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Add extensive comments on the safegcd algorithm and implementation

Browse Source 
This adds a long comment explaining the algorithm and implementation choices by building
it up step by step in Python.

Comments in the code are also reworked/added, with references to the long explanation.
This commit is contained in:
Pieter Wuille
2020-12-03 16:26:58 -08:00
parent 8e415acba2
commit d8a92fcc4c
5 changed files with 1019 additions and 188 deletions
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15
src/modinv32.h
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@@ -13,19 +13,30 @@
#include "util.h"
/* A signed 30-bit limb representation of integers.
*
* Its value is sum(v[i] * 2^(30*i), i=0..8). */
typedef struct {
int32_t v[9];
} secp256k1_modinv32_signed30;
typedef struct {
/* The modulus in signed30 notation. */
/* The modulus in signed30 notation, must be odd and in [3, 2^256]. */
secp256k1_modinv32_signed30 modulus;
/* modulus^{-1} mod 2^30 */
uint32_t modulus_inv30;
} secp256k1_modinv32_modinfo;
static void secp256k1_modinv32(secp256k1_modinv32_signed30 *x, const secp256k1_modinv32_modinfo *modinfo);
/* Replace x with its modular inverse mod modinfo->modulus. x must be in range [0, modulus).
* If x is zero, the result will be zero as well. If not, the inverse must exist (i.e., the gcd of
* x and modulus must be 1). These rules are automatically satisfied if the modulus is prime.
*
* On output, all of x's limbs will be in [0, 2^30).
*/
static void secp256k1_modinv32_var(secp256k1_modinv32_signed30 *x, const secp256k1_modinv32_modinfo *modinfo);
/* Same as secp256k1_modinv32_var, but constant time in x (not in the modulus). */
static void secp256k1_modinv32(secp256k1_modinv32_signed30 *x, const secp256k1_modinv32_modinfo *modinfo);
#endif /* SECP256K1_MODINV32_H */

209
src/modinv32_impl.h
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@@ -11,14 +11,31 @@
#include "util.h"
#include <stdlib.h>
/* This file implements modular inversion based on the paper "Fast constant-time gcd computation and
* modular inversion" by Daniel J. Bernstein and Bo-Yin Yang.
*
* For an explanation of the algorithm, see doc/safegcd_implementation.md. This file contains an
* implementation for N=30, using 30-bit signed limbs represented as int32_t.
*/
/* Take as input a signed30 number in range (-2*modulus,modulus), and add a multiple of the modulus
* to it to bring it to range [0,modulus). If sign < 0, the input will also be negated in the
* process. The input must have limbs in range (-2^30,2^30). The output will have limbs in range
* [0,2^30). */
static void secp256k1_modinv32_normalize_30(secp256k1_modinv32_signed30 *r, int32_t sign, const secp256k1_modinv32_modinfo *modinfo) {
const int32_t M30 = (int32_t)(UINT32_MAX >> 2);
int32_t r0 = r->v[0], r1 = r->v[1], r2 = r->v[2], r3 = r->v[3], r4 = r->v[4],
r5 = r->v[5], r6 = r->v[6], r7 = r->v[7], r8 = r->v[8];
int32_t cond_add, cond_negate;
/* In a first step, add the modulus if the input is negative, and then negate if requested.
* This brings r from range (-2*modulus,modulus) to range (-modulus,modulus). As all input
* limbs are in range (-2^30,2^30), this cannot overflow an int32_t. Note that the right
* shifts below are signed sign-extending shifts (see assumptions.h for tests that that is
* indeed the behavior of the right shift operator). */
cond_add = r8 >> 31;
r0 += modinfo->modulus.v[0] & cond_add;
r1 += modinfo->modulus.v[1] & cond_add;
r2 += modinfo->modulus.v[2] & cond_add;
@@ -28,9 +45,7 @@ static void secp256k1_modinv32_normalize_30(secp256k1_modinv32_signed30 *r, int3
r6 += modinfo->modulus.v[6] & cond_add;
r7 += modinfo->modulus.v[7] & cond_add;
r8 += modinfo->modulus.v[8] & cond_add;
cond_negate = sign >> 31;
r0 = (r0 ^ cond_negate) - cond_negate;
r1 = (r1 ^ cond_negate) - cond_negate;
r2 = (r2 ^ cond_negate) - cond_negate;
@@ -40,7 +55,7 @@ static void secp256k1_modinv32_normalize_30(secp256k1_modinv32_signed30 *r, int3
r6 = (r6 ^ cond_negate) - cond_negate;
r7 = (r7 ^ cond_negate) - cond_negate;
r8 = (r8 ^ cond_negate) - cond_negate;
/* Propagate the top bits, to bring limbs back to range (-2^30,2^30). */
r1 += r0 >> 30; r0 &= M30;
r2 += r1 >> 30; r1 &= M30;
r3 += r2 >> 30; r2 &= M30;
@@ -50,8 +65,9 @@ static void secp256k1_modinv32_normalize_30(secp256k1_modinv32_signed30 *r, int3
r7 += r6 >> 30; r6 &= M30;
r8 += r7 >> 30; r7 &= M30;
/* In a second step add the modulus again if the result is still negative, bringing r to range
* [0,modulus). */
cond_add = r8 >> 31;
r0 += modinfo->modulus.v[0] & cond_add;
r1 += modinfo->modulus.v[1] & cond_add;
r2 += modinfo->modulus.v[2] & cond_add;
@@ -61,7 +77,7 @@ static void secp256k1_modinv32_normalize_30(secp256k1_modinv32_signed30 *r, int3
r6 += modinfo->modulus.v[6] & cond_add;
r7 += modinfo->modulus.v[7] & cond_add;
r8 += modinfo->modulus.v[8] & cond_add;
/* And propagate again. */
r1 += r0 >> 30; r0 &= M30;
r2 += r1 >> 30; r1 &= M30;
r3 += r2 >> 30; r2 &= M30;
@@ -82,51 +98,82 @@ static void secp256k1_modinv32_normalize_30(secp256k1_modinv32_signed30 *r, int3
r->v[8] = r8;
}
/* Data type for transition matrices (see section 3 of explanation).
