Merge #1056: Save negations in var-time group addition

2f984ff Save negations in var-time group addition (Peter Dettman)

Pull request description:

  - Updated _gej_add_var, _gej_add_ge_var, _gej_add_zinv_var
  - 2 fewer _fe_negate in each method
  - Updated operation counts and standardize layout
  - Added internal benchmark for _gej_add_zinv_var

  benchmark_internal shows about 2% speedup in each method as a result (64bit).

ACKs for top commit:
  real-or-random:
    ACK 2f984ff
  jonasnick:
    ACK 2f984ff

Tree-SHA512: 01366fa23c83a8dd37c9a0a24e0acc53ce38a201607fe4da6672ea5618d82c62d1299f0e0aa50317883821539af739ea52b6561faff230c148e6fdc5bc5af30b

sage/prove_group_implementations.sage

@@ -40,29 +40,26 @@ def formula_secp256k1_gej_add_var(branch, a, b):
   s2 = s2 * a.Z
   h = -u1
   h = h + u2
-  i = -s1
-  i = i + s2
+  i = -s2
+  i = i + s1
   if branch == 2:
     r = formula_secp256k1_gej_double_var(a)
     return (constraints(), constraints(zero={h : 'h=0', i : 'i=0', a.Infinity : 'a_finite', b.Infinity : 'b_finite'}), r)
   if branch == 3:
     return (constraints(), constraints(zero={h : 'h=0', a.Infinity : 'a_finite', b.Infinity : 'b_finite'}, nonzero={i : 'i!=0'}), point_at_infinity())
-  i2 = i^2
+  t = h * b.Z
+  rz = a.Z * t
   h2 = h^2
+  h2 = -h2
   h3 = h2 * h
-  h = h * b.Z
-  rz = a.Z * h
   t = u1 * h2
-  rx = t
-  rx = rx * 2
+  rx = i^2
   rx = rx + h3
-  rx = -rx
-  rx = rx + i2
-  ry = -rx
-  ry = ry + t
-  ry = ry * i
+  rx = rx + t
+  rx = rx + t
+  t = t + rx
+  ry = t * i
   h3 = h3 * s1
-  h3 = -h3
   ry = ry + h3
   return (constraints(), constraints(zero={a.Infinity : 'a_finite', b.Infinity : 'b_finite'}, nonzero={h : 'h!=0'}), jacobianpoint(rx, ry, rz))
 
@@ -80,43 +77,41 @@ def formula_secp256k1_gej_add_ge_var(branch, a, b):
   s2 = s2 * a.Z
   h = -u1
   h = h + u2
-  i = -s1
-  i = i + s2
+  i = -s2
+  i = i + s1
   if (branch == 2):
     r = formula_secp256k1_gej_double_var(a)
     return (constraints(zero={b.Z - 1 : 'b.z=1'}), constraints(zero={a.Infinity : 'a_finite', b.Infinity : 'b_finite', h : 'h=0', i : 'i=0'}), r)
   if (branch == 3):
     return (constraints(zero={b.Z - 1 : 'b.z=1'}), constraints(zero={a.Infinity : 'a_finite', b.Infinity : 'b_finite', h : 'h=0'}, nonzero={i : 'i!=0'}), point_at_infinity())
-  i2 = i^2
-  h2 = h^2
-  h3 = h * h2
   rz = a.Z * h
+  h2 = h^2
+  h2 = -h2
+  h3 = h2 * h
   t = u1 * h2
-  rx = t
-  rx = rx * 2
+  rx = i^2
   rx = rx + h3
-  rx = -rx
-  rx = rx + i2
-  ry = -rx
-  ry = ry + t
-  ry = ry * i
+  rx = rx + t
+  rx = rx + t
+  t = t + rx
+  ry = t * i
   h3 = h3 * s1
-  h3 = -h3
   ry = ry + h3
   return (constraints(zero={b.Z - 1 : 'b.z=1'}), constraints(zero={a.Infinity : 'a_finite', b.Infinity : 'b_finite'}, nonzero={h : 'h!=0'}), jacobianpoint(rx, ry, rz))
 
