76e1a,538af074,4dc22c94,48b9b0e7,12c807b0,b92e690a,a2e068cc,e87ebbbe,aaf4bd96,9c1114bb,a54f670c)
@@ -8463,7 +8314,7 @@ static const secp256k1_ge_storage secp256k1_pre_g_128[ECMULT_TABLE_SIZE(WINDOW_G
 ,S(6f97ac0f,27b87905,6b442d13,e566978e,91f0cc1d,d6ac1e64,e9764a35,325dd1b7,83c6e70c,fac6c707,226ce1cc,691b38a0,7e937f5a,5f2d9c81,4dd0d3ff,9f433d32)
 ,S(72c5d60f,eb014e2d,ba8265ca,d454f261,2d6abcd4,b2236bad,c94c4801,561dce1f,e3119a19,7ef91963,b3b28216,3c5d3acb,97b281b6,d246cbf0,690b40da,63978fe2)
 #endif
-#if ECMULT_TABLE_SIZE(WINDOW_G) > 64
+#if WINDOW_G > 8
 ,S(a96d2da0,1b10186,6998659d,f441a1b0,2af32b94,aae8c6ea,707d9ed0,d5f33825,660d7d83,5da5235,9f7cfd41,28c370aa,5659ea71,16a91690,6c0e8108,a513f9f)
 ,S(2cce6f63,4d815ecc,1981d200,87616677,d906aa27,990c4875,17314dc5,5be3c4fa,615210dd,bd599e91,1b6f997a,fd05475,b33cb274,c9ecb6e5,d3c23323,beba4b50)
 ,S(992b0084,525dd399,d98602d9,8b8d53b2,4558fafc,758a2f46,60e89bd6,a645f0c4,83ca98d2,26545a29,8c45f40b,11420602,f5a5c70e,595eea57,2da64d61,a4e2f98d)
@@ -8529,7 +8380,7 @@ static const secp256k1_ge_storage secp256k1_pre_g_128[ECMULT_TABLE_SIZE(WINDOW_G
 ,S(7590a4a,627d5e90,e42a4b0f,6be6497a,f906182d,d2dddc48,a8b6bbfd,f56c0504,d611e21a,b5498760,5c97e878,baa464bc,cc5bd875,353b69f8,1f93ce2a,a6c38587)
 ,S(94b1da0f,b310e056,db0dff72,81db3362,1fcc555d,bf3c973b,76097908,7fe19d6a,8318893d,d5b41a56,33a2ab4,ae4b953c,45c42e9c,8f2fd159,85286de3,fd4fc217)
 #endif
-#if ECMULT_TABLE_SIZE(WINDOW_G) > 128
+#if WINDOW_G > 9
 ,S(b5af4299,bcdacef1,e07d081,daec2cfa,d6f8f821,38e151a5,f20e6d52,84c9a6cb,c0984407,8a7db82d,f572987e,b137dc09,c8cf65fd,aedcb20,43b2479b,ab95448d)
 ,S(5f49ad43,70744587,3980a153,c851829b,f8ef6141,9889bb0c,3c476847,6939c3e3,5c40d385,20f56c3d,ba08ae1,b40fc24a,2ae25c94,45cae0f7,d01d1800,747e04eb)
 ,S(f6bb067f,88ccc11c,64e30d1a,e6893942,16bea3bf,26ec9c64,5cde1b9e,487da385,315a476d,7268978c,d89d4ec8,adde4a83,28dbfdd9,f2bf44fe,ad4ed721,78288f55)
@@ -8659,7 +8510,7 @@ static const secp256k1_ge_storage secp256k1_pre_g_128[ECMULT_TABLE_SIZE(WINDOW_G
 ,S(8ac79852,75673818,69fe93c9,6141493e,72ca344,790487b,d5425ed6,9f5c5c18,bb314248,fcbfc867,d1972e04,b9ef1f90,775375f9,9d25bcec,684c72c9,bdd1c08e)
 ,S(560e9711,88cbc7ce,c61f3bf4,45f0ade3,6f3f174d,219a160,5f9c8692,3f848b9d,9e92dace,6775cc67,bfbbf21f,c64e6e9d,4d133f8a,c18ee277,bc80ef6d,d1b9cd2f)
 #endif
-#if ECMULT_TABLE_SIZE(WINDOW_G) > 256
+#if WINDOW_G > 10
 ,S(8c464204,4b95eb66,cc95ca08,8469a1c0,28eea52f,7f709fe,e6d90700,f5fb943a,bf10fc5d,7bc25181,301b3a61,5ebea597,a3492025,953c3aa5,1a11271e,689d0223)
 ,S(d293f74c,f3578b71,35377247,cd8b0367,7f2245da,f87453cb,4d8d1cc2,871eb5aa,86c2013a,61dfea14,63e45931,eb09050f,f080e3d4,ae357423,ba7afbed,a64a6f26)
 ,S(35997241,ae757b1f,c767611e,c76eb935,fefbf7a7,33666aff,4c6bd744,d7687d35,bcbea61f,246bcd4c,c3c7fd35,7bd393da,2f36e0ef,cca0df9c,5994f96f,c44b2aa0)
@@ -8917,7 +8768,7 @@ static const secp256k1_ge_storage secp256k1_pre_g_128[ECMULT_TABLE_SIZE(WINDOW_G
 ,S(6ca7e032,8bf5d67a,b804d431,e5f709e,bd3156e0,1d4511da,1dc67394,24d2e659,c428b133,f3683909,6551c2ff,2f870d80,d80aaaf4,6d2b1a69,7722057,5eca2647)
 ,S(60df19a9,83e9086c,4cbd20ba,fbafa8b2,346ae4ee,9c1ad4dd,99959e91,2df530d3,dae7b854,29f817a0,421c2f1e,e4e8e4d,a09d60f8,84701e31,d7a8b7b0,3b79cc48)
 #endif
-#if ECMULT_TABLE_SIZE(WINDOW_G) > 512
+#if WINDOW_G > 11
 ,S(6f18d50e,5ef5f2ad,bad80ea1,c59a3847,5c22ff05,6ba52c7,e1d26d6d,d3686bce,d1d7ea0f,a166efd0,facf5c83,bc3786e0,d3f3405d,5b4578f5,13a336ad,f0d7d7a9)
 ,S(b634dc9d,5ed51336,4ac9569a,8ddee9d9,fcdfe00e,65e59233,b3be8a07,2fd949e0,d72e46e2,c61b2575,f25a505c,bdc3456e,fdf18976,6562a1a7,e86d036,eb31db69)
 ,S(5a37fcf0,2dba24f3,e6646171,43dbc5ab,40d91983,69f5cf0e,4fc566fd,97445910,a960d1d8,13f8746c,662a9582,614c7847,33e6153d,2975cb59,c3342463,75c69d1e)
@@ -9431,7 +9282,7 @@ static const secp256k1_ge_storage secp256k1_pre_g_128[ECMULT_TABLE_SIZE(WINDOW_G
 ,S(edc9e8b1,3c61a1e0,8f93be12,495084ed,b5a90eca,4f11dc03,cd217b1a,56285a2c,66c2fd21,7e8fac7,4fd43b1b,58c80661,b0e8df85,a2fddfda,5e9f99fe,621e1f3d)
 ,S(dd94295f,8388a498,2aaa4164,927b81ba,cae9d002,b3ddb6b6,97082c70,f1e2c66e,838c060c,d409c1ea,9bcd9e8c,b1fac83f,3b08c2b2,482d8f4b,fd3ebc18,8d6706b)
 #endif
-#if ECMULT_TABLE_SIZE(WINDOW_G) > 1024
+#if WINDOW_G > 12
 ,S(7e068e04,dfc0be48,e5c0d3b3,5bfc734e,96e96ddd,d0ac4876,92f74535,685ab7e,df2cd146,90d225c8,d04052e6,93f14bf,69351e08,79883646,2c88401e,4ec70d0a)
 ,S(73a9bfcf,d10aac9b,46e659f,76c439a0,b7c2a073,7dec217a,21f43f39,949a5052,73b91529,3ea7b052,682062c,8f86cbf,bf379df6,91a9cec2,4a1424a9,b3be10dd)
 ,S(efddff84,c8cc754,e6e34678,d85809bc,55cc224b,f69be05a,daa847ec,d408c55b,65dd8f41,8264aab2,efe6ed7e,c45b4ac,8aac218a,3b6713fc,1736dd6d,c2f188d5)
@@ -10457,7 +10308,7 @@ static const secp256k1_ge_storage secp256k1_pre_g_128[ECMULT_TABLE_SIZE(WINDOW_G
 ,S(8a663e7,40cad237,12dad6f9,1bb5d266,30542933,853e02eb,665e198d,a5c6e53b,e206b399,51175d15,daccf711,63b0918c,786a8fe,2077812,c0c1eee6,bb5c1e99)
 ,S(2b0956cb,e0ca91ee,f44fac0a,76cc0b00,5f4a5ae8,27e63696,fe1aa8b4,a92941a1,a31e6fa8,dd454eaf,90ee1e11,62627e3,3f84b8fe,da29f423,ddf2a962,386cffd2)
 #endif
-#if ECMULT_TABLE_SIZE(WINDOW_G) > 2048
+#if WINDOW_G > 13
 ,S(e0144151,7a2ac970,eb6381df,7e11fb4c,6b009980,6a0d330,d5429126,e662c3bd,8238ce9b,a691c80c,21563934,18d18dbf,aeb3fd34,48863a1e,7f4ba360,408dfcee)
 ,S(be3213e,8b062f33,caf16c53,5a0b666,f344e0d7,1e28aeba,8b215a3,7ec86c37,552523ba,eaca38a4,cd795683,852a2643,550fb83d,d1db0adc,3f1b29c9,7d51cd1f)
 ,S(623393b6,bfbfbd64,fc7aa1db,9e58e274,1c18eb6d,b5eb30ab,c4fe167e,a9e8ff2b,7c0e4174,f0fd5bf2,a025e316,fa3b7b97,1339a197,b52b0b50,bad4dfe,34e42cd3)
@@ -12507,7 +12358,7 @@ static const secp256k1_ge_storage secp256k1_pre_g_128[ECMULT_TABLE_SIZE(WINDOW_G
 ,S(4beaa213,3c6161d0,c7da6d7c,8eb09cba,2e8ca505,a790a7d3,9d0c16b,8053bb52,da4c083c,9cbdda2e,c6626d8f,e4b2ee13,703496c,c298bffa,db1acdca,99d4f6d5)
 ,S(a8d31ec6,53ce1834,a78de363,c5f9abd1,8594b34,4d34f5fc,8a10e81f,5804831,b4d6a9cd,7cbbe370,9edb5601,cb7c8ba4,8c462c18,594a4bce,64f4c286,dac5cff9)
 #endif
-#if ECMULT_TABLE_SIZE(WINDOW_G) > 4096
+#if WINDOW_G > 14
 ,S(a8f11586,f3df4945,a753c485,ee0fd4d,e410771d,bddc1b26,c9ff10e0,77b915b7,a4ad6f16,dbd741bc,71b2dfd2,dd2d340,3816bf73,4e73cc10,abcfa6bb,d0f161a4)
 ,S(13e697c3,812d4772,254082da,372892d4,e1a66e1a,eb16bbd4,f7a0d531,c979cb2,87fa7baa,f6def12d,31e42c14,f672c0d6,a9d0e2e1,ceee2546,d65bd01,c18df57f)
 ,S(3cae4590,821a9697,a5963269,b44f0222,98f60021,b6048b3f,49c6fd4d,8650c7a,e03f8745,60418449,ad97f28,41664745,349329d,268c1c43,86e25147,6e44b234)
@@ -16606,6 +16457,4 @@ static const secp256k1_ge_storage secp256k1_pre_g_128[ECMULT_TABLE_SIZE(WINDOW_G
 ,S(1b9a142f,fc4d03ea,4b079f2d,b05fad98,8ddb2d32,b359967f,c173801f,63320825,59bda7ed,5b691c20,4fc8f8ac,f53be298,ae628954,a8134d0f,dd097e67,be9ff9b6)
 #endif
 };
-#endif
 #undef S
-#endif
diff --git a/src/precomputed_ecmult.h b/src/precomputed_ecmult.h
new file mode 100644
index 00000000000000..949b62c874b308
--- /dev/null
