/* $OpenBSD: ecp_methods.c,v 1.42 2025/01/25 13:15:21 tb Exp $ */ /* Includes code written by Lenka Fibikova * for the OpenSSL project. * Includes code written by Bodo Moeller for the OpenSSL project. */ /* ==================================================================== * Copyright (c) 1998-2002 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 this software developed by SUN MICROSYSTEMS, INC., * and contributed to the OpenSSL project. */ #include #include #include #include #include #include "bn_local.h" #include "ec_local.h" /* * Most method functions in this file are designed to work with non-trivial * representations of field elements if necessary: while standard modular * addition and subtraction are used, the field_mul and field_sqr methods will * be used for multiplication, and field_encode and field_decode (if defined) * will be used for converting between representations. * * The functions ec_points_make_affine() and ec_point_get_affine_coordinates() * assume that if a non-trivial representation is used, it is a Montgomery * representation (i.e. 'encoding' means multiplying by some factor R). */ static inline int ec_field_mul(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx) { return group->meth->field_mul(group, r, a, b, ctx); } static inline int ec_field_sqr(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a, BN_CTX *ctx) { return group->meth->field_sqr(group, r, a, ctx); } static int ec_decode_scalar(const EC_GROUP *group, BIGNUM *bn, const BIGNUM *x, BN_CTX *ctx) { if (bn == NULL) return 1; if (group->meth->field_decode != NULL) return group->meth->field_decode(group, bn, x, ctx); return bn_copy(bn, x); } static int ec_encode_scalar(const EC_GROUP *group, BIGNUM *bn, const BIGNUM *x, BN_CTX *ctx) { if (!BN_nnmod(bn, x, group->p, ctx)) return 0; if (group->meth->field_encode != NULL) return group->meth->field_encode(group, bn, bn, ctx); return 1; } static int ec_group_set_curve(EC_GROUP *group, const BIGNUM *p, const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx) { BIGNUM *a_plus_3; int ret = 0; /* p must be a prime > 3 */ if (BN_num_bits(p) <= 2 || !BN_is_odd(p)) { ECerror(EC_R_INVALID_FIELD); return 0; } BN_CTX_start(ctx); if ((a_plus_3 = BN_CTX_get(ctx)) == NULL) goto err; if (!bn_copy(group->p, p)) goto err; BN_set_negative(group->p, 0); if (!ec_encode_scalar(group, group->a, a, ctx)) goto err; if (!ec_encode_scalar(group, group->b, b, ctx)) goto err; if (!BN_set_word(a_plus_3, 3)) goto err; if (!BN_mod_add(a_plus_3, a_plus_3, a, group->p, ctx)) goto err; group->a_is_minus3 = BN_is_zero(a_plus_3); ret = 1; err: BN_CTX_end(ctx); return ret; } static int ec_group_get_curve(const EC_GROUP *group, BIGNUM *p, BIGNUM *a, BIGNUM *b, BN_CTX *ctx) { if (p != NULL) { if (!bn_copy(p, group->p)) return 0; } if (!ec_decode_scalar(group, a, group->a, ctx)) return 0; if (!ec_decode_scalar(group, b, group->b, ctx)) return 0; return 1; } static int ec_point_is_on_curve(const EC_GROUP *group, const EC_POINT *point, BN_CTX *ctx) { BIGNUM *rh, *tmp, *Z4, *Z6; int ret = -1; if (EC_POINT_is_at_infinity(group, point)) return 1; BN_CTX_start(ctx); if ((rh = BN_CTX_get(ctx)) == NULL) goto err; if ((tmp = BN_CTX_get(ctx)) == NULL) goto err; if ((Z4 = BN_CTX_get(ctx)) == NULL) goto err; if ((Z6 = BN_CTX_get(ctx)) == NULL) goto err; /* * The curve is defined by a Weierstrass equation y^2 = x^3 + a*x + b. * The point is given in Jacobian projective coordinates where (X, Y, Z) * represents (x, y) = (X/Z^2, Y/Z^3). Substituting this and multiplying * by Z^6 transforms the above into Y^2 = X^3 + a*X*Z^4 + b*Z^6. */ /* rh := X^2 */ if (!ec_field_sqr(group, rh, point->X, ctx)) goto err; if (!point->Z_is_one) { if (!ec_field_sqr(group, tmp, point->Z, ctx)) goto err; if (!ec_field_sqr(group, Z4, tmp, ctx)) goto err; if (!ec_field_mul(group, Z6, Z4, tmp, ctx)) goto err; /* rh := (rh + a*Z^4)*X */ if (group->a_is_minus3) { if (!BN_mod_lshift1_quick(tmp, Z4, group->p)) goto err; if (!BN_mod_add_quick(tmp, tmp, Z4, group->p)) goto err; if (!BN_mod_sub_quick(rh, rh, tmp, group->p)) goto err; if (!ec_field_mul(group, rh, rh, point->X, ctx)) goto err; } else { if (!ec_field_mul(group, tmp, Z4, group->a, ctx)) goto err; if (!BN_mod_add_quick(rh, rh, tmp, group->p)) goto err; if (!ec_field_mul(group, rh, rh, point->X, ctx)) goto err; } /* rh := rh + b*Z^6 */ if (!ec_field_mul(group, tmp, group->b, Z6, ctx)) goto err; if (!BN_mod_add_quick(rh, rh, tmp, group->p)) goto err; } else { /* point->Z_is_one */ /* rh := (rh + a)*X */ if (!BN_mod_add_quick(rh, rh, group->a, group->p)) goto err; if (!ec_field_mul(group, rh, rh, point->X, ctx)) goto err; /* rh := rh + b */ if (!BN_mod_add_quick(rh, rh, group->b, group->p)) goto err; } /* 'lh' := Y^2 */ if (!ec_field_sqr(group, tmp, point->Y, ctx)) goto err; ret = (0 == BN_ucmp(tmp, rh)); err: BN_CTX_end(ctx); return ret; } /* * Returns -1 on error, 0 if the points are equal, 1 if the points are distinct. */ static int ec_point_cmp(const EC_GROUP *group, const EC_POINT *a, const EC_POINT *b, BN_CTX *ctx) { BIGNUM *tmp1, *tmp2, *Za23, *Zb23; const BIGNUM *tmp1_, *tmp2_; int ret = -1; if (EC_POINT_is_at_infinity(group, a) && EC_POINT_is_at_infinity(group, b)) return 0; if (EC_POINT_is_at_infinity(group, a) || EC_POINT_is_at_infinity(group, b)) return 1; if (a->Z_is_one && b->Z_is_one) return BN_cmp(a->X, b->X) != 0 || BN_cmp(a->Y, b->Y) != 0; BN_CTX_start(ctx); if ((tmp1 = BN_CTX_get(ctx)) == NULL) goto end; if ((tmp2 = BN_CTX_get(ctx)) == NULL) goto end; if ((Za23 = BN_CTX_get(ctx)) == NULL) goto end; if ((Zb23 = BN_CTX_get(ctx)) == NULL) goto end; /* * Decide whether (X_a/Z_a^2, Y_a/Z_a^3) = (X_b/Z_b^2, Y_b/Z_b^3), or * equivalently, (X_a*Z_b^2, Y_a*Z_b^3) = (X_b*Z_a^2, Y_b*Z_a^3). */ if (!b->Z_is_one) { if (!ec_field_sqr(group, Zb23, b->Z, ctx)) goto end; if (!ec_field_mul(group, tmp1, a->X, Zb23, ctx)) goto end; tmp1_ = tmp1; } else tmp1_ = a->X; if (!a->Z_is_one) { if (!ec_field_sqr(group, Za23, a->Z, ctx)) goto end; if (!ec_field_mul(group, tmp2, b->X, Za23, ctx)) goto end; tmp2_ = tmp2; } else tmp2_ = b->X; /* compare X_a*Z_b^2 with X_b*Z_a^2 */ if (BN_cmp(tmp1_, tmp2_) != 0) { ret = 1; /* points differ */ goto end; } if (!b->Z_is_one) { if (!ec_field_mul(group, Zb23, Zb23, b->Z, ctx)) goto end; if (!ec_field_mul(group, tmp1, a->Y, Zb23, ctx)) goto end; /* tmp1_ = tmp1 */ } else tmp1_ = a->Y; if (!a->Z_is_one) { if (!ec_field_mul(group, Za23, Za23, a->Z, ctx)) goto end; if (!ec_field_mul(group, tmp2, b->Y, Za23, ctx)) goto end; /* tmp2_ = tmp2 */ } else tmp2_ = b->Y; /* compare Y_a*Z_b^3 with Y_b*Z_a^3 */ if (BN_cmp(tmp1_, tmp2_) != 0) { ret = 1; /* points differ */ goto end; } /* points are equal */ ret = 0; end: BN_CTX_end(ctx); return ret; } static int ec_point_set_affine_coordinates(const EC_GROUP *group, EC_POINT *point, const BIGNUM *x, const BIGNUM *y, BN_CTX *ctx) { int ret = 0; if (x == NULL || y == NULL) { ECerror(ERR_R_PASSED_NULL_PARAMETER); goto err; } if (!ec_encode_scalar(group, point->X, x, ctx)) goto err; if (!ec_encode_scalar(group, point->Y, y, ctx)) goto err; if (!ec_encode_scalar(group, point->Z, BN_value_one(), ctx)) goto err; point->Z_is_one = 1; ret = 1; err: return ret; } static int ec_point_get_affine_coordinates(const EC_GROUP *group, const EC_POINT *point, BIGNUM *x, BIGNUM *y, BN_CTX *ctx) { BIGNUM *z, *Z, *Z_1, *Z_2, *Z_3; int ret = 0; BN_CTX_start(ctx); if ((z = BN_CTX_get(ctx)) == NULL) goto err; if ((Z = BN_CTX_get(ctx)) == NULL) goto err; if ((Z_1 = BN_CTX_get(ctx)) == NULL) goto err; if ((Z_2 = BN_CTX_get(ctx)) == NULL) goto err; if ((Z_3 = BN_CTX_get(ctx)) == NULL) goto err; /* * Convert from Jacobian projective coordinates (X, Y, Z) into * (X/Z^2, Y/Z^3). */ if (!ec_decode_scalar(group, z, point->Z, ctx)) goto err; if (BN_is_one(z)) { if (!ec_decode_scalar(group, x, point->X, ctx)) goto err; if (!ec_decode_scalar(group, y, point->Y, ctx)) goto err; goto done; } if (BN_mod_inverse_ct(Z_1, z, group->p, ctx) == NULL) { ECerror(ERR_R_BN_LIB); goto err; } if (group->meth->field_encode == NULL) { /* field_sqr works on standard representation */ if (!ec_field_sqr(group, Z_2, Z_1, ctx)) goto err; } else { if (!BN_mod_sqr(Z_2, Z_1, group->p, ctx)) goto err; } if (x != NULL) { /* * in the Montgomery case, field_mul will cancel out * Montgomery factor in X: */ if (!ec_field_mul(group, x, point->X, Z_2, ctx)) goto err; } if (y != NULL) { if (group->meth->field_encode == NULL) { /* field_mul works on standard representation */ if (!ec_field_mul(group, Z_3, Z_2, Z_1, ctx)) goto err; } else { if (!BN_mod_mul(Z_3, Z_2, Z_1, group->p, ctx)) goto err; } /* * in the Montgomery case, field_mul will cancel out * Montgomery factor in Y: */ if (!ec_field_mul(group, y, point->Y, Z_3, ctx)) goto err; } done: ret = 1; err: BN_CTX_end(ctx); return ret; } static int ec_points_make_affine(const EC_GROUP *group, size_t num, EC_POINT **points, BN_CTX *ctx) { BIGNUM **prod_Z = NULL; BIGNUM *one, *tmp, *tmp_Z; size_t i; int ret = 0; if (num == 0) return 1; BN_CTX_start(ctx); if ((one = BN_CTX_get(ctx)) == NULL) goto err; if ((tmp = BN_CTX_get(ctx)) == NULL) goto err; if ((tmp_Z = BN_CTX_get(ctx)) == NULL) goto err; if (!ec_encode_scalar(group, one, BN_value_one(), ctx)) goto err; if ((prod_Z = calloc(num, sizeof *prod_Z)) == NULL) goto err; for (i = 0; i < num; i++) { if ((prod_Z[i] = BN_CTX_get(ctx)) == NULL) goto err; } /* * Set prod_Z[i] to the product of points[0]->Z, ..., points[i]->Z, * skipping any zero-valued inputs (pretend that they're 1). */ if (!BN_is_zero(points[0]->Z)) { if (!bn_copy(prod_Z[0], points[0]->Z)) goto err; } else { if (!bn_copy(prod_Z[0], one)) goto err; } for (i = 1; i < num; i++) { if (!BN_is_zero(points[i]->Z)) { if (!ec_field_mul(group, prod_Z[i], prod_Z[i - 1], points[i]->Z, ctx)) goto err; } else { if (!bn_copy(prod_Z[i], prod_Z[i - 1])) goto err; } } /* * Now use a single explicit inversion to replace every non-zero * points[i]->Z by its inverse. */ if (!BN_mod_inverse_nonct(tmp, prod_Z[num - 1], group->p, ctx)) { ECerror(ERR_R_BN_LIB); goto err; } if (group->meth->field_encode != NULL) { /* * In the Montgomery case we just turned R*H (representing H) * into 1/(R*H), but we need R*(1/H) (representing 1/H); i.e., * we need to multiply by the Montgomery factor twice. */ if (!group->meth->field_encode(group, tmp, tmp, ctx)) goto err; if (!group->meth->field_encode(group, tmp, tmp, ctx)) goto