diff --git a/src/scalar_4x64_impl.h b/src/scalar_4x64_impl.h
index 3eaa0418c6237..6ba38e25e7348 100644
--- a/src/scalar_4x64_impl.h
+++ b/src/scalar_4x64_impl.h
@@ -955,4 +955,183 @@ static SECP256K1_INLINE void secp256k1_scalar_cmov(secp256k1_scalar *r, const se
     r->d[3] = (r->d[3] & mask0) | (a->d[3] & mask1);
 }
 
+static void secp256k1_scalar_inverse(secp256k1_scalar *r, const secp256k1_scalar *x) {
+    secp256k1_scalar *t;
+    int i;
+    /* First compute xN as x ^ (2^N - 1) for some values of N,
+     * and uM as x ^ M for some values of M. */
+    secp256k1_scalar x2, x3, x6, x8, x14, x28, x56, x112, x126;
+    secp256k1_scalar u2, u5, u9, u11, u13;
+
+    secp256k1_scalar_sqr(&u2, x);
+    secp256k1_scalar_mul(&x2, &u2,  x);
+    secp256k1_scalar_mul(&u5, &u2, &x2);
+    secp256k1_scalar_mul(&x3, &u5,  &u2);
+    secp256k1_scalar_mul(&u9, &x3, &u2);
+    secp256k1_scalar_mul(&u11, &u9, &u2);
+    secp256k1_scalar_mul(&u13, &u11, &u2);
+
+    secp256k1_scalar_sqr(&x6, &u13);
+    secp256k1_scalar_sqr(&x6, &x6);
+    secp256k1_scalar_mul(&x6, &x6, &u11);
+
+    secp256k1_scalar_sqr(&x8, &x6);
+    secp256k1_scalar_sqr(&x8, &x8);
+    secp256k1_scalar_mul(&x8, &x8,  &x2);
+
+    secp256k1_scalar_sqr(&x14, &x8);
+    for (i = 0; i < 5; i++) {
+        secp256k1_scalar_sqr(&x14, &x14);
+    }
+    secp256k1_scalar_mul(&x14, &x14, &x6);
+
+    secp256k1_scalar_sqr(&x28, &x14);
+    for (i = 0; i < 13; i++) {
+        secp256k1_scalar_sqr(&x28, &x28);
+    }
+    secp256k1_scalar_mul(&x28, &x28, &x14);
+
+    secp256k1_scalar_sqr(&x56, &x28);
+    for (i = 0; i < 27; i++) {
+        secp256k1_scalar_sqr(&x56, &x56);
+    }
+    secp256k1_scalar_mul(&x56, &x56, &x28);
+
+    secp256k1_scalar_sqr(&x112, &x56);
+    for (i = 0; i < 55; i++) {
+        secp256k1_scalar_sqr(&x112, &x112);
+    }
+    secp256k1_scalar_mul(&x112, &x112, &x56);
+
+    secp256k1_scalar_sqr(&x126, &x112);
+    for (i = 0; i < 13; i++) {
+        secp256k1_scalar_sqr(&x126, &x126);
+    }
+    secp256k1_scalar_mul(&x126, &x126, &x14);
+
+    /* Then accumulate the final result (t starts at x126). */
+    t = &x126;
+    for (i = 0; i < 3; i++) {
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &u5); /* 101 */
+    for (i = 0; i < 4; i++) { /* 0 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &x3); /* 111 */
+    for (i = 0; i < 4; i++) { /* 0 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &u5); /* 101 */
+    for (i = 0; i < 5; i++) { /* 0 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &u11); /* 1011 */
+    for (i = 0; i < 4; i++) {
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &u11); /* 1011 */
+    for (i = 0; i < 4; i++) { /* 0 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &x3); /* 111 */
+    for (i = 0; i < 5; i++) { /* 00 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &x3); /* 111 */
+    for (i = 0; i < 6; i++) { /* 00 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &u13); /* 1101 */
+    for (i = 0; i < 4; i++) { /* 0 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &u5); /* 101 */
+    for (i = 0; i < 3; i++) {