 def formula_secp256k1_gej_add_zinv_var(branch, a, b):
   """libsecp256k1's secp256k1_gej_add_zinv_var"""
   bzinv = b.Z^(-1)
   if branch == 0:
-    return (constraints(), constraints(nonzero={b.Infinity : 'b_infinite'}), a)
-  if branch == 1:
+    rinf = b.Infinity
     bzinv2 = bzinv^2
     bzinv3 = bzinv2 * bzinv
     rx = b.X * bzinv2
     ry = b.Y * bzinv3
     rz = 1
-    return (constraints(), constraints(zero={b.Infinity : 'b_finite'}, nonzero={a.Infinity : 'a_infinite'}), jacobianpoint(rx, ry, rz))
+    return (constraints(), constraints(nonzero={a.Infinity : 'a_infinite'}), jacobianpoint(rx, ry, rz, rinf))
+  if branch == 1:
+    return (constraints(), constraints(zero={a.Infinity : 'a_finite'}, nonzero={b.Infinity : 'b_infinite'}), a)
   azz = a.Z * bzinv
   z12 = azz^2
   u1 = a.X
@@ -126,29 +121,25 @@ def formula_secp256k1_gej_add_zinv_var(branch, a, b):
   s2 = s2 * azz
   h = -u1
   h = h + u2
-  i = -s1
-  i = i + s2
+  i = -s2
+  i = i + s1
   if branch == 2:
     r = formula_secp256k1_gej_double_var(a)
     return (constraints(), constraints(zero={a.Infinity : 'a_finite', b.Infinity : 'b_finite', h : 'h=0', i : 'i=0'}), r)
   if branch == 3:
     return (constraints(), constraints(zero={a.Infinity : 'a_finite', b.Infinity : 'b_finite', h : 'h=0'}, nonzero={i : 'i!=0'}), point_at_infinity())
-  i2 = i^2
+  rz = a.Z * h
   h2 = h^2
-  h3 = h * h2
-  rz = a.Z
-  rz = rz * h
+  h2 = -h2
+  h3 = h2 * h
   t = u1 * h2
-  rx = t
-  rx = rx * 2
+  rx = i^2
   rx = rx + h3
-  rx = -rx
-  rx = rx + i2
-  ry = -rx
-  ry = ry + t
-  ry = ry * i
+  rx = rx + t
+  rx = rx + t
+  t = t + rx
+  ry = t * i
   h3 = h3 * s1
-  h3 = -h3
   ry = ry + h3
   return (constraints(), constraints(zero={a.Infinity : 'a_finite', b.Infinity : 'b_finite'}, nonzero={h : 'h!=0'}), jacobianpoint(rx, ry, rz))
 

src/bench_internal.c

@@ -254,6 +254,15 @@ void bench_group_add_affine_var(void* arg, int iters) {
     }
 }
 
+void bench_group_add_zinv_var(void* arg, int iters) {
+    int i;
+    bench_inv *data = (bench_inv*)arg;
+
+    for (i = 0; i < iters; i++) {
+        secp256k1_gej_add_zinv_var(&data->gej[0], &data->gej[0], &data->ge[1], &data->gej[0].y);
+    }
+}
+
 void bench_group_to_affine_var(void* arg, int iters) {
     int i;
     bench_inv *data = (bench_inv*)arg;
@@ -376,6 +385,7 @@ int main(int argc, char **argv) {
     if (d || have_flag(argc, argv, "group") || have_flag(argc, argv, "add")) run_benchmark("group_add_var", bench_group_add_var, bench_setup, NULL, &data, 10, iters*10);
     if (d || have_flag(argc, argv, "group") || have_flag(argc, argv, "add")) run_benchmark("group_add_affine", bench_group_add_affine, bench_setup, NULL, &data, 10, iters*10);
     if (d || have_flag(argc, argv, "group") || have_flag(argc, argv, "add")) run_benchmark("group_add_affine_var", bench_group_add_affine_var, bench_setup, NULL, &data, 10, iters*10);
+    if (d || have_flag(argc, argv, "group") || have_flag(argc, argv, "add")) run_benchmark("group_add_zinv_var", bench_group_add_zinv_var, bench_setup, NULL, &data, 10, iters*10);
     if (d || have_flag(argc, argv, "group") || have_flag(argc, argv, "to_affine")) run_benchmark("group_to_affine_var", bench_group_to_affine_var, bench_setup, NULL, &data, 10, iters);
 
