Use modified divsteps with initial delta=1/2 for constant-time

Instead of using eta=-delta, use zeta=-(delta+1/2) to represent
delta. This variant only needs at most 590 iterations for 256-bit
inputs rather than 724 (by convex hull bounds analysis).

Author: sipa

doc/safegcd_implementation.md

@@ -244,8 +244,8 @@ def modinv(M, Mi, x):
 
 This means that in practice we'll always perform a multiple of *N* divsteps. This is not a problem
 because once *g=0*, further divsteps do not affect *f*, *g*, *d*, or *e* anymore (only *δ* keeps
-increasing). For variable time code such excess iterations will be mostly optimized away in
-section 6.
+increasing). For variable time code such excess iterations will be mostly optimized away in later
+sections.
 
 
 ## 4. Avoiding modulus operations
@@ -519,6 +519,20 @@ computation:
     g >>= 1
 ```
 
+A variant of divsteps with better worst-case performance can be used instead: starting *δ* at
+*1/2* instead of *1*. This reduces the worst case number of iterations to *590* for *256*-bit inputs
+(which can be shown using convex hull analysis). In this case, the substitution *ζ=-(δ+1/2)*
+is used instead to keep the variable integral. Incrementing *δ* by *1* still translates to
+decrementing *ζ* by *1*, but negating *δ* now corresponds to going from *ζ* to *-(ζ+1)*, or
+*~ζ*. Doing that conditionally based on *c3* is simply:
+
+```python
+    ...
+    c3 = c1 & c2
+    zeta ^= c3
+    ...
+```
+
 By replacing the loop in `divsteps_n_matrix` with a variant of the divstep code above (extended to
 also apply all *f* operations to *u*, *v* and all *g* operations to *q*, *r*), a constant-time version of
 `divsteps_n_matrix` is obtained. The full code will be in section 7.
@@ -535,7 +549,8 @@ other cases, it slows down calculations unnecessarily. In this section, we will
 faster non-constant time `divsteps_n_matrix` function.
 
 To do so, first consider yet another way of writing the inner loop of divstep operations in
-`gcd` from section 1. This decomposition is also explained in the paper in section 8.2.
+`gcd` from section 1. This decomposition is also explained in the paper in section 8.2. We use
+the original version with initial *δ=1* and *η=-δ* here.
 
 ```python
 for _ in range(N):
@@ -643,24 +658,24 @@ All together we need the following functions:
   section 5, extended to handle *u*, *v*, *q*, *r*:
 
 ```python
-def divsteps_n_matrix(eta, f, g):
-    """Compute eta and transition matrix t after N divsteps (multiplied by 2^N)."""
+def divsteps_n_matrix(zeta, f, g):
+    """Compute zeta and transition matrix t after N divsteps (multiplied by 2^N)."""
     u, v, q, r = 1, 0, 0, 1 # start with identity matrix
     for _ in range(N):
-        c1 = eta >> 63
+        c1 = zeta >> 63
         # Compute x, y, z as conditionally-negated versions of f, u, v.
         x, y, z = (f ^ c1) - c1, (u ^ c1) - c1, (v ^ c1) - c1
         c2 = -(g & 1)
         # Conditionally add x, y, z to g, q, r.
         g, q, r = g + (x & c2), q + (y & c2), r + (z & c2)
         c1 &= c2                     # reusing c1 here for the earlier c3 variable
-        eta = (eta ^ c1) - (c1 + 1)  # inlining the unconditional eta decrement here
+        zeta = (zeta ^ c1) - 1       # inlining the unconditional zeta decrement here
         # Conditionally add g, q, r to f, u, v.
         f, u, v = f + (g & c1), u + (q & c1), v + (r & c1)
         # When shifting g down, don't shift q, r, as we construct a transition matrix multiplied
         # by 2^N. Instead, shift f's coefficients u and v up.
         g, u, v = g >> 1, u << 1, v << 1
-    return eta, (u, v, q, r)
+    return zeta, (u, v, q, r)
 ```
 
