Add safegcd based modular inverse modules
Refactored by: Pieter Wuille <pieter@wuille.net>
This commit is contained in:
Peter Dettman
Pieter Wuille
parent
de0a643c3d
commit
8e415acba2
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@ -34,6 +34,10 @@ noinst_HEADERS += src/field_5x52.h
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noinst_HEADERS += src/field_5x52_impl.h
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noinst_HEADERS += src/field_5x52_int128_impl.h
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noinst_HEADERS += src/field_5x52_asm_impl.h
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noinst_HEADERS += src/modinv32.h
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noinst_HEADERS += src/modinv32_impl.h
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noinst_HEADERS += src/modinv64.h
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noinst_HEADERS += src/modinv64_impl.h
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noinst_HEADERS += src/assumptions.h
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noinst_HEADERS += src/util.h
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noinst_HEADERS += src/scratch.h
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@ -0,0 +1,31 @@
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/***********************************************************************
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* Copyright (c) 2020 Peter Dettman *
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* Distributed under the MIT software license, see the accompanying *
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* file COPYING or https://www.opensource.org/licenses/mit-license.php.*
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**********************************************************************/
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#ifndef SECP256K1_MODINV32_H
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#define SECP256K1_MODINV32_H
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#if defined HAVE_CONFIG_H
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#include "libsecp256k1-config.h"
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#endif
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#include "util.h"
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typedef struct {
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int32_t v[9];
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} secp256k1_modinv32_signed30;
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typedef struct {
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/* The modulus in signed30 notation. */
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secp256k1_modinv32_signed30 modulus;
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/* modulus^{-1} mod 2^30 */
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uint32_t modulus_inv30;
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} secp256k1_modinv32_modinfo;
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static void secp256k1_modinv32(secp256k1_modinv32_signed30 *x, const secp256k1_modinv32_modinfo *modinfo);
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static void secp256k1_modinv32_var(secp256k1_modinv32_signed30 *x, const secp256k1_modinv32_modinfo *modinfo);
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#endif /* SECP256K1_MODINV32_H */
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@ -0,0 +1,364 @@
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/***********************************************************************
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* Copyright (c) 2020 Peter Dettman *
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* Distributed under the MIT software license, see the accompanying *
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* file COPYING or https://www.opensource.org/licenses/mit-license.php.*
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**********************************************************************/
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#ifndef SECP256K1_MODINV32_IMPL_H
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#define SECP256K1_MODINV32_IMPL_H
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#include "modinv32.h"
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#include "util.h"
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static void secp256k1_modinv32_normalize_30(secp256k1_modinv32_signed30 *r, int32_t sign, const secp256k1_modinv32_modinfo *modinfo) {
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const int32_t M30 = (int32_t)(UINT32_MAX >> 2);
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int32_t r0 = r->v[0], r1 = r->v[1], r2 = r->v[2], r3 = r->v[3], r4 = r->v[4],
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r5 = r->v[5], r6 = r->v[6], r7 = r->v[7], r8 = r->v[8];
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int32_t cond_add, cond_negate;
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cond_add = r8 >> 31;
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r0 += modinfo->modulus.v[0] & cond_add;
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r1 += modinfo->modulus.v[1] & cond_add;
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r2 += modinfo->modulus.v[2] & cond_add;
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r3 += modinfo->modulus.v[3] & cond_add;
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r4 += modinfo->modulus.v[4] & cond_add;
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r5 += modinfo->modulus.v[5] & cond_add;
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r6 += modinfo->modulus.v[6] & cond_add;
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r7 += modinfo->modulus.v[7] & cond_add;
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r8 += modinfo->modulus.v[8] & cond_add;
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cond_negate = sign >> 31;
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r0 = (r0 ^ cond_negate) - cond_negate;
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r1 = (r1 ^ cond_negate) - cond_negate;
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r2 = (r2 ^ cond_negate) - cond_negate;
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r3 = (r3 ^ cond_negate) - cond_negate;
