/* crypto/ec/ec2_mult.c */ /* ==================================================================== * Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED. * * The Elliptic Curve Public-Key Crypto Library (ECC Code) included * herein is developed by SUN MICROSYSTEMS, INC., and is contributed * to the OpenSSL project. * * The ECC Code is licensed pursuant to the OpenSSL open source * license provided below. * * The software is originally written by Sheueling Chang Shantz and * Douglas Stebila of Sun Microsystems Laboratories. * */ /* ==================================================================== * Copyright (c) 1998-2003 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). * */ #include <openssl/err.h> #include "ec_lcl.h" #ifndef OPENSSL_NO_EC2M /*- * Compute the x-coordinate x/z for the point 2*(x/z) in Montgomery projective * coordinates. * Uses algorithm Mdouble in appendix of * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over * GF(2^m) without precomputation" (CHES '99, LNCS 1717). * modified to not require precomputation of c=b^{2^{m-1}}. */ static int gf2m_Mdouble(const EC_GROUP *group, BIGNUM *x, BIGNUM *z, BN_CTX *ctx) { BIGNUM *t1; int ret = 0; /* Since Mdouble is static we can guarantee that ctx != NULL. */ BN_CTX_start(ctx); t1 = BN_CTX_get(ctx); if (t1 == NULL) goto err; if (!group->meth->field_sqr(group, x, x, ctx)) goto err; if (!group->meth->field_sqr(group, t1, z, ctx)) goto err; if (!group->meth->field_mul(group, z, x, t1, ctx)) goto err; if (!group->meth->field_sqr(group, x, x, ctx)) goto err; if (!group->meth->field_sqr(group, t1, t1, ctx)) goto err; if (!group->meth->field_mul(group, t1, &group->b, t1, ctx)) goto err; if (!BN_GF2m_add(x, x, t1)) goto err; ret = 1; err: BN_CTX_end(ctx); return ret; } /*- * Compute the x-coordinate x1/z1 for the point (x1/z1)+(x2/x2) in Montgomery * projective coordinates. * Uses algorithm Madd in appendix of * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over * GF(2^m) without precomputation" (CHES '99, LNCS 1717). */ static int gf2m_Madd(const EC_GROUP *group, const BIGNUM *x, BIGNUM *x1, BIGNUM *z1, const BIGNUM *x2, const BIGNUM *z2, BN_CTX *ctx) { BIGNUM *t1, *t2; int ret = 0; /* Since Madd is static we can guarantee that ctx != NULL. */ BN_CTX_start(ctx); t1 = BN_CTX_get(ctx); t2 = BN_CTX_get(ctx); if (t2 == NULL) goto err; if (!BN_copy(t1, x)) goto err; if (!group->meth->field_mul(group, x1, x1, z2, ctx)) goto err; if (!group->meth->field_mul(group, z1, z1, x2, ctx)) goto err; if (!group->meth->field_mul(group, t2, x1, z1, ctx)) goto err; if (!BN_GF2m_add(z1, z1, x1)) goto err; if (!group->meth->field_sqr(group, z1, z1, ctx)) goto err; if (!group->meth->field_mul(group, x1, z1, t1, ctx)) goto err; if (!BN_GF2m_add(x1, x1, t2)) goto err; ret = 1; err: BN_CTX_end(ctx); return ret; } /*- * Compute the x, y affine coordinates from the point (x1, z1) (x2, z2) * using Montgomery point multiplication algorithm Mxy() in appendix of * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over * GF(2^m) without precomputation" (CHES '99, LNCS 1717). * Returns: * 0 on error * 1 if return value should be the point at infinity * 2 otherwise */ static int gf2m_Mxy(const EC_GROUP *group, const BIGNUM *x, const BIGNUM *y, BIGNUM *x1, BIGNUM *z1, BIGNUM *x2, BIGNUM *z2, BN_CTX *ctx) { BIGNUM *t3, *t4, *t5; int ret = 0; if (BN_is_zero(z1)) { BN_zero(x2); BN_zero(z2); return 1; } if (BN_is_zero(z2)) { if (!BN_copy(x2, x)) return 0; if (!BN_GF2m_add(z2, x, y)) return 0; return 2; } /* Since Mxy is static we can guarantee that ctx != NULL. */ BN_CTX_start(ctx); t3 = BN_CTX_get(ctx); t4 = BN_CTX_get(ctx); t5 = BN_CTX_get(ctx); if (t5 == NULL) goto err; if (!BN_one(t5)) goto err; if (!group->meth->field_mul(group, t3, z1, z2, ctx)) goto err; if (!group->meth->field_mul(group, z1, z1, x, ctx)) goto err; if (!BN_GF2m_add(z1, z1, x1)) goto err; if (!group->meth->field_mul(group, z2, z2, x, ctx)) goto err; if (!group->meth->field_mul(group, x1, z2, x1, ctx)) goto err; if (!BN_GF2m_add(z2, z2, x2)) goto err; if (!group->meth->field_mul(group, z2, z2, z1, ctx)) goto err; if (!group->meth->field_sqr(group, t4, x, ctx)) goto err; if (!BN_GF2m_add(t4, t4, y)) goto err; if (!group->meth->field_mul(group, t4, t4, t3, ctx)) goto err; if (!BN_GF2m_add(t4, t4, z2)) goto err; if (!group->meth->field_mul(group, t3, t3, x, ctx)) goto err; if (!group->meth->field_div(group, t3, t5, t3, ctx)) goto err; if (!group->meth->field_mul(group, t4, t3, t4, ctx)) goto err; if (!group->meth->field_mul(group, x2, x1, t3, ctx)) goto err; if (!BN_GF2m_add(z2, x2, x)) goto err; if (!group->meth->field_mul(group, z2, z2, t4, ctx)) goto err; if (!BN_GF2m_add(z2, z2, y)) goto err; ret = 2; err: BN_CTX_end(ctx); return ret; } /*- * Computes scalar*point and stores the result in r. * point can not equal r. * Uses a modified algorithm 2P of * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over * GF(2^m) without precomputation" (CHES '99, LNCS 1717). * * To protect against side-channel attack the function uses constant time swap, * avoiding conditional branches. */ static int ec_GF2m_montgomery_point_multiply(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar, const EC_POINT *point, BN_CTX *ctx) { BIGNUM *x1, *x2, *z1, *z2; int ret = 0, i, group_top; BN_ULONG mask, word; if (r == point) { ECerr(EC_F_EC_GF2M_MONTGOMERY_POINT_MULTIPLY, EC_R_INVALID_ARGUMENT); return 0; } /* if result should be point at infinity */ if ((scalar == NULL) || BN_is_zero(scalar) || (point == NULL) || EC_POINT_is_at_infinity(group, point)) { return EC_POINT_set_to_infinity(group, r); } /* only support affine coordinates */ if (!point->Z_is_one) return 0; /* * Since point_multiply is static we can guarantee that ctx != NULL. */ BN_CTX_start(ctx); x1 = BN_CTX_get(ctx); z1 = BN_CTX_get(ctx); if (z1 == NULL) goto err; x2 = &r->X; z2 = &r->Y; group_top = group->field.top; if (bn_wexpand(x1, group_top) == NULL || bn_wexpand(z1, group_top) == NULL || bn_wexpand(x2, group_top) == NULL || bn_wexpand(z2, group_top) == NULL) goto err; if (!BN_GF2m_mod_arr(x1, &point->X, group->poly)) goto err; /* x1 = x */ if (!BN_one(z1)) goto err; /* z1 = 1 */ if (!group->meth->field_sqr(group, z2, x1, ctx)) goto err; /* z2 = x1^2 = x^2 */ if (!group->meth->field_sqr(group, x2, z2, ctx)) goto err; if (!BN_GF2m_add(x2, x2, &group->b)) goto err; /* x2 = x^4 + b */ /* find top most bit and go one past it */ i = scalar->top - 1; mask = BN_TBIT; word = scalar->d[i]; while (!