*
* t = [ u v ]
* [ q r ]
*/
typedef struct {
int32_t u, v, q, r;
} secp256k1_modinv32_trans2x2;
/* Compute the transition matrix and eta for 30 divsteps.
*
* Input: eta: initial eta
* f0: bottom limb of initial f
* g0: bottom limb of initial g
* Output: t: transition matrix
* Return: final eta
*
* 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) {
/* 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
* permits left shifting (which is UB for negative numbers). The range
* being inside [-2^31,2^31) means that casting to signed works correctly.
*/
uint32_t u = 1, v = 0, q = 0, r = 1;
uint32_t c1, c2, f = f0, g = g0, x, y, z;
int i;
for (i = 0; i < 30; ++i) {
VERIFY_CHECK((f & 1) == 1);
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;
c2 = -(g & 1);
/* Compute x,y,z, conditionally negated versions of f,u,v. */
x = (f ^ c1) - c1;
y = (u ^ c1) - c1;
z = (v ^ c1) - c1;
/* Conditionally add x,y,z to g,q,r. */
g += x & c2;
q += y & c2;
r += z & c2;
/* In what follows, c1 is a condition mask for (eta < 0) and (g & 1). */
c1 &= c2;
/* Conditionally negate eta, and unconditionally subtract 1. */
eta = (eta ^ c1) - (c1 + 1);
/* Conditionally add g,q,r to f,u,v. */
f += g & c1;
u += q & c1;
v += r & c1;
/* Shifts */
g >>= 1;
u <<= 1;
v <<= 1;
}
/* Return data in t and return value. */
t->u = (int32_t)u;
t->v = (int32_t)v;
t->q = (int32_t)q;
t->r = (int32_t)r;
return eta;
}
/* Compute the transition matrix and eta for 30 divsteps (variable time).
*
* Input: eta: initial eta
* f0: bottom limb of initial f
* g0: bottom limb of initial g
* Output: t: transition matrix
* Return: final eta
*
* Implements the divsteps_n_matrix_var function from the explanation.
*/
static int32_t secp256k1_modinv32_divsteps_30_var(int32_t eta, uint32_t f0, uint32_t g0, secp256k1_modinv32_trans2x2 *t) {
/* inv256[i] = -(2*i+1)^-1 (mod 256) */
static const uint8_t inv256[128] = {
@@ -143,6 +190,7 @@ static int32_t secp256k1_modinv32_divsteps_30_var(int32_t eta, uint32_t f0, uint
0xEF, 0xC5, 0xA3, 0x39, 0xB7, 0xCD, 0xAB, 0x01
};
/* Transformation matrix; see comments in secp256k1_modinv32_divsteps_30. */
uint32_t u = 1, v = 0, q = 0, r = 1;
uint32_t f = f0, g = g0, m;
uint16_t w;
@@ -151,22 +199,19 @@ static int32_t secp256k1_modinv32_divsteps_30_var(int32_t eta, uint32_t f0, uint
for (;;) {
/* Use a sentinel bit to count zeros only up to i. */
zeros = secp256k1_ctz32_var(g | (UINT32_MAX << i));
/* Perform zeros divsteps at once; they all just divide g by two. */
g >>= zeros;
u <<= zeros;
v <<= zeros;
eta -= zeros;
i -= zeros;
if (i <= 0) {
break;
}
/* We're done once we've done 30 divsteps. */
if (i == 0) break;
VERIFY_CHECK((f & 1) == 1);
VERIFY_CHECK((g & 1) == 1);
VERIFY_CHECK((u * f0 + v * g0) == f << (30 - i));
VERIFY_CHECK((q * f0 + r * g0) == g << (30 - i));
/* If eta is negative, negate it and replace f,g with g,-f. */
if (eta < 0) {
uint32_t tmp;
eta = -eta;
@@ -174,141 +219,128 @@ static int32_t secp256k1_modinv32_divsteps_30_var(int32_t eta, uint32_t f0, uint
tmp = u; u = q; q = -tmp;
tmp = v; v = r; r = -tmp;
}
/* Handle up to 8 divsteps at once, subject to eta and i. */
/* eta is now >= 0. In what follows we're going to cancel out the bottom bits of g. No more
* than i can be cancelled out (as we'd be done before that point), and no more than eta+1
* can be done as its sign will flip once that happens. */
limit = ((int)eta + 1) > i ? i : ((int)eta + 1);
/* m is a mask for the bottom min(limit, 8) bits (our table only supports 8 bits). */
m = (UINT32_MAX >> (32 - limit)) & 255U;
/* Find what multiple of f must be added to g to cancel its bottom min(limit, 8) bits. */
w = (g * inv256[(f >> 1) & 127]) & m;
/* Do so. */
g += f * w;
q += u * w;
r += v * w;
VERIFY_CHECK((g & m) == 0);
}
/* Return data in t and return value. */
t->u = (int32_t)u;
t->v = (int32_t)v;
t->q = (int32_t)q;
t->r = (int32_t)r;
return eta;
}
/* Compute (t/2^30) * [d, e] mod modulus, where t is a transition matrix for 30 divsteps.
