/* * Copyright 2002-2017 The OpenSSL Project Authors. All Rights Reserved. * Copyright (c) 2002, Oracle and/or its affiliates. All rights reserved * * Licensed under the OpenSSL license (the "License"). You may not use * this file except in compliance with the License. You can obtain a copy * in the file LICENSE in the source distribution or at * https://www.openssl.org/source/license.html */ #include #include #include #include "internal/cryptlib.h" #include "bn_lcl.h" #ifndef OPENSSL_NO_EC2M /* * Maximum number of iterations before BN_GF2m_mod_solve_quad_arr should * fail. */ # define MAX_ITERATIONS 50 static const BN_ULONG SQR_tb[16] = { 0, 1, 4, 5, 16, 17, 20, 21, 64, 65, 68, 69, 80, 81, 84, 85 }; /* Platform-specific macros to accelerate squaring. */ # if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG) # define SQR1(w) \ SQR_tb[(w) >> 60 & 0xF] << 56 | SQR_tb[(w) >> 56 & 0xF] << 48 | \ SQR_tb[(w) >> 52 & 0xF] << 40 | SQR_tb[(w) >> 48 & 0xF] << 32 | \ SQR_tb[(w) >> 44 & 0xF] << 24 | SQR_tb[(w) >> 40 & 0xF] << 16 | \ SQR_tb[(w) >> 36 & 0xF] << 8 | SQR_tb[(w) >> 32 & 0xF] # define SQR0(w) \ SQR_tb[(w) >> 28 & 0xF] << 56 | SQR_tb[(w) >> 24 & 0xF] << 48 | \ SQR_tb[(w) >> 20 & 0xF] << 40 | SQR_tb[(w) >> 16 & 0xF] << 32 | \ SQR_tb[(w) >> 12 & 0xF] << 24 | SQR_tb[(w) >> 8 & 0xF] << 16 | \ SQR_tb[(w) >> 4 & 0xF] << 8 | SQR_tb[(w) & 0xF] # endif # ifdef THIRTY_TWO_BIT # define SQR1(w) \ SQR_tb[(w) >> 28 & 0xF] << 24 | SQR_tb[(w) >> 24 & 0xF] << 16 | \ SQR_tb[(w) >> 20 & 0xF] << 8 | SQR_tb[(w) >> 16 & 0xF] # define SQR0(w) \ SQR_tb[(w) >> 12 & 0xF] << 24 | SQR_tb[(w) >> 8 & 0xF] << 16 | \ SQR_tb[(w) >> 4 & 0xF] << 8 | SQR_tb[(w) & 0xF] # endif # if !defined(OPENSSL_BN_ASM_GF2m) /* * Product of two polynomials a, b each with degree < BN_BITS2 - 1, result is * a polynomial r with degree < 2 * BN_BITS - 1 The caller MUST ensure that * the variables have the right amount of space allocated. */ # ifdef THIRTY_TWO_BIT static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a, const BN_ULONG b) { register BN_ULONG h, l, s; BN_ULONG tab[8], top2b = a >> 30; register BN_ULONG a1, a2, a4; a1 = a & (0x3FFFFFFF); a2 = a1 << 1; a4 = a2 << 1; tab[0] = 0; tab[1] = a1; tab[2] = a2; tab[3] = a1 ^ a2; tab[4] = a4; tab[5] = a1 ^ a4; tab[6] = a2 ^ a4; tab[7] = a1 ^ a2 ^ a4; s = tab[b & 0x7]; l = s; s = tab[b >> 3 & 0x7]; l ^= s << 3; h = s >> 29; s = tab[b >> 6 & 0x7]; l ^= s << 6; h ^= s >> 26; s = tab[b >> 9 & 0x7]; l ^= s << 9; h ^= s >> 23; s = tab[b >> 12 & 0x7]; l ^= s << 12; h ^= s >> 20; s = tab[b >> 15 & 0x7]; l ^= s << 15; h ^= s >> 17; s = tab[b >> 18 & 0x7]; l ^= s << 18; h ^= s >> 14; s = tab[b >> 21 & 0x7]; l ^= s << 21; h ^= s >> 11; s = tab[b >> 24 & 0x7]; l ^= s << 24; h ^= s >> 8; s = tab[b >> 27 & 0x7]; l ^= s << 27; h ^= s >> 5; s = tab[b >> 30]; l ^= s << 30; h ^= s >> 2; /* compensate for the top two bits of a */ if (top2b & 01) { l ^= b << 30; h ^= b >> 2; } if (top2b & 02) { l ^= b << 31; h ^= b >> 1; } *r1 = h; *r0 = l; } # endif # if