(word & mask)) mask >>= 1; mask >>= 1; /* if top most bit was at word break, go to next word */ if (!mask) { i--; mask = BN_TBIT; } for (; i >= 0; i--) { word = scalar->d[i]; while (mask) { BN_consttime_swap(word & mask, x1, x2, group_top); BN_consttime_swap(word & mask, z1, z2, group_top); if (!gf2m_Madd(group, &point->X, x2, z2, x1, z1, ctx)) goto err; if (!gf2m_Mdouble(group, x1, z1, ctx)) goto err; BN_consttime_swap(word & mask, x1, x2, group_top); BN_consttime_swap(word & mask, z1, z2, group_top); mask >>= 1; } mask = BN_TBIT; } /* convert out of "projective" coordinates */ i = gf2m_Mxy(group, &point->X, &point->Y, x1, z1, x2, z2, ctx); if (i == 0) goto err; else if (i == 1) { if (!EC_POINT_set_to_infinity(group, r)) goto err; } else { if (!BN_one(&r->Z)) goto err; r->Z_is_one = 1; } /* GF(2^m) field elements should always have BIGNUM::neg = 0 */ BN_set_negative(&r->X, 0); BN_set_negative(&r->Y, 0); ret = 1; err: BN_CTX_end(ctx); return ret; } /*- * Computes the sum * scalar*group->generator + scalars[0]*points[0] + ... + scalars[num-1]*points[num-1] * gracefully ignoring NULL scalar values. */ int ec_GF2m_simple_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar, size_t num, const EC_POINT *points[], const BIGNUM *scalars[], BN_CTX *ctx) { BN_CTX *new_ctx = NULL; int ret = 0; size_t i; EC_POINT *p = NULL; EC_POINT *acc = NULL; if (ctx == NULL) { ctx = new_ctx = BN_CTX_new(); if (ctx == NULL) return 0; } /* * This implementation is more efficient than the wNAF implementation for * 2 or fewer points. Use the ec_wNAF_mul implementation for 3 or more * points, or if we can perform a fast multiplication based on * precomputation. */ if ((scalar && (num > 1)) || (num > 2) || (num == 0 && EC_GROUP_have_precompute_mult(group))) { ret = ec_wNAF_mul(group, r, scalar, num, points, scalars, ctx); goto err; } if ((p = EC_POINT_new(group)) == NULL) goto err; if ((acc = EC_POINT_new(group)) == NULL) goto err; if (!EC_POINT_set_to_infinity(group, acc)) goto err; if (scalar) { if (!ec_GF2m_montgomery_point_multiply (group, p, scalar, group->generator, ctx)) goto err; if (BN_is_negative(scalar)) if (!group->meth->invert(group, p, ctx)) goto err; if (!group->meth->add(group, acc, acc, p, ctx)) goto err; } for (i = 0; i < num; i++) { if (!ec_GF2m_montgomery_point_multiply (group, p, scalars[i], points[i], ctx)) goto err; if (BN_is_negative(scalars[i])) if (!group->meth->invert(group, p, ctx)) goto err; if (!group->meth->add(group, acc, acc, p, ctx)) goto err; } if (!EC_POINT_copy(r, acc)) goto err; ret = 1; err: if (p) EC_POINT_free(p); if (acc) EC_POINT_free(acc); if (new_ctx != NULL) BN_CTX_free(new_ctx); return ret; } /* * Precomputation for point multiplication: fall back to wNAF methods because * ec_GF2m_simple_mul() uses ec_wNAF_mul() if appropriate */ int ec_GF2m_precompute_mult(EC_GROUP *group, BN_CTX *ctx) { return ec_wNAF_precompute_mult(group, ctx); } int ec_GF2m_have_precompute_mult(const EC_GROUP *group) { return ec_wNAF_have_precompute_mult(group); } #endif