*
* On input and output, d and e are in range (-2*modulus,modulus). All output limbs will be in range
* (-2^30,2^30).
*
* This implements the update_de function from the explanation.
*/
static void secp256k1_modinv32_update_de_30(secp256k1_modinv32_signed30 *d, secp256k1_modinv32_signed30 *e, const secp256k1_modinv32_trans2x2 *t, const secp256k1_modinv32_modinfo* modinfo) {
const int32_t M30 = (int32_t)(UINT32_MAX >> 2);
const int32_t u = t->u, v = t->v, q = t->q, r = t->r;
int32_t di, ei, md, me, sd, se;
int64_t cd, ce;
int i;
/*
* On input, d/e must be in the range (-2.P, P). For initially negative d (resp. e), we add
* u and/or v (resp. q and/or r) multiples of the modulus to the corresponding output (prior
* to division by 2^30). This has the same effect as if we added the modulus to the input(s).
*/
/* [md,me] start as zero; plus [u,q] if d is negative; plus [v,r] if e is negative. */
sd = d->v[8] >> 31;
se = e->v[8] >> 31;
md = (u & sd) + (v & se);
me = (q & sd) + (r & se);
/* Begin computing t*[d,e]. */
di = d->v[0];
ei = e->v[0];
cd = (int64_t)u * di + (int64_t)v * ei;
ce = (int64_t)q * di + (int64_t)r * ei;
/*
* Subtract from md/me an extra term in the range [0, 2^30) such that the low 30 bits of each
* sum of products will be 0. This allows clean division by 2^30. On output, d/e are thus in
* the range (-2.P, P), consistent with the input constraint.
*/
/* Correct md,me so that t*[d,e]+modulus*[md,me] has 30 zero bottom bits. */
md -= (modinfo->modulus_inv30 * (uint32_t)cd + md) & M30;
me -= (modinfo->modulus_inv30 * (uint32_t)ce + me) & M30;
/* Update the beginning of computation for t*[d,e]+modulus*[md,me] now md,me are known. */
cd += (int64_t)modinfo->modulus.v[0] * md;
ce += (int64_t)modinfo->modulus.v[0] * me;
/* Verify that the low 30 bits of the computation are indeed zero, and then throw them away. */
VERIFY_CHECK(((int32_t)cd & M30) == 0); cd >>= 30;
VERIFY_CHECK(((int32_t)ce & M30) == 0); ce >>= 30;
/* Now iteratively compute limb i=1..8 of t*[d,e]+modulus*[md,me], and store them in output
* limb i-1 (shifting down by 30 bits). */
for (i = 1; i < 9; ++i) {
di = d->v[i];
ei = e->v[i];
cd += (int64_t)u * di + (int64_t)v * ei;
ce += (int64_t)q * di + (int64_t)r * ei;
cd += (int64_t)modinfo->modulus.v[i] * md;
ce += (int64_t)modinfo->modulus.v[i] * me;
d->v[i - 1] = (int32_t)cd & M30; cd >>= 30;
e->v[i - 1] = (int32_t)ce & M30; ce >>= 30;
}
/* What remains is limb 9 of t*[d,e]+modulus*[md,me]; store it as output limb 8. */
d->v[8] = (int32_t)cd;
e->v[8] = (int32_t)ce;
}
/* Compute (t/2^30) * [f, g], where t is a transition matrix for 30 divsteps.
*
* This implements the update_fg function from the explanation.
