/* Originally written by Bodo Moeller and Nils Larsch for the OpenSSL project. * ==================================================================== * Copyright (c) 1998-2005 The OpenSSL Project. All rights reserved. * * Redistribution and use in source and binary forms, with or without * modification, are permitted provided that the following conditions * are met: * * 1. Redistributions of source code must retain the above copyright * notice, this list of conditions and the following disclaimer. * * 2. Redistributions in binary form must reproduce the above copyright * notice, this list of conditions and the following disclaimer in * the documentation and/or other materials provided with the * distribution. * * 3. All advertising materials mentioning features or use of this * software must display the following acknowledgment: * "This product includes software developed by the OpenSSL Project * for use in the OpenSSL Toolkit. (http://www.openssl.org/)" * * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to * endorse or promote products derived from this software without * prior written permission. For written permission, please contact * openssl-core@openssl.org. * * 5. Products derived from this software may not be called "OpenSSL" * nor may "OpenSSL" appear in their names without prior written * permission of the OpenSSL Project. * * 6. Redistributions of any form whatsoever must retain the following * acknowledgment: * "This product includes software developed by the OpenSSL Project * for use in the OpenSSL Toolkit (http://www.openssl.org/)" * * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED * OF THE POSSIBILITY OF SUCH DAMAGE. * ==================================================================== * * This product includes cryptographic software written by Eric Young * (eay@cryptsoft.com). This product includes software written by Tim * Hudson (tjh@cryptsoft.com). * */ /* ==================================================================== * Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED. * * Portions of the attached software ("Contribution") are developed by * SUN MICROSYSTEMS, INC., and are contributed to the OpenSSL project. * * The Contribution is licensed pursuant to the OpenSSL open source * license provided above. * * The elliptic curve binary polynomial software is originally written by * Sheueling Chang Shantz and Douglas Stebila of Sun Microsystems * Laboratories. */ #include #include #include #include #include "../bn/internal.h" #include "../delocate.h" #include "internal.h" static void ec_GFp_mont_felem_to_montgomery(const EC_GROUP *group, EC_FELEM *out, const EC_FELEM *in) { bn_to_montgomery_small(out->words, in->words, group->field.N.width, &group->field); } static void ec_GFp_mont_felem_from_montgomery(const EC_GROUP *group, EC_FELEM *out, const EC_FELEM *in) { bn_from_montgomery_small(out->words, group->field.N.width, in->words, group->field.N.width, &group->field); } static void ec_GFp_mont_felem_inv0(const EC_GROUP *group, EC_FELEM *out, const EC_FELEM *a) { bn_mod_inverse0_prime_mont_small(out->words, a->words, group->field.N.width, &group->field); } void ec_GFp_mont_felem_mul(const EC_GROUP *group, EC_FELEM *r, const EC_FELEM *a, const EC_FELEM *b) { bn_mod_mul_montgomery_small(r->words, a->words, b->words, group->field.N.width, &group->field); } void ec_GFp_mont_felem_sqr(const EC_GROUP *group, EC_FELEM *r, const EC_FELEM *a) { bn_mod_mul_montgomery_small(r->words, a->words, a->words, group->field.N.width, &group->field); } void ec_GFp_mont_felem_to_bytes(const EC_GROUP *group, uint8_t *out, size_t *out_len, const EC_FELEM *in) { EC_FELEM tmp; ec_GFp_mont_felem_from_montgomery(group, &tmp, in); ec_GFp_simple_felem_to_bytes(group, out, out_len, &tmp); } int ec_GFp_mont_felem_from_bytes(const EC_GROUP *group, EC_FELEM *out, const uint8_t *in, size_t len) { if (!ec_GFp_simple_felem_from_bytes(group, out, in, len)) { return 0; } ec_GFp_mont_felem_to_montgomery(group, out, out); return 1; } void ec_GFp_mont_felem_reduce(const EC_GROUP *group, EC_FELEM *out, const BN_ULONG *words, size_t num) { // Convert "from" Montgomery form so the value is reduced mod p. bn_from_montgomery_small(out->words, group->field.N.width, words, num, &group->field); // Convert "to" Montgomery form to remove the R^-1 factor added. ec_GFp_mont_felem_to_montgomery(group, out, out); // Convert to Montgomery form to match this implementation's representation. ec_GFp_mont_felem_to_montgomery(group, out, out); } void ec_GFp_mont_felem_exp(const EC_GROUP *group, EC_FELEM *out, const EC_FELEM *a, const BN_ULONG *exp, size_t num_exp) { bn_mod_exp_mont_small(out->words, a->words, group->field.N.width, exp, num_exp, &group->field); } static int ec_GFp_mont_point_get_affine_coordinates(const EC_GROUP *group, const EC_JACOBIAN *point, EC_FELEM *x, EC_FELEM *y) { if (constant_time_declassify_int( ec_GFp_simple_is_at_infinity(group, point))) { OPENSSL_PUT_ERROR(EC, EC_R_POINT_AT_INFINITY); return 0; } // Transform (X, Y, Z) into (x, y) := (X/Z^2, Y/Z^3). Note the check above // ensures |point->Z| is non-zero, so the inverse always exists. EC_FELEM z1, z2; ec_GFp_mont_felem_inv0(group, &z2, &point->Z); ec_GFp_mont_felem_sqr(group, &z1, &z2); if (x != NULL) { ec_GFp_mont_felem_mul(group, x, &point->X, &z1); } if (y != NULL) { ec_GFp_mont_felem_mul(group, &z1, &z1, &z2); ec_GFp_mont_felem_mul(group, y, &point->Y, &z1); } return 1; } static int ec_GFp_mont_jacobian_to_affine_batch(const EC_GROUP *group, EC_AFFINE *out, const EC_JACOBIAN *in, size_t num) { if (num == 0) { return 1; } // Compute prefix products of all Zs. Use |out[i].X| as scratch space // to store these values. out[0].X = in[0].Z; for (size_t i = 1; i < num; i++) { ec_GFp_mont_felem_mul(group, &out[i].X, &out[i - 1].X, &in[i].Z); } // Some input was infinity iff the product of all Zs is zero. if (ec_felem_non_zero_mask(group, &out[num - 1].X) == 0) { OPENSSL_PUT_ERROR(EC, EC_R_POINT_AT_INFINITY); return 0; } // Invert the product of all Zs. EC_FELEM zinvprod; ec_GFp_mont_felem_inv0(group, &zinvprod, &out[num - 1].X); for (size_t i = num - 1; i < num; i--) { // Our loop invariant is that |zinvprod| is Z0^-1 * Z1^-1 * ... * Zi^-1. // Recover Zi^-1 by multiplying by the previous product. EC_FELEM zinv, zinv2; if (i == 0) { zinv = zinvprod; } else { ec_GFp_mont_felem_mul(group, &zinv, &zinvprod, &out[i - 1].X); // Maintain the loop invariant for the next iteration. ec_GFp_mont_felem_mul(group, &zinvprod, &zinvprod, &in[i].Z); } // Compute affine coordinates: x = X * Z^-2 and y = Y * Z^-3. ec_GFp_mont_felem_sqr(group, &zinv2, &zinv); ec_GFp_mont_felem_mul(group, &out[i].X, &in[i].X, &zinv2); ec_GFp_mont_felem_mul(group, &out[i].Y, &in[i].Y, &zinv2); ec_GFp_mont_felem_mul(group, &out[i].Y, &out[i].Y, &zinv); } return 1; } void ec_GFp_mont_add(const EC_GROUP *group, EC_JACOBIAN *out, const EC_JACOBIAN *a, const EC_JACOBIAN *b) { if (a == b) { ec_GFp_mont_dbl(group, out, a); return; } // The method is taken from: // http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian.html#addition-add-2007-bl // // Coq transcription and correctness proof: // // EC_FELEM x_out, y_out, z_out; BN_ULONG z1nz = ec_felem_non_zero_mask(group, &a->Z); BN_ULONG z2nz = ec_felem_non_zero_mask(group, &b->Z); // z1z1 = z1z1 = z1**2 EC_FELEM z1z1; ec_GFp_mont_felem_sqr(group, &z1z1, &a->Z); // z2z2 = z2**2 EC_FELEM z2z2; ec_GFp_mont_felem_sqr(group, &z2z2, &b->Z); // u1 = x1*z2z2 EC_FELEM u1; ec_GFp_mont_felem_mul(group, &u1, &a->X, &z2z2); // two_z1z2 = (z1 + z2)**2 - (z1z1 + z2z2) = 2z1z2 EC_FELEM two_z1z2; ec_felem_add(group, &two_z1z2, &a->Z, &b->Z); ec_GFp_mont_felem_sqr(group, &two_z1z2, &two_z1z2); ec_felem_sub(group, &two_z1z2, &two_z1z2, &z1z1); ec_felem_sub(group, &two_z1z2, &two_z1z2, &z2z2); // s1 = y1 * z2**3 EC_FELEM s1; ec_GFp_mont_felem_mul(group, &s1, &b->Z, &z2z2); ec_GFp_mont_felem_mul(group, &s1, &s1, &a->Y); // u2 = x2*z1z1 EC_FELEM u2; ec_GFp_mont_felem_mul(group, &u2, &b->X, &z1z1); // h = u2 - u1 EC_FELEM h; ec_felem_sub(group, &h, &u2, &u1); BN_ULONG xneq = ec_felem_non_zero_mask(group, &h); // z_out = two_z1z2 * h ec_GFp_mont_felem_mul(group, &z_out, &h, &two_z1z2); // z1z1z1 = z1 * z1z1 EC_FELEM z1z1z1; ec_GFp_mont_felem_mul(group, &z1z1z1, &a->Z, &z1z1); // s2 = y2 * z1**3 EC_FELEM s2; ec_GFp_mont_felem_mul(group, &s2, &b->Y, &z1z1z1); // r = (s2 - s1)*2 EC_FELEM r; ec_felem_sub(group, &r, &s2, &s1); ec_felem_add(group, &r, &r, &r); BN_ULONG yneq = ec_felem_non_zero_mask(group, &r); // This case will never occur in the constant-time |ec_GFp_mont_mul|. BN_ULONG is_nontrivial_double = ~xneq & ~yneq & z1nz & z2nz; if (constant_time_declassify_w(is_nontrivial_double)) { ec_GFp_mont_dbl(group, out, a); return; } // I = (2h)**2 EC_FELEM i; ec_felem_add(group, &i, &h, &h); ec_GFp_mont_felem_sqr(group, &i, &i); // J = h * I EC_FELEM j; ec_GFp_mont_felem_mul(group, &j, &h, &i); // V = U1 * I EC_FELEM v; ec_GFp_mont_felem_mul(group, &v, &u1, &i); // x_out = r**2 - J - 2V ec_GFp_mont_felem_sqr(group, &x_out, &r); ec_felem_sub(group, &x_out, &x_out, &j); ec_felem_sub(group, &x_out, &x_out, &v); ec_felem_sub(group, &x_out, &x_out, &v); // y_out = r(V-x_out) - 