+++ b/src/precomputed_ecmult.h
@@ -0,0 +1,35 @@
+/*****************************************************************************************************
+ * Copyright (c) 2013, 2014, 2017, 2021 Pieter Wuille, Andrew Poelstra, Jonas Nick, Russell O'Connor *
+ * Distributed under the MIT software license, see the accompanying                                  *
+ * file COPYING or https://www.opensource.org/licenses/mit-license.php.                              *
+ *****************************************************************************************************/
+
+#ifndef SECP256K1_PRECOMPUTED_ECMULT_H
+#define SECP256K1_PRECOMPUTED_ECMULT_H
+
+#ifdef __cplusplus
+extern "C" {
+#endif
+
+#include "group.h"
+#if defined(EXHAUSTIVE_TEST_ORDER)
+#if EXHAUSTIVE_TEST_ORDER == 13
+#        define WINDOW_G 4
+#    elif EXHAUSTIVE_TEST_ORDER == 199
+#        define WINDOW_G 8
+#    else
+#        error No known generator for the specified exhaustive test group order.
+#    endif
+static secp256k1_ge_storage secp256k1_pre_g[ECMULT_TABLE_SIZE(WINDOW_G)];
+static secp256k1_ge_storage secp256k1_pre_g_128[ECMULT_TABLE_SIZE(WINDOW_G)];
+#else /* !defined(EXHAUSTIVE_TEST_ORDER) */
+#    define WINDOW_G ECMULT_WINDOW_SIZE
+extern const secp256k1_ge_storage secp256k1_pre_g[ECMULT_TABLE_SIZE(WINDOW_G)];
+extern const secp256k1_ge_storage secp256k1_pre_g_128[ECMULT_TABLE_SIZE(WINDOW_G)];
+#endif /* defined(EXHAUSTIVE_TEST_ORDER) */
+
+#ifdef __cplusplus
+}
+#endif
+
+#endif /* SECP256K1_PRECOMPUTED_ECMULT_H */
diff --git a/src/ecmult_gen_static_prec_table.h b/src/precomputed_ecmult_gen.c
similarity index 99%
rename from src/ecmult_gen_static_prec_table.h
rename to src/precomputed_ecmult_gen.c
index bf4e5ea248f312..d67291fcf505a1 100644
--- a/src/ecmult_gen_static_prec_table.h
+++ b/src/precomputed_ecmult_gen.c
@@ -1,13 +1,17 @@
-/* This file was automatically generated by gen_ecmult_gen_static_prec_table. */
+/* This file was automatically generated by precompute_ecmult_gen. */
 /* See ecmult_gen_impl.h for details about the contents of this file. */
-#ifndef SECP256K1_ECMULT_GEN_STATIC_PREC_TABLE_H
-#define SECP256K1_ECMULT_GEN_STATIC_PREC_TABLE_H
+#if defined HAVE_CONFIG_H
+#    include "libsecp256k1-config.h"
+#endif
+#include "../include/secp256k1.h"
 #include "group.h"
-#define S(a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p) SECP256K1_GE_STORAGE_CONST(0x##a##u,0x##b##u,0x##c##u,0x##d##u,0x##e##u,0x##f##u,0x##g##u,0x##h##u,0x##i##u,0x##j##u,0x##k##u,0x##l##u,0x##m##u,0x##n##u,0x##o##u,0x##p##u)
+#include "ecmult_gen.h"
+#include "precomputed_ecmult_gen.h"
 #ifdef EXHAUSTIVE_TEST_ORDER
-static secp256k1_ge_storage secp256k1_ecmult_gen_prec_table[ECMULT_GEN_PREC_N(ECMULT_GEN_PREC_BITS)][ECMULT_GEN_PREC_G(ECMULT_GEN_PREC_BITS)];
-#else
-static const secp256k1_ge_storage secp256k1_ecmult_gen_prec_table[ECMULT_GEN_PREC_N(ECMULT_GEN_PREC_BITS)][ECMULT_GEN_PREC_G(ECMULT_GEN_PREC_BITS)] = {
+#    error Cannot compile precomputed_ecmult_gen.c in exhaustive test mode
+#endif /* EXHAUSTIVE_TEST_ORDER */
+#define S(a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p) SECP256K1_GE_STORAGE_CONST(0x##a##u,0x##b##u,0x##c##u,0x##d##u,0x##e##u,0x##f##u,0x##g##u,0x##h##u,0x##i##u,0x##j##u,0x##k##u,0x##l##u,0x##m##u,0x##n##u,0x##o##u,0x##p##u)
+const secp256k1_ge_storage secp256k1_ecmult_gen_prec_table[ECMULT_GEN_PREC_N(ECMULT_GEN_PREC_BITS)][ECMULT_GEN_PREC_G(ECMULT_GEN_PREC_BITS)] = {
 #if ECMULT_GEN_PREC_BITS == 2
 {S(3a9ed373,6eed3eec,9aeb5ac0,21b54652,56817b1f,8de6cd0,fbcee548,ba044bb5,7bcc5928,bdc9c023,dfc663b8,9e4f6969,ab751798,8e600ec1,d242010c,45c7974a),
 S(e44d7675,c3cb2857,4e133c01,a74f4afc,5ce684f8,4a789711,603f7c4f,50abef58,25bcb62f,fe2e2ce2,196ad86c,a006e20,8c64d21b,b25320a3,b5574b9c,1e1bfb4b),
@@ -9743,6 +9747,4 @@ S(244b87a4,fcecef37,76c16c5c,24c7785,be3b3c13,46595363,b8c066ec,45bfe561,9642f5f
 S(9de52b81,157165cc,aef44485,4c2b3535,a599a79,80d024de,5334b385,ecbb2e91,74fca165,26fe2f87,a41ce510,4dd5634,5cf98c11,803c0392,3eb4b8b7,60240c02)}
 #endif
 };
-#endif /* EXHAUSTIVE_TEST_ORDER */
-#undef SC
-#endif /* SECP256K1_ECMULT_GEN_STATIC_PREC_TABLE_H */
+#undef S
diff --git a/src/precomputed_ecmult_gen.h b/src/precomputed_ecmult_gen.h
new file mode 100644
index 00000000000000..7256ad2e3049e4
--- /dev/null
+++ b/src/precomputed_ecmult_gen.h
@@ -0,0 +1,26 @@
+/*********************************************************************************
+ * Copyright (c) 2013, 2014, 2015, 2021 Thomas Daede, Cory Fields, Pieter Wuille *
+ * Distributed under the MIT software license, see the accompanying              *
+ * file COPYING or https://www.opensource.org/licenses/mit-license.php.          *
+ *********************************************************************************/
+
+#ifndef SECP256K1_PRECOMPUTED_ECMULT_GEN_H
+#define SECP256K1_PRECOMPUTED_ECMULT_GEN_H
+
+#ifdef __cplusplus
+extern "C" {
+#endif
+
+#include "group.h"
+#include "ecmult_gen.h"
+#ifdef EXHAUSTIVE_TEST_ORDER
+static secp256k1_ge_storage secp256k1_ecmult_gen_prec_table[ECMULT_GEN_PREC_N(ECMULT_GEN_PREC_BITS)][ECMULT_GEN_PREC_G(ECMULT_GEN_PREC_BITS)];
+#else
+extern const secp256k1_ge_storage secp256k1_ecmult_gen_prec_table[ECMULT_GEN_PREC_N(ECMULT_GEN_PREC_BITS)][ECMULT_GEN_PREC_G(ECMULT_GEN_PREC_BITS)];
+#endif /* defined(EXHAUSTIVE_TEST_ORDER) */
+
+#ifdef __cplusplus
+}
+#endif
+
+#endif /* SECP256K1_PRECOMPUTED_ECMULT_GEN_H */
diff --git a/src/secp256k1.c b/src/secp256k1.c
index 36fde24c3d1477..8f34c35283d3db 100644
--- a/src/secp256k1.c
+++ b/src/secp256k1.c
@@ -423,8 +423,12 @@ static int nonce_function_rfc6979(unsigned char *nonce32, const unsigned char *m
    unsigned int offset = 0;
    secp256k1_rfc6979_hmac_sha256 rng;
    unsigned int i;
+   secp256k1_scalar msg;
+   unsigned char msgmod32[32];
+   secp256k1_scalar_set_b32(&msg, msg32, NULL);
+   secp256k1_scalar_get_b32(msgmod32, &msg);
    /* We feed a byte array to the PRNG as input, consisting of:
-    * - the private key (32 bytes) and message (32 bytes), see RFC 6979 3.2d.
+    * - the private key (32 bytes) and reduced message (32 bytes), see RFC 6979 3.2d.
     * - optionally 32 extra bytes of data, see RFC 6979 3.6 Additional Data.
     * - optionally 16 extra bytes with the algorithm name.
     * Because the arguments have distinct fixed lengths it is not possible for
@@ -432,7 +436,7 @@ static int nonce_function_rfc6979(unsigned char *nonce32, const unsigned char *m
     *  nonces.
     */
    buffer_append(keydata, &offset, key32, 32);
-   buffer_append(keydata, &offset, msg32, 32);
+   buffer_append(keydata, &offset, msgmod32, 32);
    if (data != NULL) {
        buffer_append(keydata, &offset, data, 32);
    }
diff --git a/src/testrand.h b/src/testrand.h
index 667d1867bd6154..bd149bb1b486c9 100644
--- a/src/testrand.h
+++ b/src/testrand.h
@@ -17,11 +17,14 @@
 SECP256K1_INLINE static void secp256k1_testrand_seed(const unsigned char *seed16);
 