*/
static void secp256k1_modinv32_update_fg_30(secp256k1_modinv32_signed30 *f, secp256k1_modinv32_signed30 *g, const secp256k1_modinv32_trans2x2 *t) {
const int32_t M30 = (int32_t)(UINT32_MAX >> 2);
const int32_t u = t->u, v = t->v, q = t->q, r = t->r;
int32_t fi, gi;
int64_t cf, cg;
int i;
/* Start computing t*[f,g]. */
fi = f->v[0];
gi = g->v[0];
cf = (int64_t)u * fi + (int64_t)v * gi;
cg = (int64_t)q * fi + (int64_t)r * gi;
VERIFY_CHECK(((int32_t)cf & M30) == 0);
VERIFY_CHECK(((int32_t)cg & M30) == 0);
cf >>= 30;
cg >>= 30;
/* Verify that the bottom 30 bits of the result are zero, and then throw them away. */
VERIFY_CHECK(((int32_t)cf & M30) == 0); cf >>= 30;
VERIFY_CHECK(((int32_t)cg & M30) == 0); cg >>= 30;
/* Now iteratively compute limb i=1..8 of t*[f,g], and store them in output limb i-1 (shifting
* down by 30 bits). */
for (i = 1; i < 9; ++i) {
fi = f->v[i];
gi = g->v[i];
cf += (int64_t)u * fi + (int64_t)v * gi;
cg += (int64_t)q * fi + (int64_t)r * gi;
f->v[i - 1] = (int32_t)cf & M30; cf >>= 30;
g->v[i - 1] = (int32_t)cg & M30; cg >>= 30;
}
/* What remains is limb 9 of t*[f,g]; store it as output limb 8. */
f->v[8] = (int32_t)cf;
g->v[8] = (int32_t)cg;
}
/* 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) {
/* Modular inversion based on the paper "Fast constant-time gcd computation and
* modular inversion" by Daniel J. Bernstein and Bo-Yin Yang. */
/* Start with d=0, e=1, f=modulus, g=x, eta=-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;
/* The paper uses 'delta'; eta == -delta (a performance tweak).
*
* If the maximum bitlength of g is known to be less than 256, then eta can be set
* initially to -(1 + (256 - maxlen(g))), and only (741 - (256 - maxlen(g))) total
* divsteps are needed. */
eta = -1;
int32_t eta = -1;
/* 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. */
secp256k1_modinv32_trans2x2 t;
eta = secp256k1_modinv32_divsteps_30(eta, 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. */
secp256k1_modinv32_update_fg_30(&f, &g, &t);
}
@@ -317,38 +349,39 @@ static void secp256k1_modinv32(secp256k1_modinv32_signed30 *x, const secp256k1_m
* values i.e. +/- 1, and d now contains +/- the modular inverse. */
VERIFY_CHECK((g.v[0] | g.v[1] | g.v[2] | g.v[3] | g.v[4] | g.v[5] | g.v[6] | g.v[7] | g.v[8]) == 0);
secp256k1_modinv32_normalize_30(&d, f.v[8] >> 31, modinfo);
/* Optionally negate d, normalize to [0,modulus), and return it. */
secp256k1_modinv32_normalize_30(&d, f.v[8], modinfo);
*x = d;
}
/* Compute the inverse of x modulo modinfo->modulus, and replace x with it (variable time). */
static void secp256k1_modinv32_var(secp256k1_modinv32_signed30 *x, const secp256k1_modinv32_modinfo *modinfo) {
/* Modular inversion based on the paper "Fast constant-time gcd computation and
* modular inversion" by Daniel J. Bernstein and Bo-Yin Yang. */
/* Start with d=0, e=1, f=modulus, g=x, eta=-1. */
secp256k1_modinv32_signed30 d = {{0, 0, 0, 0, 0, 0, 0, 0, 0}};
secp256k1_modinv32_signed30 e = {{1, 0, 0, 0, 0, 0, 0, 0, 0}};
secp256k1_modinv32_signed30 f = modinfo->modulus;
secp256k1_modinv32_signed30 g = *x;
int j;
int32_t eta;
int32_t eta = -1;
int32_t cond;
/* The paper uses 'delta'; eta == -delta (a performance tweak).
*
* If g has leading zeros (w.r.t 256 bits), then eta can be set initially to
* -(1 + clz(g)), and the worst-case divstep count would be only (741 - clz(g)). */
eta = -1;
/* Do iterations of 30 divsteps each until g=0. */
while (1) {
/* Compute transition matrix and new eta after 30 divsteps. */
secp256k1_modinv32_trans2x2 t;
eta = secp256k1_modinv32_divsteps_30_var(eta, 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. */
secp256k1_modinv32_update_fg_30(&f, &g, &t);
/* If the bottom limb of g is 0, there is a chance g=0. */
if (g.v[0] == 0) {
cond = 0;
/* Check if the other limbs are also 0. */
for (j = 1; j < 9; ++j) {
cond |= g.v[j];
}
/* If so, we're done. */
if (cond == 0) break;
}
}
@@ -356,8 +389,8 @@ static void secp256k1_modinv32_var(secp256k1_modinv32_signed30 *x, const secp256
/* At this point g is 0 and (if g was not originally 0) f must now equal +/- GCD of
* the initial f, g values i.e. +/- 1, and d now contains +/- the modular inverse. */
secp256k1_modinv32_normalize_30(&d, f.v[8] >> 31, modinfo);
/* Optionally negate d, normalize to [0,modulus), and return it. */
secp256k1_modinv32_normalize_30(&d, f.v[8], modinfo);
*x = d;
}

15
src/modinv64.h
View File

@@ -17,19 +17,30 @@
#error "modinv64 requires 128-bit wide multiplication support"
#endif
/* A signed 62-bit limb representation of integers.