defined(SIXTY_FOUR_BIT) || defined(SIXTY_FOUR_BIT_LONG) static void bn_GF2m_mul_1x1(BN_ULONG *r1, BN_ULONG *r0, const BN_ULONG a, const BN_ULONG b) { register BN_ULONG h, l, s; BN_ULONG tab[16], top3b = a >> 61; register BN_ULONG a1, a2, a4, a8; a1 = a & (0x1FFFFFFFFFFFFFFFULL); a2 = a1 << 1; a4 = a2 << 1; a8 = a4 << 1; tab[0] = 0; tab[1] = a1; tab[2] = a2; tab[3] = a1 ^ a2; tab[4] = a4; tab[5] = a1 ^ a4; tab[6] = a2 ^ a4; tab[7] = a1 ^ a2 ^ a4; tab[8] = a8; tab[9] = a1 ^ a8; tab[10] = a2 ^ a8; tab[11] = a1 ^ a2 ^ a8; tab[12] = a4 ^ a8; tab[13] = a1 ^ a4 ^ a8; tab[14] = a2 ^ a4 ^ a8; tab[15] = a1 ^ a2 ^ a4 ^ a8; s = tab[b & 0xF]; l = s; s = tab[b >> 4 & 0xF]; l ^= s << 4; h = s >> 60; s = tab[b >> 8 & 0xF]; l ^= s << 8; h ^= s >> 56; s = tab[b >> 12 & 0xF]; l ^= s << 12; h ^= s >> 52; s = tab[b >> 16 & 0xF]; l ^= s << 16; h ^= s >> 48; s = tab[b >> 20 & 0xF]; l ^= s << 20; h ^= s >> 44; s = tab[b >> 24 & 0xF]; l ^= s << 24; h ^= s >> 40; s = tab[b >> 28 & 0xF]; l ^= s << 28; h ^= s >> 36; s = tab[b >> 32 & 0xF]; l ^= s << 32; h ^= s >> 32; s = tab[b >> 36 & 0xF]; l ^= s << 36; h ^= s >> 28; s = tab[b >> 40 & 0xF]; l ^= s << 40; h ^= s >> 24; s = tab[b >> 44 & 0xF]; l ^= s << 44; h ^= s >> 20; s = tab[b >> 48 & 0xF]; l ^= s << 48; h ^= s >> 16; s = tab[b >> 52 & 0xF]; l ^= s << 52; h ^= s >> 12; s = tab[b >> 56 & 0xF]; l ^= s << 56; h ^= s >> 8; s = tab[b >> 60]; l ^= s << 60; h ^= s >> 4; /* compensate for the top three bits of a */ if (top3b & 01) { l ^= b << 61; h ^= b >> 3; } if (top3b & 02) { l ^= b << 62; h ^= b >> 2; } if (top3b & 04) { l ^= b << 63; h ^= b >> 1; } *r1 = h; *r0 = l; } # endif /* * Product of two polynomials a, b each with degree < 2 * BN_BITS2 - 1, * result is a polynomial r with degree < 4 * BN_BITS2 - 1 The caller MUST * ensure that the variables have the right amount of space allocated. */ static void bn_GF2m_mul_2x2(BN_ULONG *r, const BN_ULONG a1, const BN_ULONG a0, const BN_ULONG b1, const BN_ULONG b0) { BN_ULONG m1, m0; /* r[3] = h1, r[2] = h0; r[1] = l1; r[0] = l0 */ bn_GF2m_mul_1x1(r + 3, r + 2, a1, b1); bn_GF2m_mul_1x1(r + 1, r, a0, b0); bn_GF2m_mul_1x1(&m1, &m0, a0 ^ a1, b0 ^ b1); /* Correction on m1 ^= l1 ^ h1; m0 ^= l0 ^ h0; */ r[2] ^= m1 ^ r[1] ^ r[3]; /* h0 ^= m1 ^ l1 ^ h1; */ r[1] = r[3] ^ r[2] ^ r[0] ^ m1 ^ m0; /* l1 ^= l0 ^ h0 ^ m0; */ } # else void bn_GF2m_mul_2x2(BN_ULONG *r, BN_ULONG a1, BN_ULONG a0, BN_ULONG b1, BN_ULONG b0); # endif /* * Add polynomials a and b and store result in r; r could be a or b, a and b * could be equal; r is the bitwise XOR of a and b. */ int BN_GF2m_add(BIGNUM *r, const BIGNUM *a, const BIGNUM *b) { int i; const BIGNUM *at, *bt; bn_check_top(a); bn_check_top(b); if (a->top < b->top) { at = b; bt = a; } else { at = a; bt = b; } if (bn_wexpand(r, at->top) == NULL) return 0; for (i = 0; i < bt->top; i++) { r->d[i] = at->d[i] ^ bt->d[i]; } for (; i < at->top; i++) { r->d[i] = at->d[i]; } r->top = at->top; bn_correct_top(r); return 1; } /*- * Some functions allow for representation of the irreducible polynomials * as an int[], say