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r4 = (r4 ^ cond_negate) - cond_negate;
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r5 = (r5 ^ cond_negate) - cond_negate;
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r6 = (r6 ^ cond_negate) - cond_negate;
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r7 = (r7 ^ cond_negate) - cond_negate;
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r8 = (r8 ^ cond_negate) - cond_negate;
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r1 += r0 >> 30; r0 &= M30;
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r2 += r1 >> 30; r1 &= M30;
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r3 += r2 >> 30; r2 &= M30;
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r4 += r3 >> 30; r3 &= M30;
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r5 += r4 >> 30; r4 &= M30;
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r6 += r5 >> 30; r5 &= M30;
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r7 += r6 >> 30; r6 &= M30;
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r8 += r7 >> 30; r7 &= M30;
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cond_add = r8 >> 31;
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r0 += modinfo->modulus.v[0] & cond_add;
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r1 += modinfo->modulus.v[1] & cond_add;
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r2 += modinfo->modulus.v[2] & cond_add;
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r3 += modinfo->modulus.v[3] & cond_add;
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r4 += modinfo->modulus.v[4] & cond_add;
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r5 += modinfo->modulus.v[5] & cond_add;
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r6 += modinfo->modulus.v[6] & cond_add;
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r7 += modinfo->modulus.v[7] & cond_add;
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r8 += modinfo->modulus.v[8] & cond_add;
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r1 += r0 >> 30; r0 &= M30;
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r2 += r1 >> 30; r1 &= M30;
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r3 += r2 >> 30; r2 &= M30;
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r4 += r3 >> 30; r3 &= M30;
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r5 += r4 >> 30; r4 &= M30;
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r6 += r5 >> 30; r5 &= M30;
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r7 += r6 >> 30; r6 &= M30;
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r8 += r7 >> 30; r7 &= M30;
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r->v[0] = r0;
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r->v[1] = r1;
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r->v[2] = r2;
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r->v[3] = r3;
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r->v[4] = r4;
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r->v[5] = r5;
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r->v[6] = r6;
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r->v[7] = r7;
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r->v[8] = r8;
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}
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typedef struct {
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int32_t u, v, q, r;
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} secp256k1_modinv32_trans2x2;
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static int32_t secp256k1_modinv32_divsteps_30(int32_t eta, uint32_t f0, uint32_t g0, secp256k1_modinv32_trans2x2 *t) {
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uint32_t u = 1, v = 0, q = 0, r = 1;
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uint32_t c1, c2, f = f0, g = g0, x, y, z;
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int i;
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for (i = 0; i < 30; ++i) {
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VERIFY_CHECK((f & 1) == 1);
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VERIFY_CHECK((u * f0 + v * g0) == f << i);
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VERIFY_CHECK((q * f0 + r * g0) == g << i);
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c1 = eta >> 31;
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c2 = -(g & 1);
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x = (f ^ c1) - c1;
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y = (u ^ c1) - c1;
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z = (v ^ c1) - c1;
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g += x & c2;
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q += y & c2;
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r += z & c2;
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c1 &= c2;
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eta = (eta ^ c1) - (c1 + 1);
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f += g & c1;
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u += q & c1;
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v += r & c1;
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g >>= 1;
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u <<= 1;
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v <<= 1;
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}
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t->u = (int32_t)u;
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t->v = (int32_t)v;
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t->q = (int32_t)q;
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t->r = (int32_t)r;
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return eta;
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}
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static int32_t secp256k1_modinv32_divsteps_30_var(int32_t eta, uint32_t f0, uint32_t g0, secp256k1_modinv32_trans2x2 *t) {
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/* inv256[i] = -(2*i+1)^-1 (mod 256) */
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static const uint8_t inv256[128] = {
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0xFF, 0x55, 0x33, 0x49, 0xC7, 0x5D, 0x3B, 0x11, 0x0F, 0xE5, 0xC3, 0x59,
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0xD7, 0xED, 0xCB, 0x21, 0x1F, 0x75, 0x53, 0x69, 0xE7, 0x7D, 0x5B, 0x31,