     if (d || have_flag(argc, argv, "ecmult") || have_flag(argc, argv, "wnaf")) run_benchmark("wnaf_const", bench_wnaf_const, bench_setup, NULL, &data, 10, iters);

src/group_impl.h

@@ -330,15 +330,14 @@ static void secp256k1_gej_double_var(secp256k1_gej *r, const secp256k1_gej *a, s
 }
 
 static void secp256k1_gej_add_var(secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_gej *b, secp256k1_fe *rzr) {
-    /* Operations: 12 mul, 4 sqr, 2 normalize, 12 mul_int/add/negate */
-    secp256k1_fe z22, z12, u1, u2, s1, s2, h, i, i2, h2, h3, t;
+    /* 12 mul, 4 sqr, 11 add/negate/normalizes_to_zero (ignoring special cases) */
+    secp256k1_fe z22, z12, u1, u2, s1, s2, h, i, h2, h3, t;
 
     if (a->infinity) {
         VERIFY_CHECK(rzr == NULL);
         *r = *b;
         return;
     }
-
     if (b->infinity) {
         if (rzr != NULL) {
             secp256k1_fe_set_int(rzr, 1);
@@ -347,15 +346,14 @@ static void secp256k1_gej_add_var(secp256k1_gej *r, const secp256k1_gej *a, cons
         return;
     }
 
-    r->infinity = 0;
     secp256k1_fe_sqr(&z22, &b->z);
     secp256k1_fe_sqr(&z12, &a->z);
     secp256k1_fe_mul(&u1, &a->x, &z22);
     secp256k1_fe_mul(&u2, &b->x, &z12);
     secp256k1_fe_mul(&s1, &a->y, &z22); secp256k1_fe_mul(&s1, &s1, &b->z);
     secp256k1_fe_mul(&s2, &b->y, &z12); secp256k1_fe_mul(&s2, &s2, &a->z);
     secp256k1_fe_negate(&h, &u1, 1); secp256k1_fe_add(&h, &u2);
-    secp256k1_fe_negate(&i, &s1, 1); secp256k1_fe_add(&i, &s2);
+    secp256k1_fe_negate(&i, &s2, 1); secp256k1_fe_add(&i, &s1);
     if (secp256k1_fe_normalizes_to_zero_var(&h)) {
         if (secp256k1_fe_normalizes_to_zero_var(&i)) {
             secp256k1_gej_double_var(r, a, rzr);
@@ -367,24 +365,33 @@ static void secp256k1_gej_add_var(secp256k1_gej *r, const secp256k1_gej *a, cons
         }
         return;
     }
-    secp256k1_fe_sqr(&i2, &i);
-    secp256k1_fe_sqr(&h2, &h);
-    secp256k1_fe_mul(&h3, &h, &h2);
-    secp256k1_fe_mul(&h, &h, &b->z);
+
+    r->infinity = 0;
+    secp256k1_fe_mul(&t, &h, &b->z);
     if (rzr != NULL) {
-        *rzr = h;
+        *rzr = t;
     }
-    secp256k1_fe_mul(&r->z, &a->z, &h);
+    secp256k1_fe_mul(&r->z, &a->z, &t);
+
+    secp256k1_fe_sqr(&h2, &h);
+    secp256k1_fe_negate(&h2, &h2, 1);
+    secp256k1_fe_mul(&h3, &h2, &h);
     secp256k1_fe_mul(&t, &u1, &h2);
-    r->x = t; secp256k1_fe_mul_int(&r->x, 2); secp256k1_fe_add(&r->x, &h3); secp256k1_fe_negate(&r->x, &r->x, 3); secp256k1_fe_add(&r->x, &i2);
-    secp256k1_fe_negate(&r->y, &r->x, 5); secp256k1_fe_add(&r->y, &t); secp256k1_fe_mul(&r->y, &r->y, &i);
-    secp256k1_fe_mul(&h3, &h3, &s1); secp256k1_fe_negate(&h3, &h3, 1);
+
+    secp256k1_fe_sqr(&r->x, &i);
+    secp256k1_fe_add(&r->x, &h3);
+    secp256k1_fe_add(&r->x, &t);
+    secp256k1_fe_add(&r->x, &t);
+
+    secp256k1_fe_add(&t, &r->x);
+    secp256k1_fe_mul(&r->y, &t, &i);
+    secp256k1_fe_mul(&h3, &h3, &s1);
     secp256k1_fe_add(&r->y, &h3);
 }
 