 /** Generate a pseudorandom number in the range [0..2**32-1]. */
-static uint32_t secp256k1_testrand32(void);
+SECP256K1_INLINE static uint32_t secp256k1_testrand32(void);
+
+/** Generate a pseudorandom number in the range [0..2**64-1]. */
+SECP256K1_INLINE static uint64_t secp256k1_testrand64(void);
 
 /** Generate a pseudorandom number in the range [0..2**bits-1]. Bits must be 1 or
  *  more. */
-static uint32_t secp256k1_testrand_bits(int bits);
+SECP256K1_INLINE static uint64_t secp256k1_testrand_bits(int bits);
 
 /** Generate a pseudorandom number in the range [0..range-1]. */
 static uint32_t secp256k1_testrand_int(uint32_t range);
diff --git a/src/testrand_impl.h b/src/testrand_impl.h
index c8d30ef6a814ba..e9b9d7ded44bc2 100644
--- a/src/testrand_impl.h
+++ b/src/testrand_impl.h
@@ -14,37 +14,64 @@
 #include "testrand.h"
 #include "hash.h"
 
-static secp256k1_rfc6979_hmac_sha256 secp256k1_test_rng;
-static uint32_t secp256k1_test_rng_precomputed[8];
-static int secp256k1_test_rng_precomputed_used = 8;
+static uint64_t secp256k1_test_state[4];
 static uint64_t secp256k1_test_rng_integer;
 static int secp256k1_test_rng_integer_bits_left = 0;
 
 SECP256K1_INLINE static void secp256k1_testrand_seed(const unsigned char *seed16) {
-    secp256k1_rfc6979_hmac_sha256_initialize(&secp256k1_test_rng, seed16, 16);
+    static const unsigned char PREFIX[19] = "secp256k1 test init";
+    unsigned char out32[32];
+    secp256k1_sha256 hash;
+    int i;
+
+    /* Use SHA256(PREFIX || seed16) as initial state. */
+    secp256k1_sha256_initialize(&hash);
+    secp256k1_sha256_write(&hash, PREFIX, sizeof(PREFIX));
+    secp256k1_sha256_write(&hash, seed16, 16);
+    secp256k1_sha256_finalize(&hash, out32);
+    for (i = 0; i < 4; ++i) {
+        uint64_t s = 0;
+        int j;
+        for (j = 0; j < 8; ++j) s = (s << 8) | out32[8*i + j];
+        secp256k1_test_state[i] = s;
+    }
+    secp256k1_test_rng_integer_bits_left = 0;
 }
 
-SECP256K1_INLINE static uint32_t secp256k1_testrand32(void) {
-    if (secp256k1_test_rng_precomputed_used == 8) {
-        secp256k1_rfc6979_hmac_sha256_generate(&secp256k1_test_rng, (unsigned char*)(&secp256k1_test_rng_precomputed[0]), sizeof(secp256k1_test_rng_precomputed));
-        secp256k1_test_rng_precomputed_used = 0;
-    }
-    return secp256k1_test_rng_precomputed[secp256k1_test_rng_precomputed_used++];
+SECP256K1_INLINE static uint64_t rotl(const uint64_t x, int k) {
+    return (x << k) | (x >> (64 - k));
+}
+
+SECP256K1_INLINE static uint64_t secp256k1_testrand64(void) {
+    /* Test-only Xoshiro256++ RNG. See https://prng.di.unimi.it/ */
+    const uint64_t result = rotl(secp256k1_test_state[0] + secp256k1_test_state[3], 23) + secp256k1_test_state[0];
+    const uint64_t t = secp256k1_test_state[1] << 17;
+    secp256k1_test_state[2] ^= secp256k1_test_state[0];
+    secp256k1_test_state[3] ^= secp256k1_test_state[1];
+    secp256k1_test_state[1] ^= secp256k1_test_state[2];
+    secp256k1_test_state[0] ^= secp256k1_test_state[3];
+    secp256k1_test_state[2] ^= t;
+    secp256k1_test_state[3] = rotl(secp256k1_test_state[3], 45);
+    return result;
 }
 
-static uint32_t secp256k1_testrand_bits(int bits) {
-    uint32_t ret;
+SECP256K1_INLINE static uint64_t secp256k1_testrand_bits(int bits) {
+    uint64_t ret;
     if (secp256k1_test_rng_integer_bits_left < bits) {
-        secp256k1_test_rng_integer |= (((uint64_t)secp256k1_testrand32()) << secp256k1_test_rng_integer_bits_left);
-        secp256k1_test_rng_integer_bits_left += 32;
+        secp256k1_test_rng_integer = secp256k1_testrand64();
+        secp256k1_test_rng_integer_bits_left = 64;
     }
     ret = secp256k1_test_rng_integer;
     secp256k1_test_rng_integer >>= bits;
     secp256k1_test_rng_integer_bits_left -= bits;
-    ret &= ((~((uint32_t)0)) >> (32 - bits));
+    ret &= ((~((uint64_t)0)) >> (64 - bits));
     return ret;
 }
 
+SECP256K1_INLINE static uint32_t secp256k1_testrand32(void) {
+    return secp256k1_testrand_bits(32);
+}
+
 static uint32_t secp256k1_testrand_int(uint32_t range) {
     /* We want a uniform integer between 0 and range-1, inclusive.
      * B is the smallest number such that range <= 2**B.
@@ -85,7 +112,19 @@ static uint32_t secp256k1_testrand_int(uint32_t range) {
 }
 
 static void secp256k1_testrand256(unsigned char *b32) {
-    secp256k1_rfc6979_hmac_sha256_generate(&secp256k1_test_rng, b32, 32);
+    int i;
+    for (i = 0; i < 4; ++i) {
+        uint64_t val = secp256k1_testrand64();
+        b32[0] = val;
+        b32[1] = val >> 8;
+        b32[2] = val >> 16;
+        b32[3] = val >> 24;
+        b32[4] = val >> 32;
+        b32[5] = val >> 40;
+        b32[6] = val >> 48;
+        b32[7] = val >> 56;
+        b32 += 8;
+    }
 }
 
 static void secp256k1_testrand_bytes_test(unsigned char *bytes, size_t len) {
@@ -109,7 +148,7 @@ static void secp256k1_testrand256_test(unsigned char *b32) {
 }
 
 static void secp256k1_testrand_flip(unsigned char *b, size_t len) {
-    b[secp256k1_testrand_int(len)] ^= (1 << secp256k1_testrand_int(8));
+    b[secp256k1_testrand_int(len)] ^= (1 << secp256k1_testrand_bits(3));
 }
 
 static void secp256k1_testrand_init(const char* hexseed) {
diff --git a/src/tests.c b/src/tests.c
index 712fc655fa135d..dd531739307962 100644
--- a/src/tests.c
+++ b/src/tests.c
@@ -28,6 +28,8 @@
 #include "modinv64_impl.h"
 #endif
 
+#define CONDITIONAL_TEST(cnt, nam) if (count < (cnt)) { printf("Skipping %s (iteration count too low)\n", nam); } else
+
 static int count = 64;
 static secp256k1_context *ctx = NULL;
 
@@ -100,6 +102,12 @@ void random_group_element_jacobian_test(secp256k1_gej *gej, const secp256k1_ge *
     gej->infinity = ge->infinity;
 }
 
+void random_gej_test(secp256k1_gej *gej) {
+    secp256k1_ge ge;
+    random_group_element_test(&ge);
+    random_group_element_jacobian_test(gej, &ge);
+}
+
 void random_scalar_order_test(secp256k1_scalar *num) {
     do {
         unsigned char b32[32];
@@ -443,14 +451,18 @@ void run_ctz_tests(void) {
 