err; } for (i = num - 1; i > 0; i--) { /* * Loop invariant: tmp is the product of the inverses of * points[0]->Z, ..., points[i]->Z (zero-valued inputs skipped). */ if (BN_is_zero(points[i]->Z)) continue; /* Set tmp_Z to the inverse of points[i]->Z. */ if (!ec_field_mul(group, tmp_Z, prod_Z[i - 1], tmp, ctx)) goto err; /* Adjust tmp to satisfy loop invariant. */ if (!ec_field_mul(group, tmp, tmp, points[i]->Z, ctx)) goto err; /* Replace points[i]->Z by its inverse. */ if (!bn_copy(points[i]->Z, tmp_Z)) goto err; } if (!BN_is_zero(points[0]->Z)) { /* Replace points[0]->Z by its inverse. */ if (!bn_copy(points[0]->Z, tmp)) goto err; } /* Finally, fix up the X and Y coordinates for all points. */ for (i = 0; i < num; i++) { EC_POINT *p = points[i]; if (BN_is_zero(p->Z)) continue; /* turn (X, Y, 1/Z) into (X/Z^2, Y/Z^3, 1) */ if (!ec_field_sqr(group, tmp, p->Z, ctx)) goto err; if (!ec_field_mul(group, p->X, p->X, tmp, ctx)) goto err; if (!ec_field_mul(group, tmp, tmp, p->Z, ctx)) goto err; if (!ec_field_mul(group, p->Y, p->Y, tmp, ctx)) goto err; if (!bn_copy(p->Z, one)) goto err; p->Z_is_one = 1; } ret = 1; err: BN_CTX_end(ctx); free(prod_Z); return ret; } static int ec_add(const EC_GROUP *group, EC_POINT *r, const EC_POINT *a, const EC_POINT *b, BN_CTX *ctx) { BIGNUM *n0, *n1, *n2, *n3, *n4, *n5, *n6; int ret = 0; if (a == b) return EC_POINT_dbl(group, r, a, ctx); if (EC_POINT_is_at_infinity(group, a)) return EC_POINT_copy(r, b); if (EC_POINT_is_at_infinity(group, b)) return EC_POINT_copy(r, a); BN_CTX_start(ctx); if ((n0 = BN_CTX_get(ctx)) == NULL) goto end; if ((n1 = BN_CTX_get(ctx)) == NULL) goto end; if ((n2 = BN_CTX_get(ctx)) == NULL) goto end; if ((n3 = BN_CTX_get(ctx)) == NULL) goto end; if ((n4 = BN_CTX_get(ctx)) == NULL) goto end; if ((n5 = BN_CTX_get(ctx)) == NULL) goto end; if ((n6 = BN_CTX_get(ctx)) == NULL) goto end; /* * Note that in this function we must not read components of 'a' or * 'b' once we have written the corresponding components of 'r'. ('r' * might be one of 'a' or 'b'.) */ /* n1, n2 */ if (b->Z_is_one) { if (!bn_copy(n1, a->X)) goto end; if (!bn_copy(n2, a->Y)) goto end; /* n1 = X_a */ /* n2 = Y_a */ } else { if (!ec_field_sqr(group, n0, b->Z, ctx)) goto end; if (!ec_field_mul(group, n1, a->X, n0, ctx)) goto end; /* n1 = X_a * Z_b^2 */ if (!ec_field_mul(group, n0, n0, b->Z, ctx)) goto end; if (!ec_field_mul(group, n2, a->Y, n0, ctx)) goto end; /* n2 = Y_a * Z_b^3 */ } /* n3, n4 */ if (a->Z_is_one) { if (!bn_copy(n3, b->X)) goto end; if (!bn_copy(n4, b->Y)) goto end; /* n3 = X_b */ /* n4 = Y_b */ } else { if (!ec_field_sqr(group, n0, a->Z, ctx)) goto end; if (!ec_field_mul(group, n3, b->X, n0, ctx)) goto end; /* n3 = X_b * Z_a^2 */ if (!ec_field_mul(group, n0, n0, a->Z, ctx)) goto end; if (!ec_field_mul(group, n4, b->Y, n0, ctx)) goto end; /* n4 = Y_b * Z_a^3 */ } /* n5, n6 */ if (!BN_mod_sub_quick(n5, n1, n3, group->p)) goto end; if (!BN_mod_sub_quick(n6, n2, n4, group->p)) goto end; /* n5 = n1 - n3 */ /* n6 = n2 - n4 */ if (BN_is_zero(n5)) { if (BN_is_zero(n6)) { /* a is the same point as b */ BN_CTX_end(ctx); ret = EC_POINT_dbl(group, r, a, ctx); ctx = NULL; goto end; } else { /* a is the inverse of b */ BN_zero(r->Z); r->Z_is_one = 0; ret = 1; goto end; } } /* 'n7', 'n8' */ if (!BN_mod_add_quick(n1, n1, n3, group->p)) goto end; if (!BN_mod_add_quick(n2, n2, n4, group->p)) goto end; /* 'n7' = n1 + n3 */ /* 'n8' = n2 + n4 */ /* Z_r */ if (a->Z_is_one && b->Z_is_one) { if (!bn_copy(r->Z, n5)) goto end; } else { if (a->Z_is_one) { if (!bn_copy(n0, b->Z)) goto end; } else if (b->Z_is_one) { if (!bn_copy(n0, a->Z)) goto end; } else { if (!ec_field_mul(group, n0, a->Z, b->Z, ctx)) goto end; } if (!ec_field_mul(group, r->Z, n0, n5, ctx)) goto end; } r->Z_is_one = 0; /* Z_r = Z_a * Z_b * n5 */ /* X_r */ if (!ec_field_sqr(group, n0, n6, ctx)) goto end; if (!ec_field_sqr(group, n4, n5, ctx)) goto end; if (!ec_field_mul(group, n3, n1, n4, ctx)) goto end; if (!BN_mod_sub_quick(r->X, n0, n3, group->p)) goto end; /* X_r = n6^2 - n5^2 * 'n7' */ /* 'n9' */ if (!BN_mod_lshift1_quick(n0, r->X, group->p)) goto end; if (!BN_mod_sub_quick(n0, n3, n0, group->p)) goto end; /* n9 = n5^2 * 'n7' - 2 * X_r */ /* Y_r */ if (!ec_field_mul(group, n0, n0, n6, ctx)) goto end; if (!ec_field_mul(group, n5, n4, n5, ctx)) goto end; /* now n5 is n5^3 */ if (!ec_field_mul(group, n1, n2, n5, ctx)) goto end; if (!BN_mod_sub_quick(n0, n0, n1, group->p)) goto end; if (BN_is_odd(n0)) if (!BN_add(n0, n0, group->p)) goto end; /* now 0 <= n0 < 2*p, and n0 is even */ if (!BN_rshift1(r->Y, n0)) goto end; /* Y_r = (n6 * 'n9' - 'n8' * 'n5^3') / 2 */ ret = 1; end: BN_CTX_end(ctx); return ret; } static int ec_dbl(const EC_GROUP *group, EC_POINT *r, const EC_POINT *a, BN_CTX *ctx) { BIGNUM *n0, *n1, *n2, *n3; int ret = 0; if (EC_POINT_is_at_infinity(group, a)) return EC_POINT_set_to_infinity(group, r); BN_CTX_start(ctx); if ((n0 = BN_CTX_get(ctx)) == NULL) goto err; if ((n1 = BN_CTX_get(ctx)) == NULL) goto err; if ((n2 = BN_CTX_get(ctx)) == NULL) goto err; if ((n3 = BN_CTX_get(ctx)) == NULL) goto err; /* * Note that in this function we must not read components of 'a' once * we have written the corresponding components of 'r'. ('r' might * the same as 'a'.) */ /* n1 */ if (a->Z_is_one) { if (!ec_field_sqr(group, n0, a->X, ctx)) goto err; if (!BN_mod_lshift1_quick(n1, n0, group->p)) goto err; if (!BN_mod_add_quick(n0, n0, n1, group->p)) goto err; if (!BN_mod_add_quick(n1, n0, group->a, group->p)) goto err; /* n1 = 3 * X_a^2 + a_curve */ } else if (group->a_is_minus3) { if (!ec_field_sqr(group, n1, a->Z, ctx)) goto err; if (!BN_mod_add_quick(n0, a->X, n1, group->p)) goto err; if (!BN_mod_sub_quick(n2, a->X, n1, group->p)) goto err; if (!ec_field_mul(group, n1, n0, n2, ctx)) goto err; if (!BN_mod_lshift1_quick(n0, n1, group->p)) goto err; if (!BN_mod_add_quick(n1, n0, n1, group->p)) goto err; /* * n1 = 3 * (X_a + Z_a^2) * (X_a - Z_a^2) = 3 * X_a^2 - 3 * * Z_a^4 */ } else { if (!ec_field_sqr(group, n0, a->X, ctx)) goto err; if (!BN_mod_lshift1_quick(n1, n0, group->p)) goto err; if (!BN_mod_add_quick(n0, n0, n1, group->p)) goto err; if (!ec_field_sqr(group, n1, a->Z, ctx)) goto err; if (!ec_field_sqr(group, n1, n1, ctx)) goto err; if (!ec_field_mul(group, n1, n1, group->a, ctx)) goto err; if (!BN_mod_add_quick(n1, n1, n0, group->p)) goto err; /* n1 = 3 * X_a^2 + a_curve * Z_a^4 */ } /* Z_r */ if (a->Z_is_one) { if (!bn_copy(n0, a->Y)) goto err; } else { if (!ec_field_mul(group, n0, a->Y, a->Z, ctx)) goto err; } if (!BN_mod_lshift1_quick(r->Z, n0, group->p)) goto err; r->Z_is_one = 0; /* Z_r = 2 * Y_a * Z_a */ /* n2 */ if (!ec_field_sqr(group, n3, a->Y, ctx)) goto err; if (!ec_field_mul(group, n2, a->X, n3, ctx)) goto err; if (!BN_mod_lshift_quick(n2, n2, 2, group->p)) goto err; /* n2 = 4 * X_a * Y_a^2 */ /* X_r */ if (!BN_mod_lshift1_quick(n0, n2, group->p)) goto err; if (!ec_field_sqr(group, r->X, n1, ctx)) goto err; if (!BN_mod_sub_quick(r->X, r->X, n0, group->p)) goto err; /* X_r = n1^2 - 2 * n2 */ /* n3 */ if (!ec_field_sqr(group, n0, n3, ctx)) goto err; if (!BN_mod_lshift_quick(n3, n0, 3, group->p)) goto err; /* n3 = 8 * Y_a^4 */ /* Y_r */ if (!BN_mod_sub_quick(n0, n2, r->X, group->p)) goto err; if (!ec_field_mul(group, n0, n1, n0, ctx)) goto err; if (!BN_mod_sub_quick(r->Y, n0, n3, group->p)) goto err; /* Y_r = n1 * (n2 - X_r) - n3 */ ret = 1; err: BN_CTX_end(ctx); return ret; } static int ec_invert(const EC_GROUP *group, EC_POINT *point, BN_CTX *ctx) { if (EC_POINT_is_at_infinity(group, point) || BN_is_zero(point->Y)) /* point is its own inverse */ return 1; return BN_usub(point->Y, group->p, point->Y); } /* * Apply randomization of EC point Jacobian projective coordinates: * * (X, Y, Z) = (lambda^2 * X, lambda^3 * Y, lambda * Z) * * where lambda is in the interval [1, p). */ static int ec_blind_coordinates(const EC_GROUP *group, EC_POINT *p, BN_CTX *ctx) { BIGNUM *lambda = NULL; BIGNUM *tmp = NULL; int ret = 0; BN_CTX_start(ctx); if ((lambda = BN_CTX_get(ctx)) == NULL) goto err; if ((tmp = BN_CTX_get(ctx)) == NULL) goto err; /* Generate lambda in [1, p). */ if (!bn_rand_interval(lambda, 1, group->p)) goto err; if (group->meth->field_encode != NULL && !group->meth->field_encode(group, lambda, lambda, ctx)) goto err; /* Z = lambda * Z */ if (!ec_field_mul(group, p->Z, lambda, p->Z, ctx)) goto err; /* tmp = lambda^2 */ if (!ec_field_sqr(group, tmp, lambda, ctx)) goto err; /* X = lambda^2 * X */ if (!ec_field_mul(group, p->X, tmp, p->X, ctx)) goto err; /* tmp = lambda^3 */ if (!ec_field_mul(group, tmp, tmp, lambda, ctx)) goto err; /* Y = lambda^3 * Y */ if (!ec_field_mul(group, p->Y, tmp, p->Y, ctx)) goto err; /* Disable optimized arithmetics after replacing Z by lambda * Z. */ p->Z_is_one = 0; ret = 1; err: BN_CTX_end(ctx); return ret; } #define EC_POINT_BN_set_flags(P, flags) do { \ BN_set_flags((P)->X, (flags)); \ BN_set_flags((P)->Y, (flags)); \ BN_set_flags((P)->Z, (flags)); \ } while(0) #define EC_POINT_CSWAP(c, a, b, w, t) do { \ if (!BN_swap_ct(c, (a)->X, (b)->X, w) || \ !BN_swap_ct(c, (a)->Y, (b)->Y, w) || \ !BN_swap_ct(c, (a)->Z, (b)->Z, w)) \ goto err; \ t = ((a)->Z_is_one ^ (b)->Z_is_one) & (c); \ (a)->Z_is_one ^= (t); \ (b)->Z_is_one ^= (t); \ } while(0) /* * This function computes (in constant time) a point multiplication over the * EC group. * * At a high level, it is Montgomery ladder with conditional swaps. * * It performs either a fixed point multiplication * (scalar * generator) * when point is NULL, or a variable point multiplication * (scalar * point) * when point is not NULL. * * scalar should be in the range [0,n) otherwise all constant time bets are off. * * NB: This says nothing about EC_POINT_add and EC_POINT_dbl, * which of course are not constant time themselves. * * The product is stored in r. * * Returns 1 on success, 0 otherwise. */ static int ec_mul_ct(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar, const EC_POINT *point, BN_CTX *ctx) { int i, cardinality_bits, group_top, kbit, pbit, Z_is_one; EC_POINT *s = NULL; BIGNUM *k = NULL; BIGNUM *lambda = NULL; BIGNUM *cardinality = NULL; int ret = 0; BN_CTX_start(ctx); if ((s = EC_POINT_dup(point, group)) == NULL) goto err; EC_POINT_BN_set_flags(s, BN_FLG_CONSTTIME); if ((cardinality = BN_CTX_get(ctx)) == NULL) goto err; if ((lambda = BN_CTX_get(ctx)) == NULL) goto