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &x3); /* 111 */
+    for (i = 0; i < 5; i++) { /* 0 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &u9); /* 1001 */
+    for (i = 0; i < 6; i++) { /* 000 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &u5); /* 101 */
+    for (i = 0; i < 10; i++) { /* 0000000 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &x3); /* 111 */
+    for (i = 0; i < 4; i++) { /* 0 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &x3); /* 111 */
+    for (i = 0; i < 9; i++) { /* 0 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &x8); /* 11111111 */
+    for (i = 0; i < 5; i++) { /* 0 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &u9); /* 1001 */
+    for (i = 0; i < 6; i++) { /* 00 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &u11); /* 1011 */
+    for (i = 0; i < 4; i++) {
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &u13); /* 1101 */
+    for (i = 0; i < 5; i++) {
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &x2); /* 11 */
+    for (i = 0; i < 6; i++) { /* 00 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &u13); /* 1101 */
+    for (i = 0; i < 10; i++) { /* 000000 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &u13); /* 1101 */
+    for (i = 0; i < 4; i++) {
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &u9); /* 1001 */
+    for (i = 0; i < 6; i++) { /* 00000 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, x); /* 1 */
+    for (i = 0; i < 8; i++) { /* 00 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(r, t, &x6); /* 111111 */
+}
+
+static void secp256k1_scalar_inverse_var(secp256k1_scalar *r, const secp256k1_scalar *x) {
+#if defined(USE_SCALAR_INV_BUILTIN)
+    secp256k1_scalar_inverse(r, x);
+#elif defined(USE_SCALAR_INV_NUM)
+    unsigned char b[32];
+    secp256k1_num n, m;
+    secp256k1_scalar t = *x;
+    secp256k1_scalar_get_b32(b, &t);
+    secp256k1_num_set_bin(&n, b, 32);
+    secp256k1_scalar_order_get_num(&m);
+    secp256k1_num_mod_inverse(&n, &n, &m);
+    secp256k1_num_get_bin(b, 32, &n);
+    secp256k1_scalar_set_b32(r, b, NULL);
+    /* Verify that the inverse was computed correctly, without GMP code. */
+    secp256k1_scalar_mul(&t, &t, r);
+    CHECK(secp256k1_scalar_is_one(&t));
+#else
+#error "Please select scalar inverse implementation"
+#endif
+}
+
+SECP256K1_INLINE static int secp256k1_scalar_is_even(const secp256k1_scalar *a) {
+    return !(a->d[0] & 1);
+}
+
 #endif /* SECP256K1_SCALAR_REPR_IMPL_H */
diff --git a/src/scalar_8x32_impl.h b/src/scalar_8x32_impl.h
index bf98e01d76284..53b8d4ec4b060 100644
--- a/src/scalar_8x32_impl.h
+++ b/src/scalar_8x32_impl.h
@@ -731,4 +731,183 @@ static SECP256K1_INLINE void secp256k1_scalar_cmov(secp256k1_scalar *r, const se
     r->d[7] = (r->d[7] & mask0) | (a->d[7] & mask1);
 }
 
+static void secp256k1_scalar_inverse(secp256k1_scalar *r, const secp256k1_scalar *x) {
+    secp256k1_scalar *t;
+    int i;
+    /* First compute xN as x ^ (2^N - 1) for some values of N,
+     * and uM as x ^ M for some values of M. */
+    secp256k1_scalar x2, x3, x6, x8, x14, x28, x56, x112, x126;
+    secp256k1_scalar u2, u5, u9, u11, u13;