 /***** HASH TESTS *****/
 
-void run_sha256_tests(void) {
-    static const char *inputs[8] = {
+void run_sha256_known_output_tests(void) {
+    static const char *inputs[] = {
         "", "abc", "message digest", "secure hash algorithm", "SHA256 is considered to be safe",
         "abcdbcdecdefdefgefghfghighijhijkijkljklmklmnlmnomnopnopq",
         "For this sample, this 63-byte string will be used as input data",
-        "This is exactly 64 bytes long, not counting the terminating byte"
+        "This is exactly 64 bytes long, not counting the terminating byte",
+        "aaaaa",
     };
-    static const unsigned char outputs[8][32] = {
+    static const unsigned int repeat[] = {
+        1, 1, 1, 1, 1, 1, 1, 1, 1000000/5
+    };
+    static const unsigned char outputs[][32] = {
         {0xe3, 0xb0, 0xc4, 0x42, 0x98, 0xfc, 0x1c, 0x14, 0x9a, 0xfb, 0xf4, 0xc8, 0x99, 0x6f, 0xb9, 0x24, 0x27, 0xae, 0x41, 0xe4, 0x64, 0x9b, 0x93, 0x4c, 0xa4, 0x95, 0x99, 0x1b, 0x78, 0x52, 0xb8, 0x55},
         {0xba, 0x78, 0x16, 0xbf, 0x8f, 0x01, 0xcf, 0xea, 0x41, 0x41, 0x40, 0xde, 0x5d, 0xae, 0x22, 0x23, 0xb0, 0x03, 0x61, 0xa3, 0x96, 0x17, 0x7a, 0x9c, 0xb4, 0x10, 0xff, 0x61, 0xf2, 0x00, 0x15, 0xad},
         {0xf7, 0x84, 0x6f, 0x55, 0xcf, 0x23, 0xe1, 0x4e, 0xeb, 0xea, 0xb5, 0xb4, 0xe1, 0x55, 0x0c, 0xad, 0x5b, 0x50, 0x9e, 0x33, 0x48, 0xfb, 0xc4, 0xef, 0xa3, 0xa1, 0x41, 0x3d, 0x39, 0x3c, 0xb6, 0x50},
@@ -458,27 +470,146 @@ void run_sha256_tests(void) {
         {0x68, 0x19, 0xd9, 0x15, 0xc7, 0x3f, 0x4d, 0x1e, 0x77, 0xe4, 0xe1, 0xb5, 0x2d, 0x1f, 0xa0, 0xf9, 0xcf, 0x9b, 0xea, 0xea, 0xd3, 0x93, 0x9f, 0x15, 0x87, 0x4b, 0xd9, 0x88, 0xe2, 0xa2, 0x36, 0x30},
         {0x24, 0x8d, 0x6a, 0x61, 0xd2, 0x06, 0x38, 0xb8, 0xe5, 0xc0, 0x26, 0x93, 0x0c, 0x3e, 0x60, 0x39, 0xa3, 0x3c, 0xe4, 0x59, 0x64, 0xff, 0x21, 0x67, 0xf6, 0xec, 0xed, 0xd4, 0x19, 0xdb, 0x06, 0xc1},
         {0xf0, 0x8a, 0x78, 0xcb, 0xba, 0xee, 0x08, 0x2b, 0x05, 0x2a, 0xe0, 0x70, 0x8f, 0x32, 0xfa, 0x1e, 0x50, 0xc5, 0xc4, 0x21, 0xaa, 0x77, 0x2b, 0xa5, 0xdb, 0xb4, 0x06, 0xa2, 0xea, 0x6b, 0xe3, 0x42},
-        {0xab, 0x64, 0xef, 0xf7, 0xe8, 0x8e, 0x2e, 0x46, 0x16, 0x5e, 0x29, 0xf2, 0xbc, 0xe4, 0x18, 0x26, 0xbd, 0x4c, 0x7b, 0x35, 0x52, 0xf6, 0xb3, 0x82, 0xa9, 0xe7, 0xd3, 0xaf, 0x47, 0xc2, 0x45, 0xf8}
+        {0xab, 0x64, 0xef, 0xf7, 0xe8, 0x8e, 0x2e, 0x46, 0x16, 0x5e, 0x29, 0xf2, 0xbc, 0xe4, 0x18, 0x26, 0xbd, 0x4c, 0x7b, 0x35, 0x52, 0xf6, 0xb3, 0x82, 0xa9, 0xe7, 0xd3, 0xaf, 0x47, 0xc2, 0x45, 0xf8},
+        {0xcd, 0xc7, 0x6e, 0x5c, 0x99, 0x14, 0xfb, 0x92, 0x81, 0xa1, 0xc7, 0xe2, 0x84, 0xd7, 0x3e, 0x67, 0xf1, 0x80, 0x9a, 0x48, 0xa4, 0x97, 0x20, 0x0e, 0x04, 0x6d, 0x39, 0xcc, 0xc7, 0x11, 0x2c, 0xd0},
     };
-    int i;
-    for (i = 0; i < 8; i++) {
+    unsigned int i, ninputs;
+
+    /* Skip last input vector for low iteration counts */
+    ninputs = sizeof(inputs)/sizeof(inputs[0]) - 1;
+    CONDITIONAL_TEST(16, "run_sha256_known_output_tests 1000000") ninputs++;
+
+    for (i = 0; i < ninputs; i++) {
         unsigned char out[32];
         secp256k1_sha256 hasher;
+        unsigned int j;
+        /* 1. Run: simply write the input bytestrings */
+        j = repeat[i];
         secp256k1_sha256_initialize(&hasher);
-        secp256k1_sha256_write(&hasher, (const unsigned char*)(inputs[i]), strlen(inputs[i]));
+        while (j > 0) {
+            secp256k1_sha256_write(&hasher, (const unsigned char*)(inputs[i]), strlen(inputs[i]));
+            j--;
+        }
         secp256k1_sha256_finalize(&hasher, out);
         CHECK(secp256k1_memcmp_var(out, outputs[i], 32) == 0);
+        /* 2. Run: split the input bytestrings randomly before writing */
         if (strlen(inputs[i]) > 0) {
             int split = secp256k1_testrand_int(strlen(inputs[i]));
             secp256k1_sha256_initialize(&hasher);
-            secp256k1_sha256_write(&hasher, (const unsigned char*)(inputs[i]), split);
-            secp256k1_sha256_write(&hasher, (const unsigned char*)(inputs[i] + split), strlen(inputs[i]) - split);
+            j = repeat[i];
+            while (j > 0) {
+                secp256k1_sha256_write(&hasher, (const unsigned char*)(inputs[i]), split);
+                secp256k1_sha256_write(&hasher, (const unsigned char*)(inputs[i] + split), strlen(inputs[i]) - split);
+                j--;
+            }
             secp256k1_sha256_finalize(&hasher, out);
             CHECK(secp256k1_memcmp_var(out, outputs[i], 32) == 0);
         }
     }
 }
 