*
* Its value is sum(v[i] * 2^(62*i), i=0..4). */
typedef struct {
int64_t v[5];
} secp256k1_modinv64_signed62;
typedef struct {
/* The modulus in signed62 notation. */
/* The modulus in signed62 notation, must be odd and in [3, 2^256]. */
secp256k1_modinv64_signed62 modulus;
/* modulus^{-1} mod 2^62 */
uint64_t modulus_inv62;
} secp256k1_modinv64_modinfo;
static void secp256k1_modinv64(secp256k1_modinv64_signed62 *x, const secp256k1_modinv64_modinfo *modinfo);
/* Replace x with its modular inverse mod modinfo->modulus. x must be in range [0, modulus).
* If x is zero, the result will be zero as well. If not, the inverse must exist (i.e., the gcd of
* x and modulus must be 1). These rules are automatically satisfied if the modulus is prime.
*
* On output, all of x's limbs will be in [0, 2^62).
*/
static void secp256k1_modinv64_var(secp256k1_modinv64_signed62 *x, const secp256k1_modinv64_modinfo *modinfo);
/* Same as secp256k1_modinv64_var, but constant time in x (not in the modulus). */
static void secp256k1_modinv64(secp256k1_modinv64_signed62 *x, const secp256k1_modinv64_modinfo *modinfo);
#endif /* SECP256K1_MODINV64_H */

218
src/modinv64_impl.h
View File

@@ -11,40 +11,54 @@
#include "util.h"
/* This file implements modular inversion based on the paper "Fast constant-time gcd computation and
* modular inversion" by Daniel J. Bernstein and Bo-Yin Yang.
*
* For an explanation of the algorithm, see doc/safegcd_implementation.md. This file contains an
* implementation for N=62, using 62-bit signed limbs represented as int64_t.
*/
/* Take as input a signed62 number in range (-2*modulus,modulus), and add a multiple of the modulus
* to it to bring it to range [0,modulus). If sign < 0, the input will also be negated in the
* process. The input must have limbs in range (-2^62,2^62). The output will have limbs in range
* [0,2^62). */
static void secp256k1_modinv64_normalize_62(secp256k1_modinv64_signed62 *r, int64_t sign, const secp256k1_modinv64_modinfo *modinfo) {
const int64_t M62 = (int64_t)(UINT64_MAX >> 2);
int64_t r0 = r->v[0], r1 = r->v[1], r2 = r->v[2], r3 = r->v[3], r4 = r->v[4];
int64_t cond_add, cond_negate;
/* In a first step, add the modulus if the input is negative, and then negate if requested.
* This brings r from range (-2*modulus,modulus) to range (-modulus,modulus). As all input
* limbs are in range (-2^62,2^62), this cannot overflow an int64_t. Note that the right
* shifts below are signed sign-extending shifts (see assumptions.h for tests that that is
* indeed the behavior of the right shift operator). */
cond_add = r4 >> 63;
r0 += modinfo->modulus.v[0] & cond_add;
r1 += modinfo->modulus.v[1] & cond_add;
r2 += modinfo->modulus.v[2] & cond_add;
r3 += modinfo->modulus.v[3] & cond_add;
r4 += modinfo->modulus.v[4] & cond_add;
cond_negate = sign >> 63;
r0 = (r0 ^ cond_negate) - cond_negate;
r1 = (r1 ^ cond_negate) - cond_negate;
r2 = (r2 ^ cond_negate) - cond_negate;
r3 = (r3 ^ cond_negate) - cond_negate;
r4 = (r4 ^ cond_negate) - cond_negate;
/* Propagate the top bits, to bring limbs back to range (-2^62,2^62). */
r1 += r0 >> 62; r0 &= M62;
r2 += r1 >> 62; r1 &= M62;
r3 += r2 >> 62; r2 &= M62;
r4 += r3 >> 62; r3 &= M62;
/* In a second step add the modulus again if the result is still negative, bringing
* r to range [0,modulus). */
cond_add = r4 >> 63;
r0 += modinfo->modulus.v[0] & cond_add;
r1 += modinfo->modulus.v[1] & cond_add;
r2 += modinfo->modulus.v[2] & cond_add;
r3 += modinfo->modulus.v[3] & cond_add;
r4 += modinfo->modulus.v[4] & cond_add;
/* And propagate again. */
r1 += r0 >> 62; r0 &= M62;
r2 += r1 >> 62; r1 &= M62;
r3 += r2 >> 62; r2 &= M62;
@@ -57,53 +71,82 @@ static void secp256k1_modinv64_normalize_62(secp256k1_modinv64_signed62 *r, int6
r->v[4] = r4;
}
/* Data type for transition matrices (see section 3 of explanation).
*
* t = [ u v ]
* [ q r ]
*/
typedef struct {
int64_t u, v, q, r;
} secp256k1_modinv64_trans2x2;
/* Compute the transition matrix and eta for 62 divsteps.