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0x2F, 0x05, 0xE3, 0x79, 0xF7, 0x0D, 0xEB, 0x41, 0x3F, 0x95, 0x73, 0x89,
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0x07, 0x9D, 0x7B, 0x51, 0x4F, 0x25, 0x03, 0x99, 0x17, 0x2D, 0x0B, 0x61,
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0x5F, 0xB5, 0x93, 0xA9, 0x27, 0xBD, 0x9B, 0x71, 0x6F, 0x45, 0x23, 0xB9,
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0x37, 0x4D, 0x2B, 0x81, 0x7F, 0xD5, 0xB3, 0xC9, 0x47, 0xDD, 0xBB, 0x91,
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0x8F, 0x65, 0x43, 0xD9, 0x57, 0x6D, 0x4B, 0xA1, 0x9F, 0xF5, 0xD3, 0xE9,
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0x67, 0xFD, 0xDB, 0xB1, 0xAF, 0x85, 0x63, 0xF9, 0x77, 0x8D, 0x6B, 0xC1,
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0xBF, 0x15, 0xF3, 0x09, 0x87, 0x1D, 0xFB, 0xD1, 0xCF, 0xA5, 0x83, 0x19,
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0x97, 0xAD, 0x8B, 0xE1, 0xDF, 0x35, 0x13, 0x29, 0xA7, 0x3D, 0x1B, 0xF1,
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0xEF, 0xC5, 0xA3, 0x39, 0xB7, 0xCD, 0xAB, 0x01
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};
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uint32_t u = 1, v = 0, q = 0, r = 1;
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uint32_t f = f0, g = g0, m;
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uint16_t w;
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int i = 30, limit, zeros;
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for (;;) {
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/* Use a sentinel bit to count zeros only up to i. */
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zeros = secp256k1_ctz32_var(g | (UINT32_MAX << i));
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g >>= zeros;
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u <<= zeros;
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v <<= zeros;
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eta -= zeros;
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i -= zeros;
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if (i <= 0) {
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break;
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}
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VERIFY_CHECK((f & 1) == 1);
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VERIFY_CHECK((g & 1) == 1);
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VERIFY_CHECK((u * f0 + v * g0) == f << (30 - i));
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VERIFY_CHECK((q * f0 + r * g0) == g << (30 - i));
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if (eta < 0) {
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uint32_t tmp;
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eta = -eta;
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tmp = f; f = g; g = -tmp;
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tmp = u; u = q; q = -tmp;
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tmp = v; v = r; r = -tmp;
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}
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/* Handle up to 8 divsteps at once, subject to eta and i. */
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limit = ((int)eta + 1) > i ? i : ((int)eta + 1);
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m = (UINT32_MAX >> (32 - limit)) & 255U;
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w = (g * inv256[(f >> 1) & 127]) & m;
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g += f * w;
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q += u * w;
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r += v * w;
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VERIFY_CHECK((g & m) == 0);
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}
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t->u = (int32_t)u;
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t->v = (int32_t)v;
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t->q = (int32_t)q;
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t->r = (int32_t)r;
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return eta;
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}
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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) {
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const int32_t M30 = (int32_t)(UINT32_MAX >> 2);
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const int32_t u = t->u, v = t->v, q = t->q, r = t->r;
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int32_t di, ei, md, me, sd, se;
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int64_t cd, ce;
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int i;
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/*
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* On input, d/e must be in the range (-2.P, P). For initially negative d (resp. e), we add
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* u and/or v (resp. q and/or r) multiples of the modulus to the corresponding output (prior
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* to division by 2^30). This has the same effect as if we added the modulus to the input(s).
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*/
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sd = d->v[8] >> 31;
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se = e->v[8] >> 31;
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md = (u & sd) + (v & se);
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me = (q & sd) + (r & se);
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di = d->v[0];
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ei = e->v[0];
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cd = (int64_t)u * di + (int64_t)v * ei;
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ce = (int64_t)q * di + (int64_t)r * ei;
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/*
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* Subtract from md/me an extra term in the range [0, 2^30) such that the low 30 bits of each
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* sum of products will be 0. This allows clean division by 2^30. On output, d/e are thus in
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* the range (-2.P, P), consistent with the input constraint.