+/** SHA256 counter tests
+
+The tests verify that the SHA256 counter doesn't wrap around at message length
+2^i bytes for i = 20, ..., 33. This wide range aims at being independent of the
+implementation of the counter and it catches multiple natural 32-bit overflows
+(e.g., counting bits, counting bytes, counting blocks, ...).
+
+The test vectors have been generated using following Python script which relies
+on https://github.com/cloudtools/sha256/ (v0.3 on Python v3.10.2).
+
+```
+from sha256 import sha256
+from copy import copy
+
+def midstate_c_definition(hasher):
+    ret  = '    {{0x' + hasher.state[0].hex('_', 4).replace('_', ', 0x') + '},\n'
+    ret += '    {0x00}, ' + str(hex(hasher.state[1])) + '}'
+    return ret
+
+def output_c_literal(hasher):
+    return '{0x' + hasher.digest().hex('_').replace('_', ', 0x') + '}'
+
+MESSAGE = b'abcdefghbcdefghicdefghijdefghijkefghijklfghijklmghijklmnhijklmno'
+assert(len(MESSAGE) == 64)
+BYTE_BOUNDARIES = [(2**b)//len(MESSAGE) - 1 for b in range(20, 34)]
+
+midstates = []
+digests = []
+hasher = sha256()
+for i in range(BYTE_BOUNDARIES[-1] + 1):
+    if i in BYTE_BOUNDARIES:
+        midstates.append(midstate_c_definition(hasher))
+        hasher_copy = copy(hasher)
+        hasher_copy.update(MESSAGE)
+        digests.append(output_c_literal(hasher_copy))
+    hasher.update(MESSAGE)
+
+for x in midstates:
+    print(x + ',')
+
+for x in digests:
+    print(x + ',')
+```
+*/
+void run_sha256_counter_tests(void) {
+    static const char *input = "abcdefghbcdefghicdefghijdefghijkefghijklfghijklmghijklmnhijklmno";
+    static const secp256k1_sha256 midstates[] = {
+        {{0xa2b5c8bb, 0x26c88bb3, 0x2abdc3d2, 0x9def99a3, 0xdfd21a6e, 0x41fe585b, 0x7ef2c440, 0x2b79adda},
+         {0x00}, 0xfffc0},
+        {{0xa0d29445, 0x9287de66, 0x76aabd71, 0x41acd765, 0x0c7528b4, 0x84e14906, 0x942faec6, 0xcc5a7b26},
+         {0x00}, 0x1fffc0},
+        {{0x50449526, 0xb9f1d657, 0xa0fc13e9, 0x50860f10, 0xa550c431, 0x3fbc97c1, 0x7bbb2d89, 0xdb67bac1},
+         {0x00}, 0x3fffc0},
+        {{0x54a6efdc, 0x46762e7b, 0x88bfe73f, 0xbbd149c7, 0x41620c43, 0x1168da7b, 0x2c5960f9, 0xeccffda6},
+         {0x00}, 0x7fffc0},
+        {{0x2515a8f5, 0x5faa2977, 0x3a850486, 0xac858cad, 0x7b7276ee, 0x235c0385, 0xc53a157c, 0x7cb3e69c},
+         {0x00}, 0xffffc0},
+        {{0x34f39828, 0x409fedb7, 0x4bbdd0fb, 0x3b643634, 0x7806bf2e, 0xe0d1b713, 0xca3f2e1e, 0xe38722c2},
+         {0x00}, 0x1ffffc0},
+        {{0x389ef5c5, 0x38c54167, 0x8f5d56ab, 0x582a75cc, 0x8217caef, 0xf10947dd, 0x6a1998a8, 0x048f0b8c},
+         {0x00}, 0x3ffffc0},
+        {{0xd6c3f394, 0x0bee43b9, 0x6783f497, 0x29fa9e21, 0x6ce491c1, 0xa81fe45e, 0x2fc3859a, 0x269012d0},
+         {0x00}, 0x7ffffc0},
+        {{0x6dd3c526, 0x44d88aa0, 0x806a1bae, 0xfbcc0d32, 0x9d6144f3, 0x9d2bd757, 0x9851a957, 0xb50430ad},
+         {0x00}, 0xfffffc0},
+        {{0x2add4021, 0xdfe8a9e6, 0xa56317c6, 0x7a15f5bb, 0x4a48aacd, 0x5d368414, 0x4f00e6f0, 0xd9355023},
+         {0x00}, 0x1fffffc0},
+        {{0xb66666b4, 0xdbeac32b, 0x0ea351ae, 0xcba9da46, 0x6278b874, 0x8c508e23, 0xe16ca776, 0x8465bac1},
+         {0x00}, 0x3fffffc0},
+        {{0xb6744789, 0x9cce87aa, 0xc4c478b7, 0xf38404d8, 0x2e38ba62, 0xa3f7019b, 0x50458fe7, 0x3047dbec},
+         {0x00}, 0x7fffffc0},
+        {{0x8b1297ba, 0xba261a80, 0x2ba1b0dd, 0xfbc67d6d, 0x61072c4e, 0x4b5a2a0f, 0x52872760, 0x2dfeb162},
+         {0x00}, 0xffffffc0},
+        {{0x24f33cf7, 0x41ad6583, 0x41c8ff5d, 0xca7ef35f, 0x50395756, 0x021b743e, 0xd7126cd7, 0xd037473a},
+         {0x00}, 0x1ffffffc0},
+    };
+    static const unsigned char outputs[][32] = {
+        {0x0e, 0x83, 0xe2, 0xc9, 0x4f, 0xb2, 0xb8, 0x2b, 0x89, 0x06, 0x92, 0x78, 0x04, 0x03, 0x48, 0x5c, 0x48, 0x44, 0x67, 0x61, 0x77, 0xa4, 0xc7, 0x90, 0x9e, 0x92, 0x55, 0x10, 0x05, 0xfe, 0x39, 0x15},
+        {0x1d, 0x1e, 0xd7, 0xb8, 0xa3, 0xa7, 0x8a, 0x79, 0xfd, 0xa0, 0x05, 0x08, 0x9c, 0xeb, 0xf0, 0xec, 0x67, 0x07, 0x9f, 0x8e, 0x3c, 0x0d, 0x8e, 0xf9, 0x75, 0x55, 0x13, 0xc1, 0xe8, 0x77, 0xf8, 0xbb},
+        {0x66, 0x95, 0x6c, 0xc9, 0xe0, 0x39, 0x65, 0xb6, 0xb0, 0x05, 0xd1, 0xaf, 0xaf, 0xf3, 0x1d, 0xb9, 0xa4, 0xda, 0x6f, 0x20, 0xcd, 0x3a, 0xae, 0x64, 0xc2, 0xdb, 0xee, 0xf5, 0xb8, 0x8d, 0x57, 0x0e},
+        {0x3c, 0xbb, 0x1c, 0x12, 0x5e, 0x17, 0xfd, 0x54, 0x90, 0x45, 0xa7, 0x7b, 0x61, 0x6c, 0x1d, 0xfe, 0xe6, 0xcc, 0x7f, 0xee, 0xcf, 0xef, 0x33, 0x35, 0x50, 0x62, 0x16, 0x70, 0x2f, 0x87, 0xc3, 0xc9},
+        {0x53, 0x4d, 0xa8, 0xe7, 0x1e, 0x98, 0x73, 0x8d, 0xd9, 0xa3, 0x54, 0xa5, 0x0e, 0x59, 0x2c, 0x25, 0x43, 0x6f, 0xaa, 0xa2, 0xf5, 0x21, 0x06, 0x3e, 0xc9, 0x82, 0x06, 0x94, 0x98, 0x72, 0x9d, 0xa7},
+        {0xef, 0x7e, 0xe9, 0x6b, 0xd3, 0xe5, 0xb7, 0x41, 0x4c, 0xc8, 0xd3, 0x07, 0x52, 0x9a, 0x5a, 0x8b, 0x4e, 0x1e, 0x75, 0xa4, 0x17, 0x78, 0xc8, 0x36, 0xcd, 0xf8, 0x2e, 0xd9, 0x57, 0xe3, 0xd7, 0x07},
+        {0x87, 0x16, 0xfb, 0xf9, 0xa5, 0xf8, 0xc4, 0x56, 0x2b, 0x48, 0x52, 0x8e, 0x2d, 0x30, 0x85, 0xb6, 0x4c, 0x56, 0xb5, 0xd1, 0x16, 0x9c, 0xcf, 0x32, 0x95, 0xad, 0x03, 0xe8, 0x05, 0x58, 0x06, 0x76},
+        {0x75, 0x03, 0x80, 0x28, 0xf2, 0xa7, 0x63, 0x22, 0x1a, 0x26, 0x9c, 0x68, 0xe0, 0x58, 0xfc, 0x73, 0xeb, 0x42, 0xf6, 0x86, 0x16, 0x24, 0x4b, 0xbc, 0x24, 0xf7, 0x02, 0xc8, 0x3d, 0x90, 0xe2, 0xb0},
+        {0xdf, 0x49, 0x0f, 0x15, 0x7b, 0x7d, 0xbf, 0xe0, 0xd4, 0xcf, 0x47, 0xc0, 0x80, 0x93, 0x4a, 0x61, 0xaa, 0x03, 0x07, 0x66, 0xb3, 0x38, 0x5d, 0xc8, 0xc9, 0x07, 0x61, 0xfb, 0x97, 0x10, 0x2f, 0xd8},
+        {0x77, 0x19, 0x40, 0x56, 0x41, 0xad, 0xbc, 0x59, 0xda, 0x1e, 0xc5, 0x37, 0x14, 0x63, 0x7b, 0xfb, 0x79, 0xe2, 0x7a, 0xb1, 0x55, 0x42, 0x99, 0x42, 0x56, 0xfe, 0x26, 0x9d, 0x0f, 0x7e, 0x80, 0xc6},
+        {0x50, 0xe7, 0x2a, 0x0e, 0x26, 0x44, 0x2f, 0xe2, 0x55, 0x2d, 0xc3, 0x93, 0x8a, 0xc5, 0x86, 0x58, 0x22, 0x8c, 0x0c, 0xbf, 0xb1, 0xd2, 0xca, 0x87, 0x2a, 0xe4, 0x35, 0x26, 0x6f, 0xcd, 0x05, 0x5e},
+        {0xe4, 0x80, 0x6f, 0xdb, 0x3d, 0x7d, 0xba, 0xde, 0x50, 0x3f, 0xea, 0x00, 0x3d, 0x46, 0x59, 0x64, 0xfd, 0x58, 0x1c, 0xa1, 0xb8, 0x7d, 0x5f, 0xac, 0x94, 0x37, 0x9e, 0xa0, 0xc0, 0x9c, 0x93, 0x8b},
+        {0x2c, 0xf3, 0xa9, 0xf6, 0x15, 0x25, 0x80, 0x70, 0x76, 0x99, 0x7d, 0xf1, 0xc3, 0x2f, 0xa3, 0x31, 0xff, 0x92, 0x35, 0x2e, 0x8d, 0x04, 0x13, 0x33, 0xd8, 0x0d, 0xdb, 0x4a, 0xf6, 0x8c, 0x03, 0x34},
+        {0xec, 0x12, 0x24, 0x9f, 0x35, 0xa4, 0x29, 0x8b, 0x9e, 0x4a, 0x95, 0xf8, 0x61, 0xaf, 0x61, 0xc5, 0x66, 0x55, 0x3e, 0x3f, 0x2a, 0x98, 0xea, 0x71, 0x16, 0x6b, 0x1c, 0xd9, 0xe4, 0x09, 0xd2, 0x8e},
+    };
+    unsigned int i;
+    for (i = 0; i < sizeof(midstates)/sizeof(midstates[0]); i++) {
+        unsigned char out[32];
+        secp256k1_sha256 hasher = midstates[i];
+        secp256k1_sha256_write(&hasher, (const unsigned char*)input, strlen(input));
+        secp256k1_sha256_finalize(&hasher, out);
+        CHECK(secp256k1_memcmp_var(out, outputs[i], 32) == 0);
+    }
+}
+
 void run_hmac_sha256_tests(void) {
     static const char *keys[6] = {
         "\x0b\x0b\x0b\x0b\x0b\x0b\x0b\x0b\x0b\x0b\x0b\x0b\x0b\x0b\x0b\x0b\x0b\x0b\x0b\x0b",
@@ -790,7 +921,7 @@ void signed30_to_uint16(uint16_t* out, const secp256k1_modinv32_signed30* in) {
 void mutate_sign_signed30(secp256k1_modinv32_signed30* x) {
     int i;
     for (i = 0; i < 16; ++i) {
-        int pos = secp256k1_testrand_int(8);
+        int pos = secp256k1_testrand_bits(3);
         if (x->v[pos] > 0 && x->v[pos + 1] <= 0x3fffffff) {
             x->v[pos] -= 0x40000000;
             x->v[pos + 1] += 1;
@@ -862,7 +993,7 @@ void mutate_sign_signed62(secp256k1_modinv64_signed62* x) {
     static const int64_t M62 = (int64_t)(UINT64_MAX >> 2);
     int i;
     for (i = 0; i < 8; ++i) {
-        int pos = secp256k1_testrand_int(4);
+        int pos = secp256k1_testrand_bits(2);
         if (x->v[pos] > 0 && x->v[pos + 1] <= M62) {
             x->v[pos] -= (M62 + 1);
             x->v[pos + 1] += 1;
@@ -2451,13 +2582,65 @@ void run_field_convert(void) {
     CHECK(secp256k1_memcmp_var(&fes2, &fes, sizeof(fes)) == 0);
 }
 
-int fe_secp256k1_memcmp_var(const secp256k1_fe *a, const secp256k1_fe *b) {
-    secp256k1_fe t = *b;
+/* Returns true if two field elements have the same representation. */
+int fe_identical(const secp256k1_fe *a, const secp256k1_fe *b) {
+    int ret = 1;
 #ifdef VERIFY
-    t.magnitude = a->magnitude;
-    t.normalized = a->normalized;
+    ret &= (a->magnitude == b->magnitude);
+    ret &= (a->normalized == b->normalized);
 #endif
-    return secp256k1_memcmp_var(a, &t, sizeof(secp256k1_fe));
+    /* Compare the struct member that holds the limbs. */
+    ret &= (secp256k1_memcmp_var(a->n, b->n, sizeof(a->n)) == 0);
+    return ret;
+}
+
+void run_field_half(void) {
+    secp256k1_fe t, u;
+    int m;
+
+    /* Check magnitude 0 input */
+    secp256k1_fe_get_bounds(&t, 0);
+    secp256k1_fe_half(&t);
+#ifdef VERIFY
+    CHECK(t.magnitude == 1);
+    CHECK(t.normalized == 0);
+#endif
+    CHECK(secp256k1_fe_normalizes_to_zero(&t));
+
+    /* Check non-zero magnitudes in the supported range */
+    for (m = 1; m < 32; m++) {
+        /* Check max-value input */
+        secp256k1_fe_get_bounds(&t, m);
+
+        u = t;
+        secp256k1_fe_half(&u);
+#ifdef VERIFY
+        CHECK(u.magnitude == (m >> 1) + 1);
+        CHECK(u.normalized == 0);
+#endif
+        secp256k1_fe_normalize_weak(&u);
+        secp256k1_fe_add(&u, &u);
+        CHECK(check_fe_equal(&t, &u));
+
+        /* Check worst-case input: ensure the LSB is 1 so that P will be added,
+         * which will also cause all carries to be 1, since all limbs that can
+         * generate a carry are initially even and all limbs of P are odd in
+         * every existing field implementation. */
+        secp256k1_fe_get_bounds(&t, m);
+        CHECK(t.n[0] > 0);
+        CHECK((t.n[0] & 1) == 0);
+        --t.n[0];
+
+        u = t;
+        secp256k1_fe_half(&u);
+#ifdef VERIFY
+        CHECK(u.magnitude == (m >> 1) + 1);
+        CHECK(u.normalized == 0);
+#endif
+        secp256k1_fe_normalize_weak(&u);
+        secp256k1_fe_add(&u, &u);
+        CHECK(check_fe_equal(&t, &u));
+    }
 }
 