p. The irreducible f(t) is then of the form: * t^p[0] + t^p[1] + ... + t^p[k] * where m = p[0] > p[1] > ... > p[k] = 0. */ /* Performs modular reduction of a and store result in r. r could be a. */ int BN_GF2m_mod_arr(BIGNUM *r, const BIGNUM *a, const int p[]) { int j, k; int n, dN, d0, d1; BN_ULONG zz, *z; bn_check_top(a); if (!p[0]) { /* reduction mod 1 => return 0 */ BN_zero(r); return 1; } /* * Since the algorithm does reduction in the r value, if a != r, copy the * contents of a into r so we can do reduction in r. */ if (a != r) { if (!bn_wexpand(r, a->top)) return 0; for (j = 0; j < a->top; j++) { r->d[j] = a->d[j]; } r->top = a->top; } z = r->d; /* start reduction */ dN = p[0] / BN_BITS2; for (j = r->top - 1; j > dN;) { zz = z[j]; if (z[j] == 0) { j--; continue; } z[j] = 0; for (k = 1; p[k] != 0; k++) { /* reducing component t^p[k] */ n = p[0] - p[k]; d0 = n % BN_BITS2; d1 = BN_BITS2 - d0; n /= BN_BITS2; z[j - n] ^= (zz >> d0); if (d0) z[j - n - 1] ^= (zz << d1); } /* reducing component t^0 */ n = dN; d0 = p[0] % BN_BITS2; d1 = BN_BITS2 - d0; z[j - n] ^= (zz >> d0); if (d0) z[j - n - 1] ^= (zz << d1); } /* final round of reduction */ while (j == dN) { d0 = p[0] % BN_BITS2; zz = z[dN] >> d0; if (zz == 0) break; d1 = BN_BITS2 - d0; /* clear up the top d1 bits */ if (d0) z[dN] = (z[dN] << d1) >> d1; else z[dN] = 0; z[0] ^= zz; /* reduction t^0 component */ for (k = 1; p[k] != 0; k++) { BN_ULONG tmp_ulong; /* reducing component t^p[k] */ n = p[k] / BN_BITS2; d0 = p[k] % BN_BITS2; d1 = BN_BITS2 - d0; z[n] ^= (zz << d0); if (d0 && (tmp_ulong = zz >> d1)) z[n + 1] ^= tmp_ulong; } } bn_correct_top(r); return 1; } /* * Performs modular reduction of a by p and store result in r. r could be a. * This function calls down to the BN_GF2m_mod_arr implementation; this wrapper * function is only provided for convenience; for best performance, use the * BN_GF2m_mod_arr function. */ int BN_GF2m_mod(BIGNUM *r, const BIGNUM *a, const BIGNUM *p) { int ret = 0; int arr[6]; bn_check_top(a); bn_check_top(p); ret = BN_GF2m_poly2arr(p, arr, OSSL_NELEM(arr)); if (!ret || ret > (int)OSSL_NELEM(arr)) { BNerr(BN_F_BN_GF2M_MOD, BN_R_INVALID_LENGTH); return 0; } ret = BN_GF2m_mod_arr(r, a, arr); bn_check_top(r); return ret; } /* * Compute the product of two polynomials a and b, reduce modulo p, and store * the result in r. r could be a or b; a could be b. */ int BN_GF2m_mod_mul_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const int p[], BN_CTX *ctx) { int zlen, i, j, k, ret = 0; BIGNUM *s; BN_ULONG x1, x0, y1, y0, zz[4]; bn_check_top(a); bn_check_top(b); if (a == b) { return BN_GF2m_mod_sqr_arr(r, a, p, ctx); } BN_CTX_start(ctx); if ((s = BN_CTX_get(ctx)) == NULL) goto err; zlen = a->top + b->top + 4; if (!bn_wexpand(s, zlen)) goto err; s->top = zlen; for (i = 0; i < zlen; i++) s->d[i] = 0; for (j = 0; j < b->top; j += 2) { y0 = b->d[j]; y1 = ((j + 1) == b->top) ? 0 : b->d[j + 1]; for (i = 0; i < a->top; i += 2) { x0 = a->d[i]; x1 = ((i + 1) == a->top) ? 