*
* Input: eta: initial eta
* f0: bottom limb of initial f
* g0: bottom limb of initial g
* Output: t: transition matrix
* Return: final eta
*
* 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) {
/* 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
* 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 c1, c2, f = f0, g = g0, x, y, z;
int i;
for (i = 0; i < 62; ++i) {
VERIFY_CHECK((f & 1) == 1);
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;
c2 = -(g & 1);
/* Compute x,y,z, conditionally negated versions of f,u,v. */
x = (f ^ c1) - c1;
y = (u ^ c1) - c1;
z = (v ^ c1) - c1;
/* Conditionally add x,y,z to g,q,r. */
g += x & c2;
q += y & c2;
r += z & c2;
/* In what follows, c1 is a condition mask for (eta < 0) and (g & 1). */
c1 &= c2;
/* Conditionally negate eta, and unconditionally subtract 1. */
eta = (eta ^ c1) - (c1 + 1);
/* Conditionally add g,q,r to f,u,v. */
f += g & c1;
u += q & c1;
v += r & c1;
/* Shifts */
g >>= 1;
u <<= 1;
v <<= 1;
}
/* Return data in t and return value. */
t->u = (int64_t)u;
t->v = (int64_t)v;
t->q = (int64_t)q;
t->r = (int64_t)r;
return eta;
}
/* Compute the transition matrix and eta for 62 divsteps (variable time).
*
* Input: eta: initial eta
* f0: bottom limb of initial f
* g0: bottom limb of initial g
* Output: t: transition matrix
* Return: final eta
*
* Implements the divsteps_n_matrix_var function from the explanation.
*/
static int64_t secp256k1_modinv64_divsteps_62_var(int64_t eta, uint64_t f0, uint64_t g0, secp256k1_modinv64_trans2x2 *t) {
/* inv256[i] = -(2*i+1)^-1 (mod 256) */
static const uint8_t inv256[128] = {
@@ -120,6 +163,7 @@ static int64_t secp256k1_modinv64_divsteps_62_var(int64_t eta, uint64_t f0, uint
0xEF, 0xC5, 0xA3, 0x39, 0xB7, 0xCD, 0xAB, 0x01
};
/* Transformation matrix; see comments in secp256k1_modinv64_divsteps_62. */
uint64_t u = 1, v = 0, q = 0, r = 1;
uint64_t f = f0, g = g0, m;
uint32_t w;
@@ -128,22 +172,19 @@ static int64_t secp256k1_modinv64_divsteps_62_var(int64_t eta, uint64_t f0, uint
for (;;) {
/* Use a sentinel bit to count zeros only up to i. */
zeros = secp256k1_ctz64_var(g | (UINT64_MAX << i));
/* Perform zeros divsteps at once; they all just divide g by two. */
g >>= zeros;
u <<= zeros;
v <<= zeros;
eta -= zeros;
i -= zeros;
if (i <= 0) {
break;
}
/* We're done once we've done 62 divsteps. */
if (i == 0) break;
VERIFY_CHECK((f & 1) == 1);
VERIFY_CHECK((g & 1) == 1);
VERIFY_CHECK((u * f0 + v * g0) == f << (62 - i));
VERIFY_CHECK((q * f0 + r * g0) == g << (62 - i));
/* If eta is negative, negate it and replace f,g with g,-f. */
if (eta < 0) {
uint64_t tmp;
eta = -eta;
@@ -151,28 +192,35 @@ static int64_t secp256k1_modinv64_divsteps_62_var(int64_t eta, uint64_t f0, uint
tmp = u; u = q; q = -tmp;
tmp = v; v = r; r = -tmp;
}
/* Handle up to 8 divsteps at once, subject to eta and i. */
/* eta is now >= 0. In what follows we're going to cancel out the bottom bits of g. No more
* than i can be cancelled out (as we'd be done before that point), and no more than eta+1
* can be done as its sign will flip once that happens. */
limit = ((int)eta + 1) > i ? i : ((int)eta + 1);
/* m is a mask for the bottom min(limit, 8) bits (our table only supports 8 bits). */
m = (UINT64_MAX >> (64 - limit)) & 255U;
/* Find what multiple of f must be added to g to cancel its bottom min(limit, 8) bits. */
w = (g * inv256[(f >> 1) & 127]) & m;
/* Do so. */
g += f * w;
q += u * w;
r += v * w;
VERIFY_CHECK((g & m) == 0);
}
/* Return data in t and return value. */
t->u = (int64_t)u;
t->v = (int64_t)v;
t->q = (int64_t)q;
t->r = (int64_t)r;
return eta;
}
/* Compute (t/2^62) * [d, e] mod modulus, where t is a transition matrix for 62 divsteps.
*
* On input and output, d and e are in range (-2*modulus,modulus). All output limbs will be in range
* (-2^62,2^62).
*
* This implements the update_de function from the explanation.
