Optimization: only do 59 hddivsteps per iteration instead of 62

Author: sipa

src/modinv64_impl.h

@@ -145,7 +145,8 @@ typedef struct {
     int64_t u, v, q, r;
 } secp256k1_modinv64_trans2x2;
 
-/* Compute the transition matrix and zeta for 62 divsteps (where zeta=-(delta+1/2)).
+/* Compute the transition matrix and eta for 59 divsteps (where zeta=-(delta+1/2)).
+ * Note that the transformation matrix is scaled by 2^62 and not 2^59.
  *
  * Input:  zeta: initial zeta
  *         f0:   bottom limb of initial f
@@ -155,18 +156,19 @@ typedef struct {
  *
  * Implements the divsteps_n_matrix function from the explanation.
  */
-static int64_t secp256k1_modinv64_divsteps_62(int64_t zeta, uint64_t f0, uint64_t g0, secp256k1_modinv64_trans2x2 *t) {
+static int64_t secp256k1_modinv64_divsteps_59(int64_t zeta, uint64_t f0, uint64_t g0, secp256k1_modinv64_trans2x2 *t) {
     /* u,v,q,r are the elements of the transformation matrix being built up,
-     * starting with the identity matrix. Semantically they are signed integers
+     * starting with the identity matrix times 8 (because the caller expects
+     * a result scaled by 2^62). Semantically they are signed integers
      * in range [-2^62,2^62], but here represented as unsigned mod 2^64. This
      * permits left shifting (which is UB for negative numbers). The range
      * being inside [-2^63,2^63) means that casting to signed works correctly.
      */
-    uint64_t u = 1, v = 0, q = 0, r = 1;
+    uint64_t u = 8, v = 0, q = 0, r = 8;
     uint64_t c1, c2, f = f0, g = g0, x, y, z;
     int i;
 
-    for (i = 0; i < 62; ++i) {
+    for (i = 3; i < 62; ++i) {
         VERIFY_CHECK((f & 1) == 1); /* f must always be odd */
         VERIFY_CHECK((u * f0 + v * g0) == f << i);
         VERIFY_CHECK((q * f0 + r * g0) == g << i);
@@ -193,8 +195,8 @@ static int64_t secp256k1_modinv64_divsteps_62(int64_t zeta, uint64_t f0, uint64_
         g >>= 1;
         u <<= 1;
         v <<= 1;
-        /* Bounds on zeta that follow from the bounds on iteration count (max 10*62 divsteps). */
-        VERIFY_CHECK(zeta >= -621 && zeta <= 621);
+        /* Bounds on zeta that follow from the bounds on iteration count (max 10*59 divsteps). */
+        VERIFY_CHECK(zeta >= -591 && zeta <= 591);
     }
     /* Return data in t and return value. */
     t->u = (int64_t)u;
@@ -204,8 +206,10 @@ static int64_t secp256k1_modinv64_divsteps_62(int64_t zeta, uint64_t f0, uint64_
     /* The determinant of t must be a power of two. This guarantees that multiplication with t
      * does not change the gcd of f and g, apart from adding a power-of-2 factor to it (which
      * will be divided out again). As each divstep's individual matrix has determinant 2, the
-     * aggregate of 62 of them will have determinant 2^62. */
-    VERIFY_CHECK((int128_t)t->u * t->r - (int128_t)t->v * t->q == ((int128_t)1) << 62);
+     * aggregate of 59 of them will have determinant 2^59. Multiplying with the initial
+     * 8*identity (which has determinant 2^6) means the overall outputs has determinant
+     * 2^65. */
+    VERIFY_CHECK((int128_t)t->u * t->r - (int128_t)t->v * t->q == ((int128_t)1) << 65);
     return zeta;
 }
 
@@ -290,7 +294,7 @@ static int64_t secp256k1_modinv64_divsteps_62_var(int64_t eta, uint64_t f0, uint
     return eta;
 }
 
-/* Compute (t/2^62) * [d, e] mod modulus, where t is a transition matrix for 62 divsteps.
+/* Compute (t/2^62) * [d, e] mod modulus, where t is a transition matrix scaled by 2^62.
  *
  * On input and output, d and e are in range (-2*modulus,modulus). All output limbs will be in range
  * (-2^62,2^62).
@@ -376,7 +380,7 @@ static void secp256k1_modinv64_update_de_62(secp256k1_modinv64_signed62 *d, secp
 #endif
 }
 
-/* Compute (t/2^62) * [f, g], where t is a transition matrix for 62 divsteps.
+/* Compute (t/2^62) * [f, g], where t is a transition matrix scaled by 2^62.
  *
  * This implements the update_fg function from the explanation.
  */
@@ -463,11 +467,11 @@ static void secp256k1_modinv64(secp256k1_modinv64_signed62 *x, const secp256k1_m
     int i;
     int64_t zeta = -1; /* zeta = -(delta+1/2); delta starts at 1/2. */
 
-    /* Do 10 iterations of 62 divsteps each = 620 divsteps. 590 suffices for 256-bit inputs. */
+    /* Do 10 iterations of 59 divsteps each = 590 divsteps. This suffices for 256-bit inputs. */
     for (i = 0; i < 10; ++i) {
-        /* Compute transition matrix and new zeta after 62 divsteps. */
+        /* Compute transition matrix and new zeta after 59 divsteps. */
         secp256k1_modinv64_trans2x2 t;
-        zeta = secp256k1_modinv64_divsteps_62(zeta, f.v[0], g.v[0], &t);
+        zeta = secp256k1_modinv64_divsteps_59(zeta, f.v[0], g.v[0], &t);
         /* Update d,e using that transition matrix. */
         secp256k1_modinv64_update_de_62(&d, &e, &t, modinfo);
         /* Update f,g using that transition matrix. */