0 : a->d[i + 1]; bn_GF2m_mul_2x2(zz, x1, x0, y1, y0); for (k = 0; k < 4; k++) s->d[i + j + k] ^= zz[k]; } } bn_correct_top(s); if (BN_GF2m_mod_arr(r, s, p)) ret = 1; bn_check_top(r); err: BN_CTX_end(ctx); return ret; } /* * Compute the product of two polynomials a and b, reduce modulo p, and store * the result in r. r could be a or b; a could equal b. This function calls * down to the BN_GF2m_mod_mul_arr implementation; this wrapper function is * only provided for convenience; for best performance, use the * BN_GF2m_mod_mul_arr function. */ int BN_GF2m_mod_mul(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p, BN_CTX *ctx) { int ret = 0; const int max = BN_num_bits(p) + 1; int *arr = NULL; bn_check_top(a); bn_check_top(b); bn_check_top(p); if ((arr = OPENSSL_malloc(sizeof(*arr) * max)) == NULL) goto err; ret = BN_GF2m_poly2arr(p, arr, max); if (!ret || ret > max) { BNerr(BN_F_BN_GF2M_MOD_MUL, BN_R_INVALID_LENGTH); goto err; } ret = BN_GF2m_mod_mul_arr(r, a, b, arr, ctx); bn_check_top(r); err: OPENSSL_free(arr); return ret; } /* Square a, reduce the result mod p, and store it in a. r could be a. */ int BN_GF2m_mod_sqr_arr(BIGNUM *r, const BIGNUM *a, const int p[], BN_CTX *ctx) { int i, ret = 0; BIGNUM *s; bn_check_top(a); BN_CTX_start(ctx); if ((s = BN_CTX_get(ctx)) == NULL) goto err; if (!bn_wexpand(s, 2 * a->top)) goto err; for (i = a->top - 1; i >= 0; i--) { s->d[2 * i + 1] = SQR1(a->d[i]); s->d[2 * i] = SQR0(a->d[i]); } s->top = 2 * a->top; bn_correct_top(s); if (!BN_GF2m_mod_arr(r, s, p)) goto err; bn_check_top(r); ret = 1; err: BN_CTX_end(ctx); return ret; } /* * Square a, reduce the result mod p, and store it in a. r could be a. This * function calls down to the BN_GF2m_mod_sqr_arr implementation; this * wrapper function is only provided for convenience; for best performance, * use the BN_GF2m_mod_sqr_arr function. */ int BN_GF2m_mod_sqr(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) { int ret = 0; const int max = BN_num_bits(p) + 1; int *arr = NULL; bn_check_top(a); bn_check_top(p); if ((arr = OPENSSL_malloc(sizeof(*arr) * max)) == NULL) goto err; ret = BN_GF2m_poly2arr(p, arr, max); if (!ret || ret > max) { BNerr(BN_F_BN_GF2M_MOD_SQR, BN_R_INVALID_LENGTH); goto err; } ret = BN_GF2m_mod_sqr_arr(r, a, arr, ctx); bn_check_top(r); err: OPENSSL_free(arr); return ret; } /* * Invert a, reduce modulo p, and store the result in r. r could be a. Uses * Modified Almost Inverse Algorithm (Algorithm 10) from Hankerson, D., * Hernandez, J.L., and Menezes, A. "Software Implementation of Elliptic * Curve Cryptography Over Binary Fields". */ int BN_GF2m_mod_inv(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) { BIGNUM *b, *c = NULL, *u = NULL, *v = NULL, *tmp; int ret = 0; bn_check_top(a); bn_check_top(p); BN_CTX_start(ctx); b = BN_CTX_get(ctx); c = BN_CTX_get(ctx); u = BN_CTX_get(ctx); v = BN_CTX_get(ctx); if (v == NULL) goto err; if (!BN_GF2m_mod(u, a, p)) goto err; if (BN_is_zero(u)) goto err; if (!BN_copy(v, p)) goto err; # if 0 if (!BN_one(b)) goto err; while (1) { while (!BN_is_odd(u)) { if (BN_is_zero(u)) goto err; if (!BN_rshift1(u, u)) goto err; if (BN_is_odd(b)) { if (!BN_GF2m_add(b, b, p)) goto err; } if (!BN_rshift1(b, b)) goto err; } if (BN_abs_is_word(u, 1)) break; if (BN_num_bits(u) < BN_num_bits(v)) { tmp = u; u = v; v = tmp; tmp = b; b = c; c = tmp; } if (!BN_GF2m_add(u, u, v)) goto err; if (!BN_GF2m_add(b, b, c)) goto err; } # else { int i; int ubits = BN_num_bits(u); int vbits = BN_num_bits(v); /* v is copy of p */ int top = p->top; BN_ULONG *udp, *bdp, *vdp, *cdp; if (!bn_wexpand(u, top)) goto err; udp = u->d; for (i = u->top; i < top; i++) udp[i] = 0; u->top = top; if (!bn_wexpand(b, top)) goto err; bdp = b->d; bdp[0] = 1; for (i = 1; i < top; i++) bdp[i] = 0; b->top = top; if (!bn_wexpand(c, top)) goto err; cdp = c->d; for (i = 0; i < top; i++) cdp[i] = 0; c->top = top; vdp = v->d; /* It pays off to "cache" *->d pointers, * because it allows optimizer to be more * aggressive. But we don't have to "cache" * p->d, because *p is declared 'const'... */ while (1) { while (ubits && !(udp[0] & 1)) { BN_ULONG u0, u1, b0, b1, mask; u0 = udp[0]; b0 = bdp[0]; mask = (BN_ULONG)0 - (b0 & 1); b0 ^= p->d[0] & mask; for (i = 0; i < top - 1; i++) { u1 = udp[i + 1]; udp[i] = ((u0 >> 1) | (u1 << (BN_BITS2 - 1))) & BN_MASK2; u0 = u1; b1 = bdp[i + 1] ^ (p->d[i + 1] & mask); bdp[i] = ((b0 >> 1) | (b1 << (BN_BITS2 - 1))) & BN_MASK2; b0 = b1; } udp[i] = u0 >> 1; bdp[i] = b0 >> 1; ubits--; } if (ubits <= BN_BITS2) { if (udp[0] == 0) /* poly was reducible */ goto err; if (udp[0] == 1) break; } if (ubits < vbits) { i = ubits; ubits = vbits; vbits = i; tmp = u; u = v; v = tmp; tmp = b; b = c; c = tmp; udp = vdp; vdp = v->d; bdp = cdp; cdp = c->d; } for (i = 0; i < top; i++) { udp[i] ^= vdp[i]; bdp[i] ^= cdp[i]; } if (ubits == vbits) { BN_ULONG ul; int utop = (ubits - 1) / BN_BITS2; while ((ul = udp[utop]) == 0 && utop) utop--; ubits = utop * BN_BITS2 + BN_num_bits_word(ul); } } bn_correct_top(b); } # endif if (!BN_copy(r, b)) goto err; bn_check_top(r); ret = 1; err: # ifdef BN_DEBUG /* BN_CTX_end would complain about the * expanded form */ bn_correct_top(c); bn_correct_top(u); bn_correct_top(v); # endif BN_CTX_end(ctx); return ret; } /* * Invert xx, reduce modulo p, and store the result in r. r could be xx. * This function calls down to the BN_GF2m_mod_inv implementation; this * wrapper function is only provided for convenience; for best performance, * use the BN_GF2m_mod_inv function. */ int BN_GF2m_mod_inv_arr(BIGNUM *r, const BIGNUM *xx, const int p[], BN_CTX *ctx) { BIGNUM *field; int ret = 0; bn_check_top(xx); BN_CTX_start(ctx); if ((field = BN_CTX_get(ctx)) == NULL) goto err; if (!BN_GF2m_arr2poly(p, field)) goto err; ret = BN_GF2m_mod_inv(r, xx, field, ctx); bn_check_top(r); err: BN_CTX_end(ctx); return ret; } # ifndef OPENSSL_SUN_GF2M_DIV /* * Divide y by x, reduce modulo p, and store the result in r. r could be x * or y, x could equal y. */ int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p, BN_CTX *ctx) { BIGNUM *xinv = NULL; int ret = 