*/
static void secp256k1_modinv64_update_de_62(secp256k1_modinv64_signed62 *d, secp256k1_modinv64_signed62 *e, const secp256k1_modinv64_trans2x2 *t, const secp256k1_modinv64_modinfo* modinfo) {
const int64_t M62 = (int64_t)(UINT64_MAX >> 2);
const int64_t d0 = d->v[0], d1 = d->v[1], d2 = d->v[2], d3 = d->v[3], d4 = d->v[4];
@@ -180,140 +228,115 @@ static void secp256k1_modinv64_update_de_62(secp256k1_modinv64_signed62 *d, secp
const int64_t u = t->u, v = t->v, q = t->q, r = t->r;
int64_t md, me, sd, se;
int128_t cd, ce;
/*
* On input, d/e must be in the range (-2.P, P). For initially negative d (resp. e), we add
* u and/or v (resp. q and/or r) multiples of the modulus to the corresponding output (prior
* to division by 2^62). This has the same effect as if we added the modulus to the input(s).
*/
/* [md,me] start as zero; plus [u,q] if d is negative; plus [v,r] if e is negative. */
sd = d4 >> 63;
se = e4 >> 63;
md = (u & sd) + (v & se);
me = (q & sd) + (r & se);
/* Begin computing t*[d,e]. */
cd = (int128_t)u * d0 + (int128_t)v * e0;
ce = (int128_t)q * d0 + (int128_t)r * e0;
/*
* Subtract from md/me an extra term in the range [0, 2^62) such that the low 62 bits of each
* sum of products will be 0. This allows clean division by 2^62. On output, d/e are thus in
* the range (-2.P, P), consistent with the input constraint.
*/
/* Correct md,me so that t*[d,e]+modulus*[md,me] has 62 zero bottom bits. */
md -= (modinfo->modulus_inv62 * (uint64_t)cd + md) & M62;
me -= (modinfo->modulus_inv62 * (uint64_t)ce + me) & M62;
/* Update the beginning of computation for t*[d,e]+modulus*[md,me] now md,me are known. */
cd += (int128_t)modinfo->modulus.v[0] * md;
ce += (int128_t)modinfo->modulus.v[0] * me;
/* Verify that the low 62 bits of the computation are indeed zero, and then throw them away. */
VERIFY_CHECK(((int64_t)cd & M62) == 0); cd >>= 62;
VERIFY_CHECK(((int64_t)ce & M62) == 0); ce >>= 62;
/* Compute limb 1 of t*[d,e]+modulus*[md,me], and store it as output limb 0 (= down shift). */
cd += (int128_t)u * d1 + (int128_t)v * e1;
ce += (int128_t)q * d1 + (int128_t)r * e1;
cd += (int128_t)modinfo->modulus.v[1] * md;
ce += (int128_t)modinfo->modulus.v[1] * me;
d->v[0] = (int64_t)cd & M62; cd >>= 62;
e->v[0] = (int64_t)ce & M62; ce >>= 62;
/* Compute limb 2 of t*[d,e]+modulus*[md,me], and store it as output limb 1. */
cd += (int128_t)u * d2 + (int128_t)v * e2;
ce += (int128_t)q * d2 + (int128_t)r * e2;
cd += (int128_t)modinfo->modulus.v[2] * md;
ce += (int128_t)modinfo->modulus.v[2] * me;
d->v[1] = (int64_t)cd & M62; cd >>= 62;
e->v[1] = (int64_t)ce & M62; ce >>= 62;
/* Compute limb 3 of t*[d,e]+modulus*[md,me], and store it as output limb 2. */
cd += (int128_t)u * d3 + (int128_t)v * e3;
ce += (int128_t)q * d3 + (int128_t)r * e3;
cd += (int128_t)modinfo->modulus.v[3] * md;
ce += (int128_t)modinfo->modulus.v[3] * me;
d->v[2] = (int64_t)cd & M62; cd >>= 62;
e->v[2] = (int64_t)ce & M62; ce >>= 62;
/* Compute limb 4 of t*[d,e]+modulus*[md,me], and store it as output limb 3. */
cd += (int128_t)u * d4 + (int128_t)v * e4;
ce += (int128_t)q * d4 + (int128_t)r * e4;
cd += (int128_t)modinfo->modulus.v[4] * md;
ce += (int128_t)modinfo->modulus.v[4] * me;
d->v[3] = (int64_t)cd & M62; cd >>= 62;
e->v[3] = (int64_t)ce & M62; ce >>= 62;
/* What remains is limb 5 of t*[d,e]+modulus*[md,me]; store it as output limb 4. */
d->v[4] = (int64_t)cd;
e->v[4] = (int64_t)ce;
}
/* Compute (t/2^62) * [f, g], where t is a transition matrix for 62 divsteps.
*
* This implements the update_fg function from the explanation.