err; if ((k = BN_CTX_get(ctx)) == NULL) goto err; if (!BN_mul(cardinality, group->order, group->cofactor, ctx)) goto err; /* * Group cardinalities are often on a word boundary. * So when we pad the scalar, some timing diff might * pop if it needs to be expanded due to carries. * So expand ahead of time. */ cardinality_bits = BN_num_bits(cardinality); group_top = cardinality->top; if (!bn_wexpand(k, group_top + 2) || !bn_wexpand(lambda, group_top + 2)) goto err; if (!bn_copy(k, scalar)) goto err; BN_set_flags(k, BN_FLG_CONSTTIME); if (BN_num_bits(k) > cardinality_bits || BN_is_negative(k)) { /* * This is an unusual input, and we don't guarantee * constant-timeness */ if (!BN_nnmod(k, k, cardinality, ctx)) goto err; } if (!BN_add(lambda, k, cardinality)) goto err; BN_set_flags(lambda, BN_FLG_CONSTTIME); if (!BN_add(k, lambda, cardinality)) goto err; /* * lambda := scalar + cardinality * k := scalar + 2*cardinality */ kbit = BN_is_bit_set(lambda, cardinality_bits); if (!BN_swap_ct(kbit, k, lambda, group_top + 2)) goto err; group_top = group->p->top; if (!bn_wexpand(s->X, group_top) || !bn_wexpand(s->Y, group_top) || !bn_wexpand(s->Z, group_top) || !bn_wexpand(r->X, group_top) || !bn_wexpand(r->Y, group_top) || !bn_wexpand(r->Z, group_top)) goto err; /* * Apply coordinate blinding for EC_POINT if the underlying EC_METHOD * implements it. */ if (!ec_blind_coordinates(group, s, ctx)) goto err; /* top bit is a 1, in a fixed pos */ if (!EC_POINT_copy(r, s)) goto err; EC_POINT_BN_set_flags(r, BN_FLG_CONSTTIME); if (!EC_POINT_dbl(group, s, s, ctx)) goto err; pbit = 0; /* * The ladder step, with branches, is * * k[i] == 0: S = add(R, S), R = dbl(R) * k[i] == 1: R = add(S, R), S = dbl(S) * * Swapping R, S conditionally on k[i] leaves you with state * * k[i] == 0: T, U = R, S * k[i] == 1: T, U = S, R * * Then perform the ECC ops. * * U = add(T, U) * T = dbl(T) * * Which leaves you with state * * k[i] == 0: U = add(R, S), T = dbl(R) * k[i] == 1: U = add(S, R), T = dbl(S) * * Swapping T, U conditionally on k[i] leaves you with state * * k[i] == 0: R, S = T, U * k[i] == 1: R, S = U, T * * Which leaves you with state * * k[i] == 0: S = add(R, S), R = dbl(R) * k[i] == 1: R = add(S, R), S = dbl(S) * * So we get the same logic, but instead of a branch it's a * conditional swap, followed by ECC ops, then another conditional swap. * * Optimization: The end of iteration i and start of i-1 looks like * * ... * CSWAP(k[i], R, S) * ECC * CSWAP(k[i], R, S) * (next iteration) * CSWAP(k[i-1], R, S) * ECC * CSWAP(k[i-1], R, S) * ... * * So instead of two contiguous swaps, you can merge the condition * bits and do a single swap. * * k[i] k[i-1] Outcome * 0 0 No Swap * 0 1 Swap * 1 0 Swap * 1 1 No Swap * * This is XOR. pbit tracks the previous bit of k. */ for (i = cardinality_bits - 1; i >= 0; i--) { kbit = BN_is_bit_set(k, i) ^ pbit; EC_POINT_CSWAP(kbit, r, s, group_top, Z_is_one); if (!EC_POINT_add(group, s, r, s, ctx)) goto err; if (!EC_POINT_dbl(group, r, r, ctx)) goto err; /* * pbit logic merges this cswap with that of the * next iteration */ pbit ^= kbit; } /* one final cswap to move the right value into r */ EC_POINT_CSWAP(pbit, r, s, group_top, Z_is_one); ret = 1; err: EC_POINT_free(s); BN_CTX_end(ctx); return ret; } #undef EC_POINT_BN_set_flags #undef EC_POINT_CSWAP static int ec_mul_single_ct(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar, const EC_POINT *point, BN_CTX *ctx) { return