0; bn_check_top(y); bn_check_top(x); bn_check_top(p); BN_CTX_start(ctx); xinv = BN_CTX_get(ctx); if (xinv == NULL) goto err; if (!BN_GF2m_mod_inv(xinv, x, p, ctx)) goto err; if (!BN_GF2m_mod_mul(r, y, xinv, p, ctx)) goto err; bn_check_top(r); ret = 1; err: BN_CTX_end(ctx); return ret; } # else /* * Divide y by x, reduce modulo p, and store the result in r. r could be x * or y, x could equal y. Uses algorithm Modular_Division_GF(2^m) from * Chang-Shantz, S. "From Euclid's GCD to Montgomery Multiplication to the * Great Divide". */ int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *y, const BIGNUM *x, const BIGNUM *p, BN_CTX *ctx) { BIGNUM *a, *b, *u, *v; int ret = 0; bn_check_top(y); bn_check_top(x); bn_check_top(p); BN_CTX_start(ctx); a = BN_CTX_get(ctx); b = BN_CTX_get(ctx); u = BN_CTX_get(ctx); v = BN_CTX_get(ctx); if (v == NULL) goto err; /* reduce x and y mod p */ if (!BN_GF2m_mod(u, y, p)) goto err; if (!BN_GF2m_mod(a, x, p)) goto err; if (!BN_copy(b, p)) goto err; while (!BN_is_odd(a)) { if (!BN_rshift1(a, a)) goto err; if (BN_is_odd(u)) if (!BN_GF2m_add(u, u, p)) goto err; if (!BN_rshift1(u, u)) goto err; } do { if (BN_GF2m_cmp(b, a) > 0) { if (!BN_GF2m_add(b, b, a)) goto err; if (!BN_GF2m_add(v, v, u)) goto err; do { if (!BN_rshift1(b, b)) goto err; if (BN_is_odd(v)) if (!BN_GF2m_add(v, v, p)) goto err; if (!BN_rshift1(v, v)) goto err; } while (!BN_is_odd(b)); } else if (BN_abs_is_word(a, 1)) break; else { if (!BN_GF2m_add(a, a, b)) goto err; if (!BN_GF2m_add(u, u, v)) goto err; do { if (!BN_rshift1(a, a)) goto err; if (BN_is_odd(u)) if (!BN_GF2m_add(u, u, p)) goto err; if (!BN_rshift1(u, u)) goto err; } while (!BN_is_odd(a)); } } while (1); if (!BN_copy(r, u)) goto err; bn_check_top(r); ret = 1; err: BN_CTX_end(ctx); return ret; } # endif /* * Divide yy by xx, reduce modulo p, and store the result in r. r could be xx * * or yy, xx could equal yy. This function calls down to the * BN_GF2m_mod_div implementation; this wrapper function is only provided for * convenience; for best performance, use the BN_GF2m_mod_div function. */ int BN_GF2m_mod_div_arr(BIGNUM *r, const BIGNUM *yy, const BIGNUM *xx, const int p[], BN_CTX *ctx) { BIGNUM *field; int ret = 0; bn_check_top(yy); bn_check_top(xx); BN_CTX_start(ctx); if ((field = BN_CTX_get(ctx)) == NULL) goto err; if (!BN_GF2m_arr2poly(p, field)) goto err; ret = BN_GF2m_mod_div(r, yy, xx, field, ctx); bn_check_top(r); err: BN_CTX_end(ctx); return ret; } /* * Compute the bth power of a, reduce modulo p, and store the result in r. r * could be a. Uses simple square-and-multiply algorithm A.5.1 from IEEE * P1363. */ int BN_GF2m_mod_exp_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const int p[], BN_CTX *ctx) { int ret = 0, i, n; BIGNUM *u; bn_check_top(a); bn_check_top(b); if (BN_is_zero(b)) return (BN_one(r)); if (BN_abs_is_word(b, 1)) return (BN_copy(r, a) != NULL); BN_CTX_start(ctx); if ((u = BN_CTX_get(ctx)) == NULL) goto err; if (!BN_GF2m_mod_arr(u, a, p)) goto err; n = BN_num_bits(b) - 1; for (i = n - 1; i >= 