 - The functions to update *f* and *g*, and *d* and *e*, from section 2 and section 4, with the constant-time
@@ -702,15 +717,15 @@ def normalize(sign, v, M):
     return v
 ```
 
-- And finally the `modinv` function too, adapted to use *&eta;* instead of *&delta;*, and using the fixed
+- And finally the `modinv` function too, adapted to use *&zeta;* instead of *&delta;*, and using the fixed
   iteration count from section 5:
 
 ```python
 def modinv(M, Mi, x):
     """Compute the modular inverse of x mod M, given Mi=1/M mod 2^N."""
-    eta, f, g, d, e = -1, M, x, 0, 1
-    for _ in range((724 + N - 1) // N):
-        eta, t = divsteps_n_matrix(-eta, f % 2**N, g % 2**N)
+    zeta, f, g, d, e = -1, M, x, 0, 1
+    for _ in range((590 + N - 1) // N):
+        zeta, t = divsteps_n_matrix(zeta, f % 2**N, g % 2**N)
         f, g = update_fg(f, g, t)
         d, e = update_de(d, e, t, M, Mi)
     return normalize(f, d, M)

src/modinv32_impl.h

@@ -168,17 +168,17 @@ typedef struct {
     int32_t u, v, q, r;
 } secp256k1_modinv32_trans2x2;
 
-/* Compute the transition matrix and eta for 30 divsteps.
+/* Compute the transition matrix and zeta for 30 divsteps.
  *
- * Input:  eta: initial eta
- *         f0:  bottom limb of initial f
- *         g0:  bottom limb of initial g
+ * Input:  zeta: initial zeta
+ *         f0:   bottom limb of initial f
+ *         g0:   bottom limb of initial g
  * Output: t: transition matrix
- * Return: final eta
+ * Return: final zeta
  *
  * Implements the divsteps_n_matrix function from the explanation.
  */
-static int32_t secp256k1_modinv32_divsteps_30(int32_t eta, uint32_t f0, uint32_t g0, secp256k1_modinv32_trans2x2 *t) {
+static int32_t secp256k1_modinv32_divsteps_30(int32_t zeta, uint32_t f0, uint32_t g0, secp256k1_modinv32_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
      * in range [-2^30,2^30], but here represented as unsigned mod 2^32. This
@@ -193,8 +193,8 @@ static int32_t secp256k1_modinv32_divsteps_30(int32_t eta, uint32_t f0, uint32_t
         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);
-        /* Compute conditional masks for (eta < 0) and for (g & 1). */
-        c1 = eta >> 31;
+        /* Compute conditional masks for (zeta < 0) and for (g & 1). */
+        c1 = zeta >> 31;
         c2 = -(g & 1);
         /* Compute x,y,z, conditionally negated versions of f,u,v. */
         x = (f ^ c1) - c1;
@@ -204,10 +204,10 @@ static int32_t secp256k1_modinv32_divsteps_30(int32_t eta, uint32_t f0, uint32_t
         g += x & c2;
         q += y & c2;
         r += z & c2;
-        /* In what follows, c1 is a condition mask for (eta < 0) and (g & 1). */
+        /* In what follows, c1 is a condition mask for (zeta < 0) and (g & 1). */
         c1 &= c2;
-        /* Conditionally negate eta, and unconditionally subtract 1. */
-        eta = (eta ^ c1) - (c1 + 1);
+        /* Conditionally change zeta into -zeta-2 or zeta-1. */
+        zeta = (zeta ^ c1) - 1;
         /* Conditionally add g,q,r to f,u,v. */
         f += g & c1;
         u += q & c1;
@@ -216,8 +216,8 @@ static int32_t secp256k1_modinv32_divsteps_30(int32_t eta, uint32_t f0, uint32_t
         g >>= 1;
         u <<= 1;
         v <<= 1;
-        /* Bounds on eta that follow from the bounds on iteration count (max 25*30 divsteps). */
-        VERIFY_CHECK(eta >= -751 && eta <= 751);
+        /* Bounds on zeta that follow from the bounds on iteration count (max 20*30 divsteps). */
+        VERIFY_CHECK(zeta >= -601 && zeta <= 601);
     }
     /* Return data in t and return value. */
     t->u = (int32_t)u;
@@ -229,7 +229,7 @@ static int32_t secp256k1_modinv32_divsteps_30(int32_t eta, uint32_t f0, uint32_t
      * will be divided out again). As each divstep's individual matrix has determinant 2, the
      * aggregate of 30 of them will have determinant 2^30. */
     VERIFY_CHECK((int64_t)t->u * t->r - (int64_t)t->v * t->q == ((int64_t)1) << 30);
-    return eta;
+    return zeta;
 }
 