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*/
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md -= (modinfo->modulus_inv30 * (uint32_t)cd + md) & M30;
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me -= (modinfo->modulus_inv30 * (uint32_t)ce + me) & M30;
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cd += (int64_t)modinfo->modulus.v[0] * md;
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ce += (int64_t)modinfo->modulus.v[0] * me;
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VERIFY_CHECK(((int32_t)cd & M30) == 0); cd >>= 30;
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VERIFY_CHECK(((int32_t)ce & M30) == 0); ce >>= 30;
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for (i = 1; i < 9; ++i) {
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di = d->v[i];
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ei = e->v[i];
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cd += (int64_t)u * di + (int64_t)v * ei;
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ce += (int64_t)q * di + (int64_t)r * ei;
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cd += (int64_t)modinfo->modulus.v[i] * md;
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ce += (int64_t)modinfo->modulus.v[i] * me;
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d->v[i - 1] = (int32_t)cd & M30; cd >>= 30;
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e->v[i - 1] = (int32_t)ce & M30; ce >>= 30;
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}
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d->v[8] = (int32_t)cd;
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e->v[8] = (int32_t)ce;
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}
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static void secp256k1_modinv32_update_fg_30(secp256k1_modinv32_signed30 *f, secp256k1_modinv32_signed30 *g, const secp256k1_modinv32_trans2x2 *t) {
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const int32_t M30 = (int32_t)(UINT32_MAX >> 2);
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const int32_t u = t->u, v = t->v, q = t->q, r = t->r;
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int32_t fi, gi;
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int64_t cf, cg;
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int i;
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fi = f->v[0];
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gi = g->v[0];
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cf = (int64_t)u * fi + (int64_t)v * gi;
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cg = (int64_t)q * fi + (int64_t)r * gi;
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VERIFY_CHECK(((int32_t)cf & M30) == 0);
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VERIFY_CHECK(((int32_t)cg & M30) == 0);
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cf >>= 30;
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cg >>= 30;
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for (i = 1; i < 9; ++i) {
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fi = f->v[i];
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gi = g->v[i];
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cf += (int64_t)u * fi + (int64_t)v * gi;
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cg += (int64_t)q * fi + (int64_t)r * gi;
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f->v[i - 1] = (int32_t)cf & M30; cf >>= 30;
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g->v[i - 1] = (int32_t)cg & M30; cg >>= 30;
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}
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f->v[8] = (int32_t)cf;
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g->v[8] = (int32_t)cg;
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}
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static void secp256k1_modinv32(secp256k1_modinv32_signed30 *x, const secp256k1_modinv32_modinfo *modinfo) {
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/* Modular inversion based on the paper "Fast constant-time gcd computation and
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* modular inversion" by Daniel J. Bernstein and Bo-Yin Yang. */
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secp256k1_modinv32_signed30 d = {{0}};
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secp256k1_modinv32_signed30 e = {{1}};