 static void secp256k1_gej_add_ge_var(secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_ge *b, secp256k1_fe *rzr) {
-    /* 8 mul, 3 sqr, 4 normalize, 12 mul_int/add/negate */
-    secp256k1_fe z12, u1, u2, s1, s2, h, i, i2, h2, h3, t;
+    /* 8 mul, 3 sqr, 13 add/negate/normalize_weak/normalizes_to_zero (ignoring special cases) */
+    secp256k1_fe z12, u1, u2, s1, s2, h, i, h2, h3, t;
     if (a->infinity) {
         VERIFY_CHECK(rzr == NULL);
         secp256k1_gej_set_ge(r, b);
@@ -397,15 +404,14 @@ static void secp256k1_gej_add_ge_var(secp256k1_gej *r, const secp256k1_gej *a, c
         *r = *a;
         return;
     }
-    r->infinity = 0;
 
     secp256k1_fe_sqr(&z12, &a->z);
     u1 = a->x; secp256k1_fe_normalize_weak(&u1);
     secp256k1_fe_mul(&u2, &b->x, &z12);
     s1 = a->y; secp256k1_fe_normalize_weak(&s1);
     secp256k1_fe_mul(&s2, &b->y, &z12); secp256k1_fe_mul(&s2, &s2, &a->z);
     secp256k1_fe_negate(&h, &u1, 1); secp256k1_fe_add(&h, &u2);
-    secp256k1_fe_negate(&i, &s1, 1); secp256k1_fe_add(&i, &s2);
+    secp256k1_fe_negate(&i, &s2, 1); secp256k1_fe_add(&i, &s1);
     if (secp256k1_fe_normalizes_to_zero_var(&h)) {
         if (secp256k1_fe_normalizes_to_zero_var(&i)) {
             secp256k1_gej_double_var(r, a, rzr);
@@ -417,28 +423,33 @@ static void secp256k1_gej_add_ge_var(secp256k1_gej *r, const secp256k1_gej *a, c
         }
         return;
     }
-    secp256k1_fe_sqr(&i2, &i);
-    secp256k1_fe_sqr(&h2, &h);
-    secp256k1_fe_mul(&h3, &h, &h2);
+
+    r->infinity = 0;
     if (rzr != NULL) {
         *rzr = h;
     }
     secp256k1_fe_mul(&r->z, &a->z, &h);
+
+    secp256k1_fe_sqr(&h2, &h);
+    secp256k1_fe_negate(&h2, &h2, 1);
+    secp256k1_fe_mul(&h3, &h2, &h);
     secp256k1_fe_mul(&t, &u1, &h2);
-    r->x = t; secp256k1_fe_mul_int(&r->x, 2); secp256k1_fe_add(&r->x, &h3); secp256k1_fe_negate(&r->x, &r->x, 3); secp256k1_fe_add(&r->x, &i2);
-    secp256k1_fe_negate(&r->y, &r->x, 5); secp256k1_fe_add(&r->y, &t); secp256k1_fe_mul(&r->y, &r->y, &i);
-    secp256k1_fe_mul(&h3, &h3, &s1); secp256k1_fe_negate(&h3, &h3, 1);
+
+    secp256k1_fe_sqr(&r->x, &i);
+    secp256k1_fe_add(&r->x, &h3);
+    secp256k1_fe_add(&r->x, &t);
+    secp256k1_fe_add(&r->x, &t);
+
+    secp256k1_fe_add(&t, &r->x);
+    secp256k1_fe_mul(&r->y, &t, &i);
+    secp256k1_fe_mul(&h3, &h3, &s1);
     secp256k1_fe_add(&r->y, &h3);
 }
 