 /* Compute the transition matrix and eta for 30 divsteps (variable time).
@@ -453,19 +453,19 @@ static void secp256k1_modinv32_update_fg_30_var(int len, secp256k1_modinv32_sign
 
 /* Compute the inverse of x modulo modinfo->modulus, and replace x with it (constant time in x). */
 static void secp256k1_modinv32(secp256k1_modinv32_signed30 *x, const secp256k1_modinv32_modinfo *modinfo) {
-    /* Start with d=0, e=1, f=modulus, g=x, eta=-1. */
+    /* Start with d=0, e=1, f=modulus, g=x, zeta=-1. */
     secp256k1_modinv32_signed30 d = {{0}};
     secp256k1_modinv32_signed30 e = {{1}};
     secp256k1_modinv32_signed30 f = modinfo->modulus;
     secp256k1_modinv32_signed30 g = *x;
     int i;
-    int32_t eta = -1;
+    int32_t zeta = -1; /* zeta = -(delta+1/2); delta is initially 1/2. */
 
-    /* Do 25 iterations of 30 divsteps each = 750 divsteps. 724 suffices for 256-bit inputs. */
-    for (i = 0; i < 25; ++i) {
-        /* Compute transition matrix and new eta after 30 divsteps. */
+    /* Do 20 iterations of 30 divsteps each = 600 divsteps. 590 suffices for 256-bit inputs. */
+    for (i = 0; i < 20; ++i) {
+        /* Compute transition matrix and new zeta after 30 divsteps. */
         secp256k1_modinv32_trans2x2 t;
-        eta = secp256k1_modinv32_divsteps_30(eta, f.v[0], g.v[0], &t);
+        zeta = secp256k1_modinv32_divsteps_30(zeta, f.v[0], g.v[0], &t);
         /* Update d,e using that transition matrix. */
         secp256k1_modinv32_update_de_30(&d, &e, &t, modinfo);
         /* Update f,g using that transition matrix. */
@@ -515,7 +515,7 @@ static void secp256k1_modinv32_var(secp256k1_modinv32_signed30 *x, const secp256
     int i = 0;
 #endif
     int j, len = 9;
-    int32_t eta = -1;
+    int32_t eta = -1; /* eta = -delta; delta is initially 1 (faster for the variable-time code) */
     int32_t cond, fn, gn;
 
     /* Do iterations of 30 divsteps each until g=0. */

src/modinv64_impl.h

@@ -145,17 +145,17 @@ typedef struct {
     int64_t u, v, q, r;
 } secp256k1_modinv64_trans2x2;
 