 void run_field_misc(void) {
@@ -2467,9 +2650,13 @@ void run_field_misc(void) {
     secp256k1_fe q;
     secp256k1_fe fe5 = SECP256K1_FE_CONST(0, 0, 0, 0, 0, 0, 0, 5);
     int i, j;
-    for (i = 0; i < 5*count; i++) {
+    for (i = 0; i < 1000 * count; i++) {
         secp256k1_fe_storage xs, ys, zs;
-        random_fe(&x);
+        if (i & 1) {
+            random_fe(&x);
+        } else {
+            random_fe_test(&x);
+        }
         random_fe_non_zero(&y);
         /* Test the fe equality and comparison operations. */
         CHECK(secp256k1_fe_cmp_var(&x, &x) == 0);
@@ -2483,13 +2670,13 @@ void run_field_misc(void) {
         CHECK(x.normalized && x.magnitude == 1);
 #endif
         secp256k1_fe_cmov(&x, &x, 1);
-        CHECK(fe_secp256k1_memcmp_var(&x, &z) != 0);
-        CHECK(fe_secp256k1_memcmp_var(&x, &q) == 0);
+        CHECK(!fe_identical(&x, &z));
+        CHECK(fe_identical(&x, &q));
         secp256k1_fe_cmov(&q, &z, 1);
 #ifdef VERIFY
         CHECK(!q.normalized && q.magnitude == z.magnitude);
 #endif
-        CHECK(fe_secp256k1_memcmp_var(&q, &z) == 0);
+        CHECK(fe_identical(&q, &z));
         secp256k1_fe_normalize_var(&x);
         secp256k1_fe_normalize_var(&z);
         CHECK(!secp256k1_fe_equal_var(&x, &z));
@@ -2537,6 +2724,14 @@ void run_field_misc(void) {
         secp256k1_fe_add(&q, &x);
         CHECK(check_fe_equal(&y, &z));
         CHECK(check_fe_equal(&q, &y));
+        /* Check secp256k1_fe_half. */
+        z = x;
+        secp256k1_fe_half(&z);
+        secp256k1_fe_add(&z, &z);
+        CHECK(check_fe_equal(&x, &z));
+        secp256k1_fe_add(&z, &z);
+        secp256k1_fe_half(&z);
+        CHECK(check_fe_equal(&x, &z));
     }
 }
 
@@ -3338,6 +3533,37 @@ void run_ge(void) {
     test_intialized_inf();
 }
 
+void test_gej_cmov(const secp256k1_gej *a, const secp256k1_gej *b) {
+    secp256k1_gej t = *a;
+    secp256k1_gej_cmov(&t, b, 0);
+    CHECK(gej_xyz_equals_gej(&t, a));
+    secp256k1_gej_cmov(&t, b, 1);
+    CHECK(gej_xyz_equals_gej(&t, b));
+}
+
+void run_gej(void) {
+    int i;
+    secp256k1_gej a, b;
+
+    /* Tests for secp256k1_gej_cmov */
+    for (i = 0; i < count; i++) {
+        secp256k1_gej_set_infinity(&a);
+        secp256k1_gej_set_infinity(&b);
+        test_gej_cmov(&a, &b);
+
+        random_gej_test(&a);
+        test_gej_cmov(&a, &b);
+        test_gej_cmov(&b, &a);
+
+        b = a;
+        test_gej_cmov(&a, &b);
+
+        random_gej_test(&b);
+        test_gej_cmov(&a, &b);
+        test_gej_cmov(&b, &a);
+    }
+}
+
 void test_ec_combine(void) {
     secp256k1_scalar sum = SECP256K1_SCALAR_CONST(0, 0, 0, 0, 0, 0, 0, 0);
     secp256k1_pubkey data[6];
@@ -4052,6 +4278,174 @@ void test_ecmult_multi(secp256k1_scratch *scratch, secp256k1_ecmult_multi_func e
     }
 }
 
+int test_ecmult_multi_random(secp256k1_scratch *scratch) {
+    /* Large random test for ecmult_multi_* functions which exercises:
+     * - Few or many inputs (0 up to 128, roughly exponentially distributed).
+     * - Few or many 0*P or a*INF inputs (roughly uniformly distributed).
+     * - Including or excluding an nonzero a*G term (or such a term at all).
+     * - Final expected result equal to infinity or not (roughly 50%).
+     * - ecmult_multi_var, ecmult_strauss_single_batch, ecmult_pippenger_single_batch
+     */
+
+    /* These 4 variables define the eventual input to the ecmult_multi function.
+     * g_scalar is the G scalar fed to it (or NULL, possibly, if g_scalar=0), and
+     * scalars[0..filled-1] and gejs[0..filled-1] are the scalars and points
+     * which form its normal inputs. */
+    int filled = 0;
+    secp256k1_scalar g_scalar = SECP256K1_SCALAR_CONST(0, 0, 0, 0, 0, 0, 0, 0);
+    secp256k1_scalar scalars[128];
+    secp256k1_gej gejs[128];
+    /* The expected result, and the computed result. */
+    secp256k1_gej expected, computed;
+    /* Temporaries. */
+    secp256k1_scalar sc_tmp;
+    secp256k1_ge ge_tmp;
+    /* Variables needed for the actual input to ecmult_multi. */
+    secp256k1_ge ges[128];
+    ecmult_multi_data data;
+
+    int i;
+    /* Which multiplication function to use */
+    int fn = secp256k1_testrand_int(3);
+    secp256k1_ecmult_multi_func ecmult_multi = fn == 0 ? secp256k1_ecmult_multi_var :
+                                               fn == 1 ? secp256k1_ecmult_strauss_batch_single :
+                                               secp256k1_ecmult_pippenger_batch_single;
+    /* Simulate exponentially distributed num. */
+    int num_bits = 2 + secp256k1_testrand_int(6);
+    /* Number of (scalar, point) inputs (excluding g). */
+    int num = secp256k1_testrand_int((1 << num_bits) + 1);
+    /* Number of those which are nonzero. */
+    int num_nonzero = secp256k1_testrand_int(num + 1);
+    /* Whether we're aiming to create an input with nonzero expected result. */
+    int nonzero_result = secp256k1_testrand_bits(1);
+    /* Whether we will provide nonzero g multiplicand. In some cases our hand
+     * is forced here based on num_nonzero and nonzero_result. */
+    int g_nonzero = num_nonzero == 0 ? nonzero_result :
+                    num_nonzero == 1 && !nonzero_result ? 1 :
+                    (int)secp256k1_testrand_bits(1);
+    /* Which g_scalar pointer to pass into ecmult_multi(). */
+    const secp256k1_scalar* g_scalar_ptr = (g_nonzero || secp256k1_testrand_bits(1)) ? &g_scalar : NULL;
+    /* How many EC multiplications were performed in this function. */
+    int mults = 0;
+    /* How many randomization steps to apply to the input list. */
+    int rands = (int)secp256k1_testrand_bits(3);
+    if (rands > num_nonzero) rands = num_nonzero;
+
+    secp256k1_gej_set_infinity(&expected);
+    secp256k1_gej_set_infinity(&gejs[0]);
+    secp256k1_scalar_set_int(&scalars[0], 0);
+
+    if (g_nonzero) {
+        /* If g_nonzero, set g_scalar to nonzero value r. */
+        random_scalar_order_test(&g_scalar);
+        if (!nonzero_result) {
+            /* If expected=0 is desired, add a (a*r, -(1/a)*g) term to compensate. */
+            CHECK(num_nonzero > filled);
+            random_scalar_order_test(&sc_tmp);
+            secp256k1_scalar_mul(&scalars[filled], &sc_tmp, &g_scalar);
+            secp256k1_scalar_inverse_var(&sc_tmp, &sc_tmp);
+            secp256k1_scalar_negate(&sc_tmp, &sc_tmp);
+            secp256k1_ecmult_gen(&ctx->ecmult_gen_ctx, &gejs[filled], &sc_tmp);
+            ++filled;
+            ++mults;
+        }
+    }
+
+    if (nonzero_result && filled < num_nonzero) {
+        /* If a nonzero result is desired, and there is space, add a random nonzero term. */
+        random_scalar_order_test(&scalars[filled]);
+        random_group_element_test(&ge_tmp);
+        secp256k1_gej_set_ge(&gejs[filled], &ge_tmp);
+        ++filled;
+    }
+
+    if (nonzero_result) {
+        /* Compute the expected result using normal ecmult. */
+        CHECK(filled <= 1);
+        secp256k1_ecmult(&expected, &gejs[0], &scalars[0], &g_scalar);
+        mults += filled + g_nonzero;
+    }
+
+    /* At this point we have expected = scalar_g*G + sum(scalars[i]*gejs[i] for i=0..filled-1). */
+    CHECK(filled <= 1 + !nonzero_result);
+    CHECK(filled <= num_nonzero);
+
+    /* Add entries to scalars,gejs so that there are num of them. All the added entries
+     * either have scalar=0 or point=infinity, so these do not change the expected result. */
+    while (filled < num) {
+        if (secp256k1_testrand_bits(1)) {
+            secp256k1_gej_set_infinity(&gejs[filled]);
+            random_scalar_order_test(&scalars[filled]);
+        } else {
+            secp256k1_scalar_set_int(&scalars[filled], 0);
+            random_group_element_test(&ge_tmp);
+            secp256k1_gej_set_ge(&gejs[filled], &ge_tmp);
+        }
+        ++filled;
+    }
+
+    /* Now perform cheapish transformations on gejs and scalars, for indices
+     * 0..num_nonzero-1, which do not change the expected result, but may
+     * convert some of them to be both non-0-scalar and non-infinity-point. */
+    for (i = 0; i < rands; ++i) {
+        int j;
+        secp256k1_scalar v, iv;
+        /* Shuffle the entries. */
+        for (j = 0; j < num_nonzero; ++j) {
+            int k = secp256k1_testrand_int(num_nonzero - j);
+            if (k != 0) {
+                secp256k1_gej gej = gejs[j];
+                secp256k1_scalar sc = scalars[j];
+                gejs[j] = gejs[j + k];
+                scalars[j] = scalars[j + k];
+                gejs[j + k] = gej;
+                scalars[j + k] = sc;
+            }
+        }
+        /* Perturb all consecutive pairs of inputs:
+         * a*P + b*Q -> (a+b)*P + b*(Q-P). */
+        for (j = 0; j + 1 < num_nonzero; j += 2) {
+            secp256k1_gej gej;
+            secp256k1_scalar_add(&scalars[j], &scalars[j], &scalars[j+1]);
+            secp256k1_gej_neg(&gej, &gejs[j]);
+            secp256k1_gej_add_var(&gejs[j+1], &gejs[j+1], &gej, NULL);
+        }
+        /* Transform the last input: a*P -> (v*a) * ((1/v)*P). */
+        CHECK(num_nonzero >= 1);
+        random_scalar_order_test(&v);
+        secp256k1_scalar_inverse(&iv, &v);
+        secp256k1_scalar_mul(&scalars[num_nonzero - 1], &scalars[num_nonzero - 1], &v);
+        secp256k1_ecmult(&gejs[num_nonzero - 1], &gejs[num_nonzero - 1], &iv, NULL);
+        ++mults;
+    }
+
+    /* Shuffle all entries (0..num-1). */
+    for (i = 0; i < num; ++i) {
+        int j = secp256k1_testrand_int(num - i);
+        if (j != 0) {
+            secp256k1_gej gej = gejs[i];
+            secp256k1_scalar sc = scalars[i];
+            gejs[i] = gejs[i + j];
+            scalars[i] = scalars[i + j];
+            gejs[i + j] = gej;
+            scalars[i + j] = sc;
+        }
+    }
+
+    /* Compute affine versions of all inputs. */
+    secp256k1_ge_set_all_gej_var(ges, gejs, filled);
+    /* Invoke ecmult_multi code. */
+    data.sc = scalars;
+    data.pt = ges;
+    CHECK(ecmult_multi(&ctx->error_callback, scratch, &computed, g_scalar_ptr, ecmult_multi_callback, &data, filled));
+    mults += num_nonzero + g_nonzero;
+    /* Compare with expected result. */
+    secp256k1_gej_neg(&computed, &computed);
+    secp256k1_gej_add_var(&computed, &computed, &expected, NULL);
+    CHECK(secp256k1_gej_is_infinity(&computed));
+    return mults;
+}
+
 void test_ecmult_multi_batch_single(secp256k1_ecmult_multi_func ecmult_multi) {
     secp256k1_scalar szero;
     secp256k1_scalar sc;
@@ -4093,7 +4487,7 @@ void test_secp256k1_pippenger_bucket_window_inv(void) {
  * for a given scratch space.
  */
 void test_ecmult_multi_pippenger_max_points(void) {
-    size_t scratch_size = secp256k1_testrand_int(256);
+    size_t scratch_size = secp256k1_testrand_bits(8);
     size_t max_size = secp256k1_pippenger_scratch_size(secp256k1_pippenger_bucket_window_inv(PIPPENGER_MAX_BUCKET_WINDOW-1)+512, 12);
     secp256k1_scratch *scratch;
     size_t n_points_supported;
@@ -4242,6 +4636,7 @@ void test_ecmult_multi_batching(void) {
 