+
+    secp256k1_scalar_sqr(&u2, x);
+    secp256k1_scalar_mul(&x2, &u2,  x);
+    secp256k1_scalar_mul(&u5, &u2, &x2);
+    secp256k1_scalar_mul(&x3, &u5,  &u2);
+    secp256k1_scalar_mul(&u9, &x3, &u2);
+    secp256k1_scalar_mul(&u11, &u9, &u2);
+    secp256k1_scalar_mul(&u13, &u11, &u2);
+
+    secp256k1_scalar_sqr(&x6, &u13);
+    secp256k1_scalar_sqr(&x6, &x6);
+    secp256k1_scalar_mul(&x6, &x6, &u11);
+
+    secp256k1_scalar_sqr(&x8, &x6);
+    secp256k1_scalar_sqr(&x8, &x8);
+    secp256k1_scalar_mul(&x8, &x8,  &x2);
+
+    secp256k1_scalar_sqr(&x14, &x8);
+    for (i = 0; i < 5; i++) {
+        secp256k1_scalar_sqr(&x14, &x14);
+    }
+    secp256k1_scalar_mul(&x14, &x14, &x6);
+
+    secp256k1_scalar_sqr(&x28, &x14);
+    for (i = 0; i < 13; i++) {
+        secp256k1_scalar_sqr(&x28, &x28);
+    }
+    secp256k1_scalar_mul(&x28, &x28, &x14);
+
+    secp256k1_scalar_sqr(&x56, &x28);
+    for (i = 0; i < 27; i++) {
+        secp256k1_scalar_sqr(&x56, &x56);
+    }
+    secp256k1_scalar_mul(&x56, &x56, &x28);
+
+    secp256k1_scalar_sqr(&x112, &x56);
+    for (i = 0; i < 55; i++) {
+        secp256k1_scalar_sqr(&x112, &x112);
+    }
+    secp256k1_scalar_mul(&x112, &x112, &x56);
+
+    secp256k1_scalar_sqr(&x126, &x112);
+    for (i = 0; i < 13; i++) {
+        secp256k1_scalar_sqr(&x126, &x126);
+    }
+    secp256k1_scalar_mul(&x126, &x126, &x14);
+
+    /* Then accumulate the final result (t starts at x126). */
+    t = &x126;
+    for (i = 0; i < 3; i++) {
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &u5); /* 101 */
+    for (i = 0; i < 4; i++) { /* 0 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &x3); /* 111 */
+    for (i = 0; i < 4; i++) { /* 0 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &u5); /* 101 */
+    for (i = 0; i < 5; i++) { /* 0 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &u11); /* 1011 */
+    for (i = 0; i < 4; i++) {
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &u11); /* 1011 */
+    for (i = 0; i < 4; i++) { /* 0 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &x3); /* 111 */
+    for (i = 0; i < 5; i++) { /* 00 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &x3); /* 111 */
+    for (i = 0; i < 6; i++) { /* 00 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &u13); /* 1101 */
+    for (i = 0; i < 4; i++) { /* 0 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &u5); /* 101 */
+    for (i = 0; i < 3; i++) {
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &x3); /* 111 */
+    for (i = 0; i < 5; i++) { /* 0 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &u9); /* 1001 */
+    for (i = 0; i < 6; i++) { /* 000 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &u5); /* 101 */
+    for (i = 0; i < 10; i++) { /* 0000000 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &x3); /* 111 */
+    for (i = 0; i < 4; i++) { /* 0 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &x3); /* 111 */
+    for (i = 0; i < 9; i++) { /* 0 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &x8); /* 11111111 */
+    for (i = 0; i < 5; i++) { /* 0 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &u9); /* 1001 */