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secp256k1_modinv32_signed30 f = modinfo->modulus;
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secp256k1_modinv32_signed30 g = *x;
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int i;
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int32_t eta;
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/* The paper uses 'delta'; eta == -delta (a performance tweak).
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*
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* If the maximum bitlength of g is known to be less than 256, then eta can be set
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* initially to -(1 + (256 - maxlen(g))), and only (741 - (256 - maxlen(g))) total
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* divsteps are needed. */
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eta = -1;
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for (i = 0; i < 25; ++i) {
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secp256k1_modinv32_trans2x2 t;
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eta = secp256k1_modinv32_divsteps_30(eta, f.v[0], g.v[0], &t);
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secp256k1_modinv32_update_de_30(&d, &e, &t, modinfo);
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secp256k1_modinv32_update_fg_30(&f, &g, &t);
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}
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/* At this point sufficient iterations have been performed that g must have reached 0
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* and (if g was not originally 0) f must now equal +/- GCD of the initial f, g
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* values i.e. +/- 1, and d now contains +/- the modular inverse. */
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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);
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secp256k1_modinv32_normalize_30(&d, f.v[8] >> 31, modinfo);
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*x = d;
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}
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static void secp256k1_modinv32_var(secp256k1_modinv32_signed30 *x, const secp256k1_modinv32_modinfo *modinfo) {
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/* Modular inversion based on the paper "Fast constant-time gcd computation and
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* modular inversion" by Daniel J. Bernstein and Bo-Yin Yang. */
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secp256k1_modinv32_signed30 d = {{0, 0, 0, 0, 0, 0, 0, 0, 0}};
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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 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;
|
||||
|
||||
while (1) {
|
||||
secp256k1_modinv32_trans2x2 t;
|
||||
eta = secp256k1_modinv32_divsteps_30_var(eta, f.v[0], g.v[0], &t);
|
||||
secp256k1_modinv32_update_de_30(&d, &e, &t, modinfo);
|
||||
secp256k1_modinv32_update_fg_30(&f, &g, &t);
|
||||
if (g.v[0] == 0) {
|
||||
cond = 0;
|
||||
for (j = 1; j < 9; ++j) {
|
||||
cond |= g.v[j];
|
||||
}
|
||||
if (cond == 0) break;
|
||||
}
|
||||
}
|
||||
|
||||
/* 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);
|
||||
|
||||
*x = d;
|
||||
}
|
||||
|
||||
#endif /* SECP256K1_MODINV32_IMPL_H */
|
||||
|
|
@ -0,0 +1,35 @@
|
|||
/***********************************************************************
|
||||
* Copyright (c) 2020 Peter Dettman *
|
||||
* Distributed under the MIT software license, see the accompanying *
|
||||
* file COPYING or https://www.opensource.org/licenses/mit-license.php.*
|
||||
**********************************************************************/
|
||||
|
||||
#ifndef SECP256K1_MODINV64_H
|
||||
#define SECP256K1_MODINV64_H
|
||||
|
||||
#if defined HAVE_CONFIG_H
|
||||
#include "libsecp256k1-config.h"
|
||||
#endif
|
||||
|
||||
#include "util.h"
|
||||
|
||||
#ifndef SECP256K1_WIDEMUL_INT128
|
||||
#error "modinv64 requires 128-bit wide multiplication support"
|
||||
#endif
|
||||
|
||||
typedef struct {
|
||||
int64_t v[5];
|
||||
} secp256k1_modinv64_signed62;
|
||||
|
||||
typedef struct {
|
||||
/* The modulus in signed62 notation. */
|
||||
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);
|
||||
static void secp256k1_modinv64_var(secp256k1_modinv64_signed62 *x, const secp256k1_modinv64_modinfo *modinfo);