0; i--) { if (!BN_GF2m_mod_sqr_arr(u, u, p, ctx)) goto err; if (BN_is_bit_set(b, i)) { if (!BN_GF2m_mod_mul_arr(u, u, a, p, ctx)) goto err; } } if (!BN_copy(r, u)) goto err; bn_check_top(r); ret = 1; err: BN_CTX_end(ctx); return ret; } /* * Compute the bth power of a, reduce modulo p, and store the result in r. r * could be a. This function calls down to the BN_GF2m_mod_exp_arr * implementation; this wrapper function is only provided for convenience; * for best performance, use the BN_GF2m_mod_exp_arr function. */ int BN_GF2m_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p, BN_CTX *ctx) { int ret = 0; const int max = BN_num_bits(p) + 1; int *arr = NULL; bn_check_top(a); bn_check_top(b); bn_check_top(p); if ((arr = OPENSSL_malloc(sizeof(*arr) * max)) == NULL) goto err; ret = BN_GF2m_poly2arr(p, arr, max); if (!ret || ret > max) { BNerr(BN_F_BN_GF2M_MOD_EXP, BN_R_INVALID_LENGTH); goto err; } ret = BN_GF2m_mod_exp_arr(r, a, b, arr, ctx); bn_check_top(r); err: OPENSSL_free(arr); return ret; } /* * Compute the square root of a, reduce modulo p, and store the result in r. * r could be a. Uses exponentiation as in algorithm A.4.1 from IEEE P1363. */ int BN_GF2m_mod_sqrt_arr(BIGNUM *r, const BIGNUM *a, const int p[], BN_CTX *ctx) { int ret = 0; BIGNUM *u; bn_check_top(a); if (!p[0]) { /* reduction mod 1 => return 0 */ BN_zero(r); return 1; } BN_CTX_start(ctx); if ((u = BN_CTX_get(ctx)) == NULL) goto err; if (!BN_set_bit(u, p[0] - 1)) goto err; ret = BN_GF2m_mod_exp_arr(r, a, u, p, ctx); bn_check_top(r); err: BN_CTX_end(ctx); return ret; } /* * Compute the square root of a, reduce modulo p, and store the result in r. * r could be a. This function calls down to the BN_GF2m_mod_sqrt_arr * implementation; this wrapper function is only provided for convenience; * for best performance, use the BN_GF2m_mod_sqrt_arr function. */ int BN_GF2m_mod_sqrt(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) { int ret = 0; const int max = BN_num_bits(p) + 1; int *arr = NULL; bn_check_top(a); bn_check_top(p); if ((arr = OPENSSL_malloc(sizeof(*arr) * max)) == NULL) goto err; ret = BN_GF2m_poly2arr(p, arr, max); if (!ret || ret > max) { BNerr(BN_F_BN_GF2M_MOD_SQRT, BN_R_INVALID_LENGTH); goto err; } ret = BN_GF2m_mod_sqrt_arr(r, a, arr, ctx); bn_check_top(r); err: OPENSSL_free(arr); return ret; } /* * Find r such that r^2 + r = a mod p. r could be a. If no r exists returns * 0. Uses algorithms A.4.7 and A.4.6 from IEEE P1363. */ int BN_GF2m_mod_solve_quad_arr(BIGNUM *r, const BIGNUM *a_, const int p[], BN_CTX *ctx) { int ret = 0, count = 0, j; BIGNUM *a, *z, *rho, *w, *w2, *tmp; bn_check_top(a_); if (!p[0]) { /* reduction mod 1 => return 0 */ BN_zero(r); return 1; } BN_CTX_start(ctx); a = BN_CTX_get(ctx); z = BN_CTX_get(ctx); w = BN_CTX_get(ctx); if (w == NULL) goto err; if (!BN_GF2m_mod_arr(a, a_, p)) goto err; if (BN_is_zero(a)) { BN_zero(r); ret = 1; goto err; } if (p[0] & 0x1) { /* m is odd */ /* compute half-trace of a */ if (!BN_copy(z, a)) goto err; for (j = 1; j <= (p[0] - 1) / 