+    for (i = 0; i < 6; i++) { /* 00 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &u11); /* 1011 */
+    for (i = 0; i < 4; i++) {
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &u13); /* 1101 */
+    for (i = 0; i < 5; i++) {
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &x2); /* 11 */
+    for (i = 0; i < 6; i++) { /* 00 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &u13); /* 1101 */
+    for (i = 0; i < 10; i++) { /* 000000 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &u13); /* 1101 */
+    for (i = 0; i < 4; i++) {
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, &u9); /* 1001 */
+    for (i = 0; i < 6; i++) { /* 00000 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(t, t, x); /* 1 */
+    for (i = 0; i < 8; i++) { /* 00 */
+        secp256k1_scalar_sqr(t, t);
+    }
+    secp256k1_scalar_mul(r, t, &x6); /* 111111 */
+}
+
+static void secp256k1_scalar_inverse_var(secp256k1_scalar *r, const secp256k1_scalar *x) {
+#if defined(USE_SCALAR_INV_BUILTIN)
+    secp256k1_scalar_inverse(r, x);
+#elif defined(USE_SCALAR_INV_NUM)
+    unsigned char b[32];
+    secp256k1_num n, m;
+    secp256k1_scalar t = *x;
+    secp256k1_scalar_get_b32(b, &t);
+    secp256k1_num_set_bin(&n, b, 32);
+    secp256k1_scalar_order_get_num(&m);
+    secp256k1_num_mod_inverse(&n, &n, &m);
+    secp256k1_num_get_bin(b, 32, &n);
+    secp256k1_scalar_set_b32(r, b, NULL);
+    /* Verify that the inverse was computed correctly, without GMP code. */
+    secp256k1_scalar_mul(&t, &t, r);
+    CHECK(secp256k1_scalar_is_one(&t));
+#else
+#error "Please select scalar inverse implementation"
+#endif
+}
+
+SECP256K1_INLINE static int secp256k1_scalar_is_even(const secp256k1_scalar *a) {
+    return !(a->d[0] & 1);
+}
+
 #endif /* SECP256K1_SCALAR_REPR_IMPL_H */
diff --git a/src/scalar_impl.h b/src/scalar_impl.h
index 61c1fbd58e28c..b328afdb9b937 100644
--- a/src/scalar_impl.h
+++ b/src/scalar_impl.h
@@ -65,197 +65,6 @@ static int secp256k1_scalar_set_b32_seckey(secp256k1_scalar *r, const unsigned c
     return (!overflow) & (!secp256k1_scalar_is_zero(r));
 }
 
-static void secp256k1_scalar_inverse(secp256k1_scalar *r, const secp256k1_scalar *x) {
-#if defined(EXHAUSTIVE_TEST_ORDER)
-    int i;
-    *r = 0;
-    for (i = 0; i < EXHAUSTIVE_TEST_ORDER; i++)
-        if ((i * *x) % EXHAUSTIVE_TEST_ORDER == 1)
-            *r = i;
-    /* If this VERIFY_CHECK triggers we were given a noninvertible scalar (and thus
-     * have a composite group order; fix it in exhaustive_tests.c). */
-    VERIFY_CHECK(*r != 0);
-}
-#else
-    secp256k1_scalar *t;
-    int i;
-    /* First compute xN as x ^ (2^N - 1) for some values of N,
-     * and uM as x ^ M for some values of M. */
-    secp256k1_scalar x2, x3, x6, x8, x14, x28, x56, x112, x126;
-    secp256k1_scalar u2, u5, u9, u11, u13;
-
-    secp256k1_scalar_sqr(&u2, x);
-    secp256k1_scalar_mul(&x2, &u2,  x);
-    secp256k1_scalar_mul(&u5, &u2, &x2);
-    secp256k1_scalar_mul(&x3, &u5,  &u2);
-    secp256k1_scalar_mul(&u9, &x3, &u2);
-    secp256k1_scalar_mul(&u11, &u9, &u2);
-    secp256k1_scalar_mul(&u13, &u11, &u2);