ec_mul_ct(group, r, scalar, point, ctx); } static int ec_mul_double_nonct(const EC_GROUP *group, EC_POINT *r, const BIGNUM *g_scalar, const BIGNUM *p_scalar, const EC_POINT *point, BN_CTX *ctx) { return ec_wnaf_mul(group, r, g_scalar, point, p_scalar, ctx); } static int ec_simple_field_mul(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx) { return BN_mod_mul(r, a, b, group->p, ctx); } static int ec_simple_field_sqr(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a, BN_CTX *ctx) { return BN_mod_sqr(r, a, group->p, ctx); } static int ec_mont_group_set_curve(EC_GROUP *group, const BIGNUM *p, const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx) { BN_MONT_CTX *mont = NULL; int ret = 0; BN_MONT_CTX_free(group->mont_ctx); group->mont_ctx = NULL; if ((mont = BN_MONT_CTX_new()) == NULL) goto err; if (!BN_MONT_CTX_set(mont, p, ctx)) { ECerror(ERR_R_BN_LIB); goto err; } group->mont_ctx = mont; mont = NULL; if (!ec_group_set_curve(group, p, a, b, ctx)) { BN_MONT_CTX_free(group->mont_ctx); group->mont_ctx = NULL; goto err; } ret = 1; err: BN_MONT_CTX_free(mont); return ret; } static int ec_mont_field_mul(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx) { if (group->mont_ctx == NULL) { ECerror(EC_R_NOT_INITIALIZED); return 0; } return BN_mod_mul_montgomery(r, a, b, group->mont_ctx, ctx); } static int ec_mont_field_sqr(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a, BN_CTX *ctx) { if (group->mont_ctx == NULL) { ECerror(EC_R_NOT_INITIALIZED); return 0; } return BN_mod_mul_montgomery(r, a, a, group->mont_ctx, ctx); } static int ec_mont_field_encode(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a, BN_CTX *ctx) { if (group->mont_ctx == NULL) { ECerror(EC_R_NOT_INITIALIZED); return 0; } return BN_to_montgomery(r, a, group->mont_ctx, ctx); } static int ec_mont_field_decode(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a, BN_CTX *ctx) { if (group->mont_ctx == NULL) { ECerror(EC_R_NOT_INITIALIZED); return 0; } return BN_from_montgomery(r, a, group->mont_ctx, ctx); } static const EC_METHOD ec_GFp_simple_method = { .group_set_curve = ec_group_set_curve, .group_get_curve = ec_group_get_curve, .point_is_on_curve = ec_point_is_on_curve, .point_cmp = ec_point_cmp, .point_set_affine_coordinates = ec_point_set_affine_coordinates, .point_get_affine_coordinates = ec_point_get_affine_coordinates, .points_make_affine = ec_points_make_affine, .add = ec_add, .dbl = ec_dbl, .invert = ec_invert, .mul_single_ct = ec_mul_single_ct, .mul_double_nonct = ec_mul_double_nonct, .field_mul = ec_simple_field_mul, .field_sqr = ec_simple_field_sqr, }; const EC_METHOD * EC_GFp_simple_method(void) { return &ec_GFp_simple_method; } LCRYPTO_ALIAS(EC_GFp_simple_method); static const EC_METHOD ec_GFp_mont_method = { .group_set_curve = ec_mont_group_set_curve, .group_get_curve = ec_group_get_curve, .point_is_on_curve = ec_point_is_on_curve, .point_cmp = ec_point_cmp, .point_set_affine_coordinates = ec_point_set_affine_coordinates, .point_get_affine_coordinates = ec_point_get_affine_coordinates, .points_make_affine = ec_points_make_affine, .add = ec_add, .dbl = ec_dbl, .invert = ec_invert, .mul_single_ct = ec_mul_single_ct, .mul_double_nonct = ec_mul_double_nonct, .field_mul = ec_mont_field_mul, .field_sqr = ec_mont_field_sqr, .field_encode = ec_mont_field_encode, .field_decode = ec_mont_field_decode, }; const EC_METHOD * EC_GFp_mont_method(void) { return &ec_GFp_mont_method; } LCRYPTO_ALIAS(EC_GFp_mont_method);