|
||||
|
||||
#endif /* SECP256K1_MODINV64_H */
|
||||
|
|
@ -0,0 +1,367 @@
|
|||
/***********************************************************************
|
||||
* Copyright (c) 2020 Peter Dettman *
|
||||
* Distributed under the MIT software license, see the accompanying *
|
||||
* file COPYING or https://www.opensource.org/licenses/mit-license.php.*
|
||||
**********************************************************************/
|
||||
|
||||
#ifndef SECP256K1_MODINV64_IMPL_H
|
||||
#define SECP256K1_MODINV64_IMPL_H
|
||||
|
||||
#include "modinv64.h"
|
||||
|
||||
#include "util.h"
|
||||
|
||||
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;
|
||||
|
||||
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;
|
||||
|
||||
r1 += r0 >> 62; r0 &= M62;
|
||||
r2 += r1 >> 62; r1 &= M62;
|
||||
r3 += r2 >> 62; r2 &= M62;
|
||||
r4 += r3 >> 62; r3 &= M62;
|
||||
|
||||
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;
|
||||
|
||||
r1 += r0 >> 62; r0 &= M62;
|
||||
r2 += r1 >> 62; r1 &= M62;
|
||||
r3 += r2 >> 62; r2 &= M62;
|
||||
r4 += r3 >> 62; r3 &= M62;
|
||||
|
||||
r->v[0] = r0;
|
||||
r->v[1] = r1;
|
||||
r->v[2] = r2;
|
||||
r->v[3] = r3;
|
||||
r->v[4] = r4;
|
||||
}
|
||||
|
||||
typedef struct {
|
||||
int64_t u, v, q, r;
|
||||
} secp256k1_modinv64_trans2x2;
|
||||
|
||||
static int64_t secp256k1_modinv64_divsteps_62(int64_t eta, uint64_t f0, uint64_t g0, secp256k1_modinv64_trans2x2 *t) {
|
||||
|
||||
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((u * f0 + v * g0) == f << i);
|
||||
VERIFY_CHECK((q * f0 + r * g0) == g << i);
|
||||
|
||||
c1 = eta >> 63;
|
||||
c2 = -(g & 1);
|
||||
|
||||
x = (f ^ c1) - c1;
|
||||
y = (u ^ c1) - c1;
|
||||
z = (v ^ c1) - c1;
|
||||
|
||||
g += x & c2;
|
||||
q += y & c2;
|
||||
r += z & c2;
|
||||
|
||||
c1 &= c2;
|
||||
eta = (eta ^ c1) - (c1 + 1);
|
||||
|
||||
f += g & c1;
|
||||
u += q & c1;
|
||||
v += r & c1;
|
||||
|
||||
g >>= 1;
|
||||
u <<= 1;
|
||||
v <<= 1;
|
||||
}
|
||||
|
||||
t->u = (int64_t)u;
|
||||
t->v = (int64_t)v;
|
||||
t->q = (int64_t)q;
|
||||
t->r = (int64_t)r;
|
||||
|
||||
return eta;
|
||||
}
|
||||
|
||||
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] = {
|
||||
0xFF, 0x55, 0x33, 0x49, 0xC7, 0x5D, 0x3B, 0x11, 0x0F, 0xE5, 0xC3, 0x59,
|
||||
0xD7, 0xED, 0xCB, 0x21, 0x1F, 0x75, 0x53, 0x69, 0xE7, 0x7D, 0x5B, 0x31,
|
||||
0x2F, 0x05, 0xE3, 0x79, 0xF7, 0x0D, 0xEB, 0x41, 0x3F, 0x95, 0x73, 0x89,
|
||||
0x07, 0x9D, 0x7B, 0x51, 0x4F, 0x25, 0x03, 0x99, 0x17, 0x2D, 0x0B, 0x61,
|
||||
0x5F, 0xB5, 0x93, 0xA9, 0x27, 0xBD, 0x9B, 0x71, 0x6F, 0x45, 0x23, 0xB9,
|
||||
0x37, 0x4D, 0x2B, 0x81, 0x7F, 0xD5, 0xB3, 0xC9, 0x47, 0xDD, 0xBB, 0x91,
|
||||
0x8F, 0x65, 0x43, 0xD9, 0x57, 0x6D, 0x4B, 0xA1, 0x9F, 0xF5, 0xD3, 0xE9,
|
||||
0x67, 0xFD, 0xDB, 0xB1, 0xAF, 0x85, 0x63, 0xF9, 0x77, 0x8D, 0x6B, 0xC1,
|
||||
0xBF, 0x15, 0xF3, 0x09, 0x87, 0x1D, 0xFB, 0xD1, 0xCF, 0xA5, 0x83, 0x19,
|
||||
0x97, 0xAD, 0x8B, 0xE1, 0xDF, 0x35, 0x13, 0x29, 0xA7, 0x3D, 0x1B, 0xF1,
|
||||
0xEF, 0xC5, 0xA3, 0x39, 0xB7, 0xCD, 0xAB, 0x01
|
||||
};
|
||||
|
||||
uint64_t u = 1, v = 0, q = 0, r = 1;
|
||||
uint64_t f = f0, g = g0, m;
|
||||
uint32_t w;
|
||||
int i = 62, limit, zeros;
|
||||
|
||||
for (;;) {
|
||||
/* Use a sentinel bit to count zeros only up to i. */
|
||||
zeros = secp256k1_ctz64_var(g | (UINT64_MAX << i));
|
||||
|
||||
g >>= zeros;
|
||||
u <<= zeros;
|
||||
v <<= zeros;
|
||||
eta -= zeros;
|
||||
i -= zeros;
|
||||
|
||||
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 < 0) {
|
||||
uint64_t tmp;
|
||||
eta = -eta;
|
||||
tmp = f; f = g; g = -tmp;
|
||||
tmp = u; u = q; q = -tmp;
|
||||
tmp = v; v = r; r = -tmp;
|
||||
}
|
||||
|
||||
/* Handle up to 8 divsteps at once, subject to eta and i. */
|
||||
limit = ((int)eta + 1) > i ? i : ((int)eta + 1);
|
||||
m = (UINT64_MAX >> (64 - limit)) & 255U;
|
||||
|
||||
w = (g * inv256[(f >> 1) & 127]) & m;
|
||||
|
||||
g += f * w;
|
||||
q += u * w;
|
||||
r += v * w;
|
||||
|
||||
VERIFY_CHECK((g & m) == 0);
|
||||
}
|
||||
|
||||
t->u = (int64_t)u;
|
||||
t->v = (int64_t)v;
|
||||
t->q = (int64_t)q;
|
||||
t->r = (int64_t)r;
|
||||
|
||||
return eta;
|
||||
}
|
||||
|
||||
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];
|
||||
const int64_t e0 = e->v[0], e1 = e->v[1], e2 = e->v[2], e3 = e->v[3], e4 = e->v[4];
|
||||
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).