2; j++) { if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err; if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err; if (!BN_GF2m_add(z, z, a)) goto err; } } else { /* m is even */ rho = BN_CTX_get(ctx); w2 = BN_CTX_get(ctx); tmp = BN_CTX_get(ctx); if (tmp == NULL) goto err; do { if (!BN_rand(rho, p[0], BN_RAND_TOP_ONE, BN_RAND_BOTTOM_ANY)) goto err; if (!BN_GF2m_mod_arr(rho, rho, p)) goto err; BN_zero(z); if (!BN_copy(w, rho)) goto err; for (j = 1; j <= p[0] - 1; j++) { if (!BN_GF2m_mod_sqr_arr(z, z, p, ctx)) goto err; if (!BN_GF2m_mod_sqr_arr(w2, w, p, ctx)) goto err; if (!BN_GF2m_mod_mul_arr(tmp, w2, a, p, ctx)) goto err; if (!BN_GF2m_add(z, z, tmp)) goto err; if (!BN_GF2m_add(w, w2, rho)) goto err; } count++; } while (BN_is_zero(w) && (count < MAX_ITERATIONS)); if (BN_is_zero(w)) { BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_TOO_MANY_ITERATIONS); goto err; } } if (!BN_GF2m_mod_sqr_arr(w, z, p, ctx)) goto err; if (!BN_GF2m_add(w, z, w)) goto err; if (BN_GF2m_cmp(w, a)) { BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD_ARR, BN_R_NO_SOLUTION); goto err; } if (!BN_copy(r, z)) goto err; bn_check_top(r); ret = 1; err: BN_CTX_end(ctx); return ret; } /* * Find r such that r^2 + r = a mod p. r could be a. If no r exists returns * 0. This function calls down to the BN_GF2m_mod_solve_quad_arr * implementation; this wrapper function is only provided for convenience; * for best performance, use the BN_GF2m_mod_solve_quad_arr function. */ int BN_GF2m_mod_solve_quad(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) { int ret = 0; const int max = BN_num_bits(p) + 1; int *arr = NULL; bn_check_top(a); bn_check_top(p); if ((arr = OPENSSL_malloc(sizeof(*arr) * max)) == NULL) goto err; ret = BN_GF2m_poly2arr(p, arr, max); if (!ret || ret > max) { BNerr(BN_F_BN_GF2M_MOD_SOLVE_QUAD, BN_R_INVALID_LENGTH); goto err; } ret = BN_GF2m_mod_solve_quad_arr(r, a, arr, ctx); bn_check_top(r); err: OPENSSL_free(arr); return ret; } /* * Convert the bit-string representation of a polynomial ( \sum_{i=0}^n a_i * * x^i) into an array of integers corresponding to the bits with non-zero * coefficient. Array is terminated with -1. Up to max elements of the array * will be filled. Return value is total number of array elements that would * be filled if array was large enough. */ int BN_GF2m_poly2arr(const BIGNUM *a, int p[], int max) { int i, j, k = 0; BN_ULONG mask; if (BN_is_zero(a)) return 0; for (i = a->top - 1; i >= 0; i--) { if (!a->d[i]) /* skip word if a->d[i] == 0 */ continue; mask = BN_TBIT; for (j = BN_BITS2 - 1; j >= 0; j--) { if (a->d[i] & mask) { if (k < max) p[k] = BN_BITS2 * i + j; k++; } mask >>= 1; } } if (k < max) { p[k] = -1; k++; } return k; } /* * Convert the coefficient array representation of a polynomial to a * bit-string. The array must be terminated by -1. */ int BN_GF2m_arr2poly(const int p[], BIGNUM *a) { int i; bn_check_top(a); BN_zero(a); for (i = 0; p[i] != -1; i++) { if (BN_set_bit(a, p[i]) == 0) return 0; } bn_check_top(a); return 1; } #endif