 void run_ecmult_multi_tests(void) {
     secp256k1_scratch *scratch;
+    int64_t todo = (int64_t)320 * count;
 
     test_secp256k1_pippenger_bucket_window_inv();
     test_ecmult_multi_pippenger_max_points();
@@ -4252,6 +4647,9 @@ void run_ecmult_multi_tests(void) {
     test_ecmult_multi_batch_single(secp256k1_ecmult_pippenger_batch_single);
     test_ecmult_multi(scratch, secp256k1_ecmult_strauss_batch_single);
     test_ecmult_multi_batch_single(secp256k1_ecmult_strauss_batch_single);
+    while (todo > 0) {
+        todo -= test_ecmult_multi_random(scratch);
+    }
     secp256k1_scratch_destroy(&ctx->error_callback, scratch);
 
     /* Run test_ecmult_multi with space for exactly one point */
@@ -4347,7 +4745,7 @@ void test_constant_wnaf(const secp256k1_scalar *number, int w) {
         secp256k1_scalar_add(&x, &x, &t);
     }
     /* Skew num because when encoding numbers as odd we use an offset */
-    secp256k1_scalar_set_int(&scalar_skew, 1 << (skew == 2));
+    secp256k1_scalar_set_int(&scalar_skew, skew);
     secp256k1_scalar_add(&num, &num, &scalar_skew);
     CHECK(secp256k1_scalar_eq(&x, &num));
 }
@@ -4540,8 +4938,8 @@ void test_ecmult_accumulate(secp256k1_sha256* acc, const secp256k1_scalar* x, se
     }
 }
 
-void test_ecmult_constants(void) {
-    /* Test ecmult_gen for:
+void test_ecmult_constants_2bit(void) {
+    /* Using test_ecmult_accumulate, test ecmult for:
      * - For i in 0..36:
      *   - Key i
      *   - Key -i
@@ -4584,8 +4982,81 @@ void test_ecmult_constants(void) {
     secp256k1_scratch_space_destroy(ctx, scratch);
 }
 
+void test_ecmult_constants_sha(uint32_t prefix, size_t iter, const unsigned char* expected32) {
+    /* Using test_ecmult_accumulate, test ecmult for:
+     * - Key 0
+     * - Key 1
+     * - Key -1
+     * - For i in range(iter):
+     *   - Key SHA256(LE32(prefix) || LE16(i))
+     */
+    secp256k1_scalar x;
+    secp256k1_sha256 acc;
+    unsigned char b32[32];
+    unsigned char inp[6];
+    size_t i;
+    secp256k1_scratch_space *scratch = secp256k1_scratch_space_create(ctx, 65536);
+
+    inp[0] = prefix & 0xFF;
+    inp[1] = (prefix >> 8) & 0xFF;
+    inp[2] = (prefix >> 16) & 0xFF;
+    inp[3] = (prefix >> 24) & 0xFF;
+    secp256k1_sha256_initialize(&acc);
+    secp256k1_scalar_set_int(&x, 0);
+    test_ecmult_accumulate(&acc, &x, scratch);
+    secp256k1_scalar_set_int(&x, 1);
+    test_ecmult_accumulate(&acc, &x, scratch);
+    secp256k1_scalar_negate(&x, &x);
+    test_ecmult_accumulate(&acc, &x, scratch);
+
+    for (i = 0; i < iter; ++i) {
+        secp256k1_sha256 gen;
+        inp[4] = i & 0xff;
+        inp[5] = (i >> 8) & 0xff;
+        secp256k1_sha256_initialize(&gen);
+        secp256k1_sha256_write(&gen, inp, sizeof(inp));
+        secp256k1_sha256_finalize(&gen, b32);
+        secp256k1_scalar_set_b32(&x, b32, NULL);
+        test_ecmult_accumulate(&acc, &x, scratch);
+    }
+    secp256k1_sha256_finalize(&acc, b32);
+    CHECK(secp256k1_memcmp_var(b32, expected32, 32) == 0);
+
+    secp256k1_scratch_space_destroy(ctx, scratch);
+}
+
 void run_ecmult_constants(void) {
-    test_ecmult_constants();
+    /* Expected hashes of all points in the tests below. Computed using an
+     * independent implementation. */
+    static const unsigned char expected32_6bit20[32] = {
+        0x68, 0xb6, 0xed, 0x6f, 0x28, 0xca, 0xc9, 0x7f,
+        0x8e, 0x8b, 0xd6, 0xc0, 0x61, 0x79, 0x34, 0x6e,
+        0x5a, 0x8f, 0x2b, 0xbc, 0x3e, 0x1f, 0xc5, 0x2e,
+        0x2a, 0xd0, 0x45, 0x67, 0x7f, 0x95, 0x95, 0x8e
+    };
+    static const unsigned char expected32_8bit8[32] = {
+        0x8b, 0x65, 0x8e, 0xea, 0x86, 0xae, 0x3c, 0x95,
+        0x90, 0xb6, 0x77, 0xa4, 0x8c, 0x76, 0xd9, 0xec,
+        0xf5, 0xab, 0x8a, 0x2f, 0xfd, 0xdb, 0x19, 0x12,
+        0x1a, 0xee, 0xe6, 0xb7, 0x6e, 0x05, 0x3f, 0xc6
+    };
+    /* For every combination of 6 bit positions out of 256, restricted to
+     * 20-bit windows (i.e., the first and last bit position are no more than
+     * 19 bits apart), all 64 bit patterns occur in the input scalars used in
+     * this test. */
+    CONDITIONAL_TEST(1, "test_ecmult_constants_sha 1024") {
+        test_ecmult_constants_sha(4808378u, 1024, expected32_6bit20);
+    }
+
+    /* For every combination of 8 consecutive bit positions, all 256 bit
+     * patterns occur in the input scalars used in this test. */
+    CONDITIONAL_TEST(3, "test_ecmult_constants_sha 2048") {
+        test_ecmult_constants_sha(1607366309u, 2048, expected32_8bit8);
+    }
+
+    CONDITIONAL_TEST(35, "test_ecmult_constants_2bit") {
+        test_ecmult_constants_2bit();
+    }
 }
 
 void test_ecmult_gen_blind(void) {
@@ -5851,14 +6322,14 @@ static void random_ber_signature(unsigned char *sig, size_t *len, int* certainly
         /* We generate two classes of numbers: nlow==1 "low" ones (up to 32 bytes), nlow==0 "high" ones (32 bytes with 129 top bits set, or larger than 32 bytes) */
         nlow[n] = der ? 1 : (secp256k1_testrand_bits(3) != 0);
         /* The length of the number in bytes (the first byte of which will always be nonzero) */
-        nlen[n] = nlow[n] ? secp256k1_testrand_int(33) : 32 + secp256k1_testrand_int(200) * secp256k1_testrand_int(8) / 8;
+        nlen[n] = nlow[n] ? secp256k1_testrand_int(33) : 32 + secp256k1_testrand_int(200) * secp256k1_testrand_bits(3) / 8;
         CHECK(nlen[n] <= 232);
         /* The top bit of the number. */
         nhbit[n] = (nlow[n] == 0 && nlen[n] == 32) ? 1 : (nlen[n] == 0 ? 0 : secp256k1_testrand_bits(1));
         /* The top byte of the number (after the potential hardcoded 16 0xFF characters for "high" 32 bytes numbers) */
         nhbyte[n] = nlen[n] == 0 ? 0 : (nhbit[n] ? 128 + secp256k1_testrand_bits(7) : 1 + secp256k1_testrand_int(127));
         /* The number of zero bytes in front of the number (which is 0 or 1 in case of DER, otherwise we extend up to 300 bytes) */
-        nzlen[n] = der ? ((nlen[n] == 0 || nhbit[n]) ? 1 : 0) : (nlow[n] ? secp256k1_testrand_int(3) : secp256k1_testrand_int(300 - nlen[n]) * secp256k1_testrand_int(8) / 8);
+        nzlen[n] = der ? ((nlen[n] == 0 || nhbit[n]) ? 1 : 0) : (nlow[n] ? secp256k1_testrand_int(3) : secp256k1_testrand_int(300 - nlen[n]) * secp256k1_testrand_bits(3) / 8);
         if (nzlen[n] > ((nlen[n] == 0 || nhbit[n]) ? 1 : 0)) {
             *certainly_not_der = 1;
         }
@@ -5867,7 +6338,7 @@ static void random_ber_signature(unsigned char *sig, size_t *len, int* certainly
         nlenlen[n] = nlen[n] + nzlen[n] < 128 ? 0 : (nlen[n] + nzlen[n] < 256 ? 1 : 2);
         if (!der) {
             /* nlenlen[n] max 127 bytes */
-            int add = secp256k1_testrand_int(127 - nlenlen[n]) * secp256k1_testrand_int(16) * secp256k1_testrand_int(16) / 256;
+            int add = secp256k1_testrand_int(127 - nlenlen[n]) * secp256k1_testrand_bits(4) * secp256k1_testrand_bits(4) / 256;
             nlenlen[n] += add;
             if (add != 0) {
                 *certainly_not_der = 1;
@@ -5881,7 +6352,7 @@ static void random_ber_signature(unsigned char *sig, size_t *len, int* certainly
     CHECK(tlen <= 856);
 