-
-    secp256k1_scalar_sqr(&x6, &u13);
-    secp256k1_scalar_sqr(&x6, &x6);
-    secp256k1_scalar_mul(&x6, &x6, &u11);
-
-    secp256k1_scalar_sqr(&x8, &x6);
-    secp256k1_scalar_sqr(&x8, &x8);
-    secp256k1_scalar_mul(&x8, &x8,  &x2);
-
-    secp256k1_scalar_sqr(&x14, &x8);
-    for (i = 0; i < 5; i++) {
-        secp256k1_scalar_sqr(&x14, &x14);
-    }
-    secp256k1_scalar_mul(&x14, &x14, &x6);
-
-    secp256k1_scalar_sqr(&x28, &x14);
-    for (i = 0; i < 13; i++) {
-        secp256k1_scalar_sqr(&x28, &x28);
-    }
-    secp256k1_scalar_mul(&x28, &x28, &x14);
-
-    secp256k1_scalar_sqr(&x56, &x28);
-    for (i = 0; i < 27; i++) {
-        secp256k1_scalar_sqr(&x56, &x56);
-    }
-    secp256k1_scalar_mul(&x56, &x56, &x28);
-
-    secp256k1_scalar_sqr(&x112, &x56);
-    for (i = 0; i < 55; i++) {
-        secp256k1_scalar_sqr(&x112, &x112);
-    }
-    secp256k1_scalar_mul(&x112, &x112, &x56);
-
-    secp256k1_scalar_sqr(&x126, &x112);
-    for (i = 0; i < 13; i++) {
-        secp256k1_scalar_sqr(&x126, &x126);
-    }
-    secp256k1_scalar_mul(&x126, &x126, &x14);
-
-    /* Then accumulate the final result (t starts at x126). */
-    t = &x126;
-    for (i = 0; i < 3; i++) {
-        secp256k1_scalar_sqr(t, t);
-    }
-    secp256k1_scalar_mul(t, t, &u5); /* 101 */
-    for (i = 0; i < 4; i++) { /* 0 */
-        secp256k1_scalar_sqr(t, t);
-    }
-    secp256k1_scalar_mul(t, t, &x3); /* 111 */
-    for (i = 0; i < 4; i++) { /* 0 */
-        secp256k1_scalar_sqr(t, t);
-    }
-    secp256k1_scalar_mul(t, t, &u5); /* 101 */
-    for (i = 0; i < 5; i++) { /* 0 */
-        secp256k1_scalar_sqr(t, t);
-    }
-    secp256k1_scalar_mul(t, t, &u11); /* 1011 */
-    for (i = 0; i < 4; i++) {
-        secp256k1_scalar_sqr(t, t);
-    }
-    secp256k1_scalar_mul(t, t, &u11); /* 1011 */
-    for (i = 0; i < 4; i++) { /* 0 */
-        secp256k1_scalar_sqr(t, t);
-    }
-    secp256k1_scalar_mul(t, t, &x3); /* 111 */
-    for (i = 0; i < 5; i++) { /* 00 */
-        secp256k1_scalar_sqr(t, t);
-    }
-    secp256k1_scalar_mul(t, t, &x3); /* 111 */
-    for (i = 0; i < 6; i++) { /* 00 */
-        secp256k1_scalar_sqr(t, t);
-    }
-    secp256k1_scalar_mul(t, t, &u13); /* 1101 */
-    for (i = 0; i < 4; i++) { /* 0 */
-        secp256k1_scalar_sqr(t, t);
-    }
-    secp256k1_scalar_mul(t, t, &u5); /* 101 */
-    for (i = 0; i < 3; i++) {
-        secp256k1_scalar_sqr(t, t);
-    }
-    secp256k1_scalar_mul(t, t, &x3); /* 111 */
-    for (i = 0; i < 5; i++) { /* 0 */
-        secp256k1_scalar_sqr(t, t);
-    }
-    secp256k1_scalar_mul(t, t, &u9); /* 1001 */
-    for (i = 0; i < 6; i++) { /* 000 */
-        secp256k1_scalar_sqr(t, t);
-    }
-    secp256k1_scalar_mul(t, t, &u5); /* 101 */
-    for (i = 0; i < 10; i++) { /* 0000000 */
-        secp256k1_scalar_sqr(t, t);
-    }
-    secp256k1_scalar_mul(t, t, &x3); /* 111 */
-    for (i = 0; i < 4; i++) { /* 0 */
-        secp256k1_scalar_sqr(t, t);
-    }
-    secp256k1_scalar_mul(t, t, &x3); /* 111 */
-    for (i = 0; i < 9; i++) { /* 0 */
-        secp256k1_scalar_sqr(t, t);
-    }
-    secp256k1_scalar_mul(t, t, &x8); /* 11111111 */