 static void secp256k1_gej_add_zinv_var(secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_ge *b, const secp256k1_fe *bzinv) {
-    /* 9 mul, 3 sqr, 4 normalize, 12 mul_int/add/negate */
-    secp256k1_fe az, z12, u1, u2, s1, s2, h, i, i2, h2, h3, t;
+    /* 9 mul, 3 sqr, 13 add/negate/normalize_weak/normalizes_to_zero (ignoring special cases) */
+    secp256k1_fe az, z12, u1, u2, s1, s2, h, i, h2, h3, t;
 
-    if (b->infinity) {
-        *r = *a;
-        return;
-    }
     if (a->infinity) {
         secp256k1_fe bzinv2, bzinv3;
         r->infinity = b->infinity;
@@ -449,7 +460,10 @@ static void secp256k1_gej_add_zinv_var(secp256k1_gej *r, const secp256k1_gej *a,
         secp256k1_fe_set_int(&r->z, 1);
         return;
     }
-    r->infinity = 0;
+    if (b->infinity) {
+        *r = *a;
+        return;
+    }
 
     /** We need to calculate (rx,ry,rz) = (ax,ay,az) + (bx,by,1/bzinv). Due to
      *  secp256k1's isomorphism we can multiply the Z coordinates on both sides
@@ -467,7 +481,7 @@ static void secp256k1_gej_add_zinv_var(secp256k1_gej *r, const secp256k1_gej *a,
     s1 = a->y; secp256k1_fe_normalize_weak(&s1);
     secp256k1_fe_mul(&s2, &b->y, &z12); secp256k1_fe_mul(&s2, &s2, &az);
     secp256k1_fe_negate(&h, &u1, 1); secp256k1_fe_add(&h, &u2);
-    secp256k1_fe_negate(&i, &s1, 1); secp256k1_fe_add(&i, &s2);
+    secp256k1_fe_negate(&i, &s2, 1); secp256k1_fe_add(&i, &s1);
     if (secp256k1_fe_normalizes_to_zero_var(&h)) {
         if (secp256k1_fe_normalizes_to_zero_var(&i)) {
             secp256k1_gej_double_var(r, a, NULL);
@@ -476,14 +490,23 @@ static void secp256k1_gej_add_zinv_var(secp256k1_gej *r, const secp256k1_gej *a,
         }
         return;
     }
-    secp256k1_fe_sqr(&i2, &i);
+
+    r->infinity = 0;
+    secp256k1_fe_mul(&r->z, &a->z, &h);
+
     secp256k1_fe_sqr(&h2, &h);
-    secp256k1_fe_mul(&h3, &h, &h2);
-    r->z = a->z; secp256k1_fe_mul(&r->z, &r->z, &h);
+    secp256k1_fe_negate(&h2, &h2, 1);
+    secp256k1_fe_mul(&h3, &h2, &h);
     secp256k1_fe_mul(&t, &u1, &h2);
-    r->x = t; secp256k1_fe_mul_int(&r->x, 2); secp256k1_fe_add(&r->x, &h3); secp256k1_fe_negate(&r->x, &r->x, 3); secp256k1_fe_add(&r->x, &i2);
-    secp256k1_fe_negate(&r->y, &r->x, 5); secp256k1_fe_add(&r->y, &t); secp256k1_fe_mul(&r->y, &r->y, &i);
-    secp256k1_fe_mul(&h3, &h3, &s1); secp256k1_fe_negate(&h3, &h3, 1);
+
+    secp256k1_fe_sqr(&r->x, &i);
+    secp256k1_fe_add(&r->x, &h3);
+    secp256k1_fe_add(&r->x, &t);
+    secp256k1_fe_add(&r->x, &t);
+
+    secp256k1_fe_add(&t, &r->x);
+    secp256k1_fe_mul(&r->y, &t, &i);
+    secp256k1_fe_mul(&h3, &h3, &s1);
     secp256k1_fe_add(&r->y, &h3);
 }