-/* Compute the transition matrix and eta for 62 divsteps.
+/* Compute the transition matrix and zeta for 62 divsteps (where zeta=-(delta+1/2)).
  *
- * Input:  eta: initial eta
- *         f0:  bottom limb of initial f
- *         g0:  bottom limb of initial g
+ * Input:  zeta: initial zeta
+ *         f0:   bottom limb of initial f
+ *         g0:   bottom limb of initial g
  * Output: t: transition matrix
- * Return: final eta
+ * Return: final zeta
  *
  * Implements the divsteps_n_matrix function from the explanation.
  */
-static int64_t secp256k1_modinv64_divsteps_62(int64_t eta, uint64_t f0, uint64_t g0, secp256k1_modinv64_trans2x2 *t) {
+static int64_t secp256k1_modinv64_divsteps_62(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
      * in range [-2^62,2^62], but here represented as unsigned mod 2^64. This
@@ -170,8 +170,8 @@ static int64_t secp256k1_modinv64_divsteps_62(int64_t eta, uint64_t f0, uint64_t
         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);
-        /* Compute conditional masks for (eta < 0) and for (g & 1). */
-        c1 = eta >> 63;
+        /* Compute conditional masks for (zeta < 0) and for (g & 1). */
+        c1 = zeta >> 63;
         c2 = -(g & 1);
         /* Compute x,y,z, conditionally negated versions of f,u,v. */
         x = (f ^ c1) - c1;
@@ -181,10 +181,10 @@ static int64_t secp256k1_modinv64_divsteps_62(int64_t eta, uint64_t f0, uint64_t
         g += x & c2;
         q += y & c2;
         r += z & c2;
-        /* In what follows, c1 is a condition mask for (eta < 0) and (g & 1). */
+        /* In what follows, c1 is a condition mask for (zeta < 0) and (g & 1). */
         c1 &= c2;
-        /* Conditionally negate eta, and unconditionally subtract 1. */
-        eta = (eta ^ c1) - (c1 + 1);
+        /* Conditionally change zeta into -zeta-2 or zeta-1. */
+        zeta = (zeta ^ c1) - 1;
         /* Conditionally add g,q,r to f,u,v. */
         f += g & c1;
         u += q & c1;
@@ -193,8 +193,8 @@ static int64_t secp256k1_modinv64_divsteps_62(int64_t eta, uint64_t f0, uint64_t
         g >>= 1;
         u <<= 1;
         v <<= 1;
-        /* Bounds on eta that follow from the bounds on iteration count (max 12*62 divsteps). */
-        VERIFY_CHECK(eta >= -745 && eta <= 745);
+        /* Bounds on zeta that follow from the bounds on iteration count (max 10*62 divsteps). */
+        VERIFY_CHECK(zeta >= -621 && zeta <= 621);
     }
     /* Return data in t and return value. */
     t->u = (int64_t)u;
@@ -206,10 +206,10 @@ static int64_t secp256k1_modinv64_divsteps_62(int64_t eta, uint64_t f0, uint64_t
      * 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);
-    return eta;
+    return zeta;
 }
 
-/* Compute the transition matrix and eta for 62 divsteps (variable time).
+/* Compute the transition matrix and eta for 62 divsteps (variable time, eta=-delta).
  *
  * Input:  eta: initial eta
  *         f0:  bottom limb of initial f
@@ -455,19 +455,19 @@ static void secp256k1_modinv64_update_fg_62_var(int len, secp256k1_modinv64_sign
 
 /* Compute the inverse of x modulo modinfo->modulus, and replace x with it (constant time in x). */
 static void secp256k1_modinv64(secp256k1_modinv64_signed62 *x, const secp256k1_modinv64_modinfo *modinfo) {
-    /* Start with d=0, e=1, f=modulus, g=x, eta=-1. */
+    /* Start with d=0, e=1, f=modulus, g=x, zeta=-1. */
     secp256k1_modinv64_signed62 d = {{0, 0, 0, 0, 0}};
     secp256k1_modinv64_signed62 e = {{1, 0, 0, 0, 0}};
     secp256k1_modinv64_signed62 f = modinfo->modulus;
     secp256k1_modinv64_signed62 g = *x;
     int i;
-    int64_t eta = -1;
+    int64_t zeta = -1; /* zeta = -(delta+1/2); delta starts at 1/2. */
 
-    /* Do 12 iterations of 62 divsteps each = 744 divsteps. 724 suffices for 256-bit inputs. */
-    for (i = 0; i < 12; ++i) {
-        /* Compute transition matrix and new eta after 62 divsteps. */
+    /* Do 10 iterations of 62 divsteps each = 620 divsteps. 590 suffices for 256-bit inputs. */
+    for (i = 0; i < 10; ++i) {
+        /* Compute transition matrix and new zeta after 62 divsteps. */
         secp256k1_modinv64_trans2x2 t;
-        eta = secp256k1_modinv64_divsteps_62(eta, f.v[0], g.v[0], &t);
+        zeta = secp256k1_modinv64_divsteps_62(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. */
@@ -517,7 +517,7 @@ static void secp256k1_modinv64_var(secp256k1_modinv64_signed62 *x, const secp256
     int i = 0;
 #endif
     int j, len = 5;
-    int64_t eta = -1;
+    int64_t eta = -1; /* eta = -delta; delta is initially 1 */
     int64_t cond, fn, gn;
 
     /* Do iterations of 62 divsteps each until g=0. */