-    for (i = 0; i < 5; i++) { /* 0 */
-        secp256k1_scalar_sqr(t, t);
-    }
-    secp256k1_scalar_mul(t, t, &u9); /* 1001 */
-    for (i = 0; i < 6; i++) { /* 00 */
-        secp256k1_scalar_sqr(t, t);
-    }
-    secp256k1_scalar_mul(t, t, &u11); /* 1011 */
-    for (i = 0; i < 4; i++) {
-        secp256k1_scalar_sqr(t, t);
-    }
-    secp256k1_scalar_mul(t, t, &u13); /* 1101 */
-    for (i = 0; i < 5; i++) {
-        secp256k1_scalar_sqr(t, t);
-    }
-    secp256k1_scalar_mul(t, t, &x2); /* 11 */
-    for (i = 0; i < 6; i++) { /* 00 */
-        secp256k1_scalar_sqr(t, t);
-    }
-    secp256k1_scalar_mul(t, t, &u13); /* 1101 */
-    for (i = 0; i < 10; i++) { /* 000000 */
-        secp256k1_scalar_sqr(t, t);
-    }
-    secp256k1_scalar_mul(t, t, &u13); /* 1101 */
-    for (i = 0; i < 4; i++) {
-        secp256k1_scalar_sqr(t, t);
-    }
-    secp256k1_scalar_mul(t, t, &u9); /* 1001 */
-    for (i = 0; i < 6; i++) { /* 00000 */
-        secp256k1_scalar_sqr(t, t);
-    }
-    secp256k1_scalar_mul(t, t, x); /* 1 */
-    for (i = 0; i < 8; i++) { /* 00 */
-        secp256k1_scalar_sqr(t, t);
-    }
-    secp256k1_scalar_mul(r, t, &x6); /* 111111 */
-}
-
-SECP256K1_INLINE static int secp256k1_scalar_is_even(const secp256k1_scalar *a) {
-    return !(a->d[0] & 1);
-}
-#endif
-
-static void secp256k1_scalar_inverse_var(secp256k1_scalar *r, const secp256k1_scalar *x) {
-#if defined(USE_SCALAR_INV_BUILTIN)
-    secp256k1_scalar_inverse(r, x);
-#elif defined(USE_SCALAR_INV_NUM)
-    unsigned char b[32];
-    secp256k1_num n, m;
-    secp256k1_scalar t = *x;
-    secp256k1_scalar_get_b32(b, &t);
-    secp256k1_num_set_bin(&n, b, 32);
-    secp256k1_scalar_order_get_num(&m);
-    secp256k1_num_mod_inverse(&n, &n, &m);
-    secp256k1_num_get_bin(b, 32, &n);
-    secp256k1_scalar_set_b32(r, b, NULL);
-    /* Verify that the inverse was computed correctly, without GMP code. */
-    secp256k1_scalar_mul(&t, &t, r);
-    CHECK(secp256k1_scalar_is_one(&t));
-#else
-#error "Please select scalar inverse implementation"
-#endif
-}
-
 /* These parameters are generated using sage/gen_exhaustive_groups.sage. */
 #if defined(EXHAUSTIVE_TEST_ORDER)
 #  if EXHAUSTIVE_TEST_ORDER == 13
diff --git a/src/scalar_low_impl.h b/src/scalar_low_impl.h
index 98ffd1536eb2d..eff270720a80e 100644
--- a/src/scalar_low_impl.h
+++ b/src/scalar_low_impl.h
@@ -125,4 +125,19 @@ static SECP256K1_INLINE void secp256k1_scalar_cmov(secp256k1_scalar *r, const se
     *r = (*r & mask0) | (*a & mask1);
 }
 
+static void secp256k1_scalar_inverse(secp256k1_scalar *r, const secp256k1_scalar *x) {
+    int i;
+    *r = 0;
+    for (i = 0; i < EXHAUSTIVE_TEST_ORDER; i++)
+        if ((i * *x) % EXHAUSTIVE_TEST_ORDER == 1)
+            *r = i;
+    /* If this VERIFY_CHECK triggers we were given a noninvertible scalar (and thus
+     * have a composite group order; fix it in exhaustive_tests.c). */
+    VERIFY_CHECK(*r != 0);
+}
+
+static void secp256k1_scalar_inverse_var(secp256k1_scalar *r, const secp256k1_scalar *x) {
+    secp256k1_scalar_inverse(r, x);
+}
+
 #endif /* SECP256K1_SCALAR_REPR_IMPL_H */