2 * s1 * J ec_felem_sub(group, &y_out, &v, &x_out); ec_GFp_mont_felem_mul(group, &y_out, &y_out, &r); EC_FELEM s1j; ec_GFp_mont_felem_mul(group, &s1j, &s1, &j); ec_felem_sub(group, &y_out, &y_out, &s1j); ec_felem_sub(group, &y_out, &y_out, &s1j); ec_felem_select(group, &x_out, z1nz, &x_out, &b->X); ec_felem_select(group, &out->X, z2nz, &x_out, &a->X); ec_felem_select(group, &y_out, z1nz, &y_out, &b->Y); ec_felem_select(group, &out->Y, z2nz, &y_out, &a->Y); ec_felem_select(group, &z_out, z1nz, &z_out, &b->Z); ec_felem_select(group, &out->Z, z2nz, &z_out, &a->Z); } void ec_GFp_mont_dbl(const EC_GROUP *group, EC_JACOBIAN *r, const EC_JACOBIAN *a) { if (group->a_is_minus3) { // The method is taken from: // http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b // // Coq transcription and correctness proof: // // EC_FELEM delta, gamma, beta, ftmp, ftmp2, tmptmp, alpha, fourbeta; // delta = z^2 ec_GFp_mont_felem_sqr(group, &delta, &a->Z); // gamma = y^2 ec_GFp_mont_felem_sqr(group, &gamma, &a->Y); // beta = x*gamma ec_GFp_mont_felem_mul(group, &beta, &a->X, &gamma); // alpha = 3*(x-delta)*(x+delta) ec_felem_sub(group, &ftmp, &a->X, &delta); ec_felem_add(group, &ftmp2, &a->X, &delta); ec_felem_add(group, &tmptmp, &ftmp2, &ftmp2); ec_felem_add(group, &ftmp2, &ftmp2, &tmptmp); ec_GFp_mont_felem_mul(group, &alpha, &ftmp, &ftmp2); // x' = alpha^2 - 8*beta ec_GFp_mont_felem_sqr(group, &r->X, &alpha); ec_felem_add(group, &fourbeta, &beta, &beta); ec_felem_add(group, &fourbeta, &fourbeta, &fourbeta); ec_felem_add(group, &tmptmp, &fourbeta, &fourbeta); ec_felem_sub(group, &r->X, &r->X, &tmptmp); // z' = (y + z)^2 - gamma - delta ec_felem_add(group, &delta, &gamma, &delta); ec_felem_add(group, &ftmp, &a->Y, &a->Z); ec_GFp_mont_felem_sqr(group, &r->Z, &ftmp); ec_felem_sub(group, &r->Z, &r->Z, &delta); // y' = alpha*(4*beta - x') - 8*gamma^2 ec_felem_sub(group, &r->Y, &fourbeta, &r->X); ec_felem_add(group, &gamma, &gamma, &gamma); ec_GFp_mont_felem_sqr(group, &gamma, &gamma); ec_GFp_mont_felem_mul(group, &r->Y, &alpha, &r->Y); ec_felem_add(group, &gamma, &gamma, &gamma); ec_felem_sub(group, &r->Y, &r->Y, &gamma); } else { // The method is taken from: // http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian.html#doubling-dbl-2007-bl // // Coq transcription and correctness proof: // // EC_FELEM xx, yy, yyyy, zz; ec_GFp_mont_felem_sqr(group, &xx, &a->X); ec_GFp_mont_felem_sqr(group, &yy, &a->Y); ec_GFp_mont_felem_sqr(group, &yyyy, &yy); ec_GFp_mont_felem_sqr(group, &zz, &a->Z); // s = 2*((x_in + yy)^2 - xx - yyyy) EC_FELEM s; ec_felem_add(group, &s, &a->X, &yy); ec_GFp_mont_felem_sqr(group, &s, &s); ec_felem_sub(group, &s, &s, &xx); ec_felem_sub(group, &s, &s, &yyyy); ec_felem_add(group, &s, &s, &s); // m = 3*xx + a*zz^2 EC_FELEM m; ec_GFp_mont_felem_sqr(group, &m, &zz); ec_GFp_mont_felem_mul(group, &m, &group->a, &m); ec_felem_add(group, &m, &m, &xx); ec_felem_add(group, &m, &m, &xx); ec_felem_add(group, &m, &m, &xx); // x_out = m^2 - 2*s ec_GFp_mont_felem_sqr(group, &r->X, &m); ec_felem_sub(group, &r->X, &r->X, &s); ec_felem_sub(group, &r->X, &r->X, &s); // z_out = (y_in + z_in)^2 - yy - zz ec_felem_add(group, &r->Z, &a->Y, &a->Z); ec_GFp_mont_felem_sqr(group, &r->Z, &r->Z); ec_felem_sub(group, &r->Z, &r->Z, &yy); ec_felem_sub(group, &r->Z, &r->Z, &zz); // y_out = m*(s-x_out) - 8*yyyy ec_felem_add(group, &yyyy, &yyyy, &yyyy); ec_felem_add(group, &yyyy, &yyyy, &yyyy); ec_felem_add(group, &yyyy, &yyyy, &yyyy); ec_felem_sub(group, &r->Y, &s, &r->X); ec_GFp_mont_felem_mul(group, &r->Y, &r->Y, &m); ec_felem_sub(group, &r->Y, &r->Y, &yyyy); } } static int ec_GFp_mont_cmp_x_coordinate(const EC_GROUP *group, const EC_JACOBIAN *p, const EC_SCALAR *r) { if (!group->field_greater_than_order || group->field.N.width != group->order.N.width) { // Do not bother optimizing this case. p > order in all commonly-used // curves. return ec_GFp_simple_cmp_x_coordinate(group, p, r); } if (ec_GFp_simple_is_at_infinity(group, p)) { return 0; } // We wish to compare X/Z^2 with r. This is equivalent to comparing X with // r*Z^2. Note that X and Z are represented in Montgomery form, while r is // not. EC_FELEM r_Z2, Z2_mont, X; ec_GFp_mont_felem_mul(group, &Z2_mont, &p->Z, &p->Z); // r < order < p, so this is valid. OPENSSL_memcpy(r_Z2.words, r->words, group->field.N.width * sizeof(BN_ULONG)); ec_GFp_mont_felem_mul(group, &r_Z2, &r_Z2, &Z2_mont); ec_GFp_mont_felem_from_montgomery(group, &X, &p->X); if (ec_felem_equal(group, &r_Z2, &X)) { return 1; } // During signing the x coefficient is reduced modulo the group order. // Therefore there is a small possibility, less than 1/2^128, that group_order // < p.x < P. in that case we need not only to compare against |r| but also to // compare against r+group_order. BN_ULONG carry = bn_add_words(r_Z2.words, r->words, group->order.N.d, group->field.N.width); if (carry == 0 && bn_less_than_words(r_Z2.words, group->field.N.d, group->field.N.width)) { // r + group_order < p, so compare (r + group_order) * Z^2 against X. ec_GFp_mont_felem_mul(group, &r_Z2, &r_Z2, &Z2_mont); if (ec_felem_equal(group, &r_Z2, &X)) { return 1; } } return 0; } DEFINE_METHOD_FUNCTION(EC_METHOD, EC_GFp_mont_method) { out->point_get_affine_coordinates = ec_GFp_mont_point_get_affine_coordinates; out->jacobian_to_affine_batch = ec_GFp_mont_jacobian_to_affine_batch; out->add = ec_GFp_mont_add; out->dbl = ec_GFp_mont_dbl; out->mul = ec_GFp_mont_mul; out->mul_base = ec_GFp_mont_mul_base; out->mul_batch = ec_GFp_mont_mul_batch; out->mul_public_batch = ec_GFp_mont_mul_public_batch; out->init_precomp = ec_GFp_mont_init_precomp; out->mul_precomp = ec_GFp_mont_mul_precomp; out->felem_mul = ec_GFp_mont_felem_mul; out->felem_sqr = ec_GFp_mont_felem_sqr; out->felem_to_bytes = ec_GFp_mont_felem_to_bytes; out->felem_from_bytes = ec_GFp_mont_felem_from_bytes; out->felem_reduce = ec_GFp_mont_felem_reduce; out->felem_exp = ec_GFp_mont_felem_exp; out->scalar_inv0_montgomery = ec_simple_scalar_inv0_montgomery; out->scalar_to_montgomery_inv_vartime = ec_simple_scalar_to_montgomery_inv_vartime; out->cmp_x_coordinate = ec_GFp_mont_cmp_x_coordinate; }