     /* The length of the garbage inside the tuple. */
-    elen = (der || indet) ? 0 : secp256k1_testrand_int(980 - tlen) * secp256k1_testrand_int(8) / 8;
+    elen = (der || indet) ? 0 : secp256k1_testrand_int(980 - tlen) * secp256k1_testrand_bits(3) / 8;
     if (elen != 0) {
         *certainly_not_der = 1;
     }
@@ -5889,7 +6360,7 @@ static void random_ber_signature(unsigned char *sig, size_t *len, int* certainly
     CHECK(tlen <= 980);
 
     /* The length of the garbage after the end of the tuple. */
-    glen = der ? 0 : secp256k1_testrand_int(990 - tlen) * secp256k1_testrand_int(8) / 8;
+    glen = der ? 0 : secp256k1_testrand_int(990 - tlen) * secp256k1_testrand_bits(3) / 8;
     if (glen != 0) {
         *certainly_not_der = 1;
     }
@@ -5904,7 +6375,7 @@ static void random_ber_signature(unsigned char *sig, size_t *len, int* certainly
     } else {
         int tlenlen = tlen < 128 ? 0 : (tlen < 256 ? 1 : 2);
         if (!der) {
-            int add = secp256k1_testrand_int(127 - tlenlen) * secp256k1_testrand_int(16) * secp256k1_testrand_int(16) / 256;
+            int add = secp256k1_testrand_int(127 - tlenlen) * secp256k1_testrand_bits(4) * secp256k1_testrand_bits(4) / 256;
             tlenlen += add;
             if (add != 0) {
                 *certainly_not_der = 1;
@@ -6416,6 +6887,19 @@ void run_secp256k1_memczero_test(void) {
     CHECK(secp256k1_memcmp_var(buf1, buf2, sizeof(buf1)) == 0);
 }
 
+void run_secp256k1_byteorder_tests(void) {
+    const uint32_t x = 0xFF03AB45;
+    const unsigned char x_be[4] = {0xFF, 0x03, 0xAB, 0x45};
+    unsigned char buf[4];
+    uint32_t x_;
+
+    secp256k1_write_be32(buf, x);
+    CHECK(secp256k1_memcmp_var(buf, x_be, sizeof(buf)) == 0);
+
+    x_ = secp256k1_read_be32(buf);
+    CHECK(x == x_);
+}
+
 void int_cmov_test(void) {
     int r = INT_MAX;
     int a = 0;
@@ -6616,7 +7100,8 @@ int main(int argc, char **argv) {
     run_modinv_tests();
     run_inverse_tests();
 
-    run_sha256_tests();
+    run_sha256_known_output_tests();
+    run_sha256_counter_tests();
     run_hmac_sha256_tests();
     run_rfc6979_hmac_sha256_tests();
     run_tagged_sha256_tests();
@@ -6625,6 +7110,7 @@ int main(int argc, char **argv) {
     run_scalar_tests();
 
     /* field tests */
+    run_field_half();
     run_field_misc();
     run_field_convert();
     run_fe_mul();
@@ -6633,6 +7119,7 @@ int main(int argc, char **argv) {
 
     /* group tests */
     run_ge();
+    run_gej();
     run_group_decompress();
 
     /* ecmult tests */
@@ -6687,6 +7174,7 @@ int main(int argc, char **argv) {
 
     /* util tests */
     run_secp256k1_memczero_test();
+    run_secp256k1_byteorder_tests();
 
     run_cmov_tests();
 
diff --git a/src/tests_exhaustive.c b/src/tests_exhaustive.c
index 6bae7a47780c6b..6a4e2340f2856b 100644
--- a/src/tests_exhaustive.c
+++ b/src/tests_exhaustive.c
@@ -22,7 +22,8 @@
 #include "assumptions.h"
 #include "group.h"
 #include "testrand_impl.h"
-#include "ecmult_gen_prec_impl.h"
+#include "ecmult_compute_table_impl.h"
+#include "ecmult_gen_compute_table_impl.h"
 
 static int count = 2;
 
@@ -389,8 +390,9 @@ int main(int argc, char** argv) {
         printf("running tests for core %lu (out of [0..%lu])\n", (unsigned long)this_core, (unsigned long)num_cores - 1);
     }
 
-    /* Recreate the ecmult_gen table using the right generator (as selected via EXHAUSTIVE_TEST_ORDER) */
-    secp256k1_ecmult_gen_create_prec_table(&secp256k1_ecmult_gen_prec_table[0][0], &secp256k1_ge_const_g, ECMULT_GEN_PREC_BITS);
+    /* Recreate the ecmult{,_gen} tables using the right generator (as selected via EXHAUSTIVE_TEST_ORDER) */
+    secp256k1_ecmult_gen_compute_table(&secp256k1_ecmult_gen_prec_table[0][0], &secp256k1_ge_const_g, ECMULT_GEN_PREC_BITS);
+    secp256k1_ecmult_compute_two_tables(secp256k1_pre_g, secp256k1_pre_g_128, WINDOW_G, &secp256k1_ge_const_g);
 
     while (count--) {
         /* Build context */
diff --git a/src/util.h b/src/util.h
index 04227a7c9b76b8..dac86bd77f2289 100644
--- a/src/util.h
+++ b/src/util.h
@@ -173,31 +173,6 @@ static SECP256K1_INLINE void *checked_realloc(const secp256k1_callback* cb, void
 # define SECP256K1_GNUC_EXT
 #endif
 
-/* If SECP256K1_{LITTLE,BIG}_ENDIAN is not explicitly provided, infer from various other system macros. */
-#if !defined(SECP256K1_LITTLE_ENDIAN) && !defined(SECP256K1_BIG_ENDIAN)
-/* Inspired by https://github.com/rofl0r/endianness.h/blob/9853923246b065a3b52d2c43835f3819a62c7199/endianness.h#L52L73 */
-# if (defined(__BYTE_ORDER__) && defined(__ORDER_LITTLE_ENDIAN__) && __BYTE_ORDER__ == __ORDER_LITTLE_ENDIAN__) || \
-     defined(_X86_) || defined(__x86_64__) || defined(__i386__) || \
-     defined(__i486__) || defined(__i586__) || defined(__i686__) || \
-     defined(__MIPSEL) || defined(_MIPSEL) || defined(MIPSEL) || \
-     defined(__ARMEL__) || defined(__AARCH64EL__) || \
-     (defined(__LITTLE_ENDIAN__) && __LITTLE_ENDIAN__ == 1) || \
-     (defined(_LITTLE_ENDIAN) && _LITTLE_ENDIAN == 1) || \
-     defined(_M_IX86) || defined(_M_AMD64) || defined(_M_ARM) /* MSVC */
-#  define SECP256K1_LITTLE_ENDIAN
-# endif
-# if (defined(__BYTE_ORDER__) && defined(__ORDER_BIG_ENDIAN__) && __BYTE_ORDER__ == __ORDER_BIG_ENDIAN__) || \
-     defined(__MIPSEB) || defined(_MIPSEB) || defined(MIPSEB) || \
-     defined(__MICROBLAZEEB__) || defined(__ARMEB__) || defined(__AARCH64EB__) || \
-     (defined(__BIG_ENDIAN__) && __BIG_ENDIAN__ == 1) || \
-     (defined(_BIG_ENDIAN) && _BIG_ENDIAN == 1)
-#  define SECP256K1_BIG_ENDIAN
-# endif
-#endif
-#if defined(SECP256K1_LITTLE_ENDIAN) == defined(SECP256K1_BIG_ENDIAN)
-# error Please make sure that either SECP256K1_LITTLE_ENDIAN or SECP256K1_BIG_ENDIAN is set, see src/util.h.
-#endif
-
 /* Zero memory if flag == 1. Flag must be 0 or 1. Constant time. */
 static SECP256K1_INLINE void secp256k1_memczero(void *s, size_t len, int flag) {
     unsigned char *p = (unsigned char *)s;
@@ -338,4 +313,20 @@ static SECP256K1_INLINE int secp256k1_ctz64_var(uint64_t x) {
 #endif
 }
 
+/* Read a uint32_t in big endian */
+SECP256K1_INLINE static uint32_t secp256k1_read_be32(const unsigned char* p) {
+    return (uint32_t)p[0] << 24 |
+           (uint32_t)p[1] << 16 |
+           (uint32_t)p[2] << 8  |
+           (uint32_t)p[3];
+}
+
+/* Write a uint32_t in big endian */
+SECP256K1_INLINE static void secp256k1_write_be32(unsigned char* p, uint32_t x) {
+    p[3] = x;
+    p[2] = x >>  8;
+    p[1] = x >> 16;
+    p[0] = x >> 24;
+}
+
 #endif /* SECP256K1_UTIL_H */
diff --git a/src/valgrind_ctime_test.c b/src/valgrind_ctime_test.c
index ea6d4b3deb3bcb..6ff0085d348f36 100644
--- a/src/valgrind_ctime_test.c
+++ b/src/valgrind_ctime_test.c
@@ -166,7 +166,7 @@ void run_tests(secp256k1_context *ctx, unsigned char *key) {
     ret = secp256k1_keypair_create(ctx, &keypair, key);
     VALGRIND_MAKE_MEM_DEFINED(&ret, sizeof(ret));
     CHECK(ret == 1);
-    ret = secp256k1_schnorrsig_sign(ctx, sig, msg, &keypair, NULL);
+    ret = secp256k1_schnorrsig_sign32(ctx, sig, msg, &keypair, NULL);
     VALGRIND_MAKE_MEM_DEFINED(&ret, sizeof(ret));
     CHECK(ret == 1);
 #endif