*/
static void secp256k1_modinv64_update_fg_62(secp256k1_modinv64_signed62 *f, secp256k1_modinv64_signed62 *g, const secp256k1_modinv64_trans2x2 *t) {
const int64_t M62 = (int64_t)(UINT64_MAX >> 2);
const int64_t f0 = f->v[0], f1 = f->v[1], f2 = f->v[2], f3 = f->v[3], f4 = f->v[4];
const int64_t g0 = g->v[0], g1 = g->v[1], g2 = g->v[2], g3 = g->v[3], g4 = g->v[4];
const int64_t u = t->u, v = t->v, q = t->q, r = t->r;
int128_t cf, cg;
/* Start computing t*[f,g]. */
cf = (int128_t)u * f0 + (int128_t)v * g0;
cg = (int128_t)q * f0 + (int128_t)r * g0;
/* Verify that the bottom 62 bits of the result are zero, and then throw them away. */
VERIFY_CHECK(((int64_t)cf & M62) == 0); cf >>= 62;
VERIFY_CHECK(((int64_t)cg & M62) == 0); cg >>= 62;
/* Compute limb 1 of t*[f,g], and store it as output limb 0 (= down shift). */
cf += (int128_t)u * f1 + (int128_t)v * g1;
cg += (int128_t)q * f1 + (int128_t)r * g1;
f->v[0] = (int64_t)cf & M62; cf >>= 62;
g->v[0] = (int64_t)cg & M62; cg >>= 62;
/* Compute limb 2 of t*[f,g], and store it as output limb 1. */
cf += (int128_t)u * f2 + (int128_t)v * g2;
cg += (int128_t)q * f2 + (int128_t)r * g2;
f->v[1] = (int64_t)cf & M62; cf >>= 62;
g->v[1] = (int64_t)cg & M62; cg >>= 62;
/* Compute limb 3 of t*[f,g], and store it as output limb 2. */
cf += (int128_t)u * f3 + (int128_t)v * g3;
cg += (int128_t)q * f3 + (int128_t)r * g3;
f->v[2] = (int64_t)cf & M62; cf >>= 62;
g->v[2] = (int64_t)cg & M62; cg >>= 62;
/* Compute limb 4 of t*[f,g], and store it as output limb 3. */
cf += (int128_t)u * f4 + (int128_t)v * g4;
cg += (int128_t)q * f4 + (int128_t)r * g4;
f->v[3] = (int64_t)cf & M62; cf >>= 62;
g->v[3] = (int64_t)cg & M62; cg >>= 62;
/* What remains is limb 5 of t*[f,g]; store it as output limb 4. */
f->v[4] = (int64_t)cf;
g->v[4] = (int64_t)cg;
}
/* 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) {
/* Modular inversion based on the paper "Fast constant-time gcd computation and
* modular inversion" by Daniel J. Bernstein and Bo-Yin Yang. */
/* Start with d=0, e=1, f=modulus, g=x, eta=-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;
/* The paper uses 'delta'; eta == -delta (a performance tweak).
*
* If the maximum bitlength of g is known to be less than 256, then eta can be set
* initially to -(1 + (256 - maxlen(g))), and only (741 - (256 - maxlen(g))) total
* divsteps are needed. */
eta = -1;
int64_t eta = -1;
/* 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. */
secp256k1_modinv64_trans2x2 t;
eta = secp256k1_modinv64_divsteps_62(eta, 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. */
secp256k1_modinv64_update_fg_62(&f, &g, &t);
}
@@ -322,45 +345,48 @@ static void secp256k1_modinv64(secp256k1_modinv64_signed62 *x, const secp256k1_m
* values i.e. +/- 1, and d now contains +/- the modular inverse. */
VERIFY_CHECK((g.v[0] | g.v[1] | g.v[2] | g.v[3] | g.v[4]) == 0);
/* Optionally negate d, normalize to [0,modulus), and return it. */
secp256k1_modinv64_normalize_62(&d, f.v[4], modinfo);
*x = d;
}
/* Compute the inverse of x modulo modinfo->modulus, and replace x with it (variable time). */
static void secp256k1_modinv64_var(secp256k1_modinv64_signed62 *x, const secp256k1_modinv64_modinfo *modinfo) {
/* Modular inversion based on the paper "Fast constant-time gcd computation and
* modular inversion" by Daniel J. Bernstein and Bo-Yin Yang. */
/* Start with d=0, e=1, f=modulus, g=x, eta=-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 j;
uint64_t eta;
int64_t eta = -1;
int64_t cond;
/* The paper uses 'delta'; eta == -delta (a performance tweak).
*
* If g has leading zeros (w.r.t 256 bits), then eta can be set initially to
* -(1 + clz(g)), and the worst-case divstep count would be only (741 - clz(g)). */
eta = -1;
/* Do iterations of 62 divsteps each until g=0. */
while (1) {
/* Compute transition matrix and new eta after 62 divsteps. */
secp256k1_modinv64_trans2x2 t;
eta = secp256k1_modinv64_divsteps_62_var(eta, 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. */
secp256k1_modinv64_update_fg_62(&f, &g, &t);
/* If the bottom limb of g is zero, there is a chance that g=0. */
if (g.v[0] == 0) {
cond = 0;
/* Check if the other limbs are also 0. */
for (j = 1; j < 5; ++j) {
cond |= g.v[j];
}
/* If so, we're done. */
if (cond == 0) break;
}
}
secp256k1_modinv64_normalize_62(&d, f.v[4], modinfo);
/* At this point g is 0 and (if g was not originally 0) f must now equal +/- GCD of
* the initial f, g values i.e. +/- 1, and d now contains +/- the modular inverse. */
/* Optionally negate d, normalize to [0,modulus), and return it. */
secp256k1_modinv64_normalize_62(&d, f.v[4], modinfo);
*x = d;
}