|
||||
*/
|
||||
|
||||
sd = d4 >> 63;
|
||||
se = e4 >> 63;
|
||||
|
||||
md = (u & sd) + (v & se);
|
||||
me = (q & sd) + (r & se);
|
||||
|
||||
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.
|
||||
*/
|
||||
|
||||
md -= (modinfo->modulus_inv62 * (uint64_t)cd + md) & M62;
|
||||
me -= (modinfo->modulus_inv62 * (uint64_t)ce + me) & M62;
|
||||
|
||||
cd += (int128_t)modinfo->modulus.v[0] * md;
|
||||
ce += (int128_t)modinfo->modulus.v[0] * me;
|
||||
|
||||
VERIFY_CHECK(((int64_t)cd & M62) == 0); cd >>= 62;
|
||||
VERIFY_CHECK(((int64_t)ce & M62) == 0); ce >>= 62;
|
||||
|
||||
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;
|
||||
|
||||
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;
|
||||
|
||||
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;
|
||||
|
||||
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;
|
||||
|
||||
d->v[4] = (int64_t)cd;
|
||||
e->v[4] = (int64_t)ce;
|
||||
}
|
||||
|
||||
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;
|
||||
|
||||
cf = (int128_t)u * f0 + (int128_t)v * g0;
|
||||
cg = (int128_t)q * f0 + (int128_t)r * g0;
|
||||
|
||||
VERIFY_CHECK(((int64_t)cf & M62) == 0); cf >>= 62;
|
||||
VERIFY_CHECK(((int64_t)cg & M62) == 0); cg >>= 62;
|
||||
|
||||
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;
|
||||
|
||||
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;
|
||||
|
||||
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;
|
||||
|
||||
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;
|
||||
|
||||
f->v[4] = (int64_t)cf;
|
||||
g->v[4] = (int64_t)cg;
|
||||
}
|
||||
|
||||
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. */
|
||||
|
||||
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;
|
||||
|
||||
for (i = 0; i < 12; ++i) {
|
||||
secp256k1_modinv64_trans2x2 t;
|
||||
eta = secp256k1_modinv64_divsteps_62(eta, f.v[0], g.v[0], &t);
|
||||
secp256k1_modinv64_update_de_62(&d, &e, &t, modinfo);
|
||||
secp256k1_modinv64_update_fg_62(&f, &g, &t);
|
||||
}
|
||||
|
||||
/* At this point sufficient iterations have been performed that g must have reached 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. */
|
||||
VERIFY_CHECK((g.v[0] | g.v[1] | g.v[2] | g.v[3] | g.v[4]) == 0);
|
||||
|
||||
secp256k1_modinv64_normalize_62(&d, f.v[4], modinfo);
|
||||
|
||||
*x = d;
|
||||
}
|
||||
|
||||
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. */
|
||||
|
||||
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 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;
|
||||
|
||||
while (1) {
|
||||
secp256k1_modinv64_trans2x2 t;
|
||||
eta = secp256k1_modinv64_divsteps_62_var(eta, f.v[0], g.v[0], &t);
|
||||
secp256k1_modinv64_update_de_62(&d, &e, &t, modinfo);
|
||||
secp256k1_modinv64_update_fg_62(&f, &g, &t);
|
||||
if (g.v[0] == 0) {
|
||||
cond = 0;
|
||||
for (j = 1; j < 5; ++j) {
|
||||
cond |= g.v[j];
|
||||
}
|
||||
if (cond == 0) break;
|
||||
}
|
||||
}
|
||||
|
||||
secp256k1_modinv64_normalize_62(&d, f.v[4], modinfo);
|
||||
|
||||
*x = d;
|
||||
}
|
||||
|
||||
#endif /* SECP256K1_MODINV64_IMPL_H */
|
||||
Loading…
Reference in New Issue