[331] | 1 | /*
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| 2 | * Copyright 2002-2016 The OpenSSL Project Authors. All Rights Reserved.
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| 3 | *
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| 4 | * Licensed under the OpenSSL license (the "License"). You may not use
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| 5 | * this file except in compliance with the License. You can obtain a copy
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| 6 | * in the file LICENSE in the source distribution or at
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| 7 | * https://www.openssl.org/source/license.html
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| 8 | */
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| 9 |
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| 10 | /* ====================================================================
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| 11 | * Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED.
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| 12 | *
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| 13 | * The Elliptic Curve Public-Key Crypto Library (ECC Code) included
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| 14 | * herein is developed by SUN MICROSYSTEMS, INC., and is contributed
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| 15 | * to the OpenSSL project.
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| 16 | *
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| 17 | * The ECC Code is licensed pursuant to the OpenSSL open source
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| 18 | * license provided below.
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| 19 | *
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| 20 | * The software is originally written by Sheueling Chang Shantz and
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| 21 | * Douglas Stebila of Sun Microsystems Laboratories.
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| 22 | *
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| 23 | */
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| 24 |
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| 25 | #include <openssl/err.h>
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| 26 |
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| 27 | #include "internal/bn_int.h"
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| 28 | #include "ec_lcl.h"
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| 29 |
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| 30 | #ifndef OPENSSL_NO_EC2M
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| 31 |
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| 32 | /*-
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| 33 | * Compute the x-coordinate x/z for the point 2*(x/z) in Montgomery projective
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| 34 | * coordinates.
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| 35 | * Uses algorithm Mdouble in appendix of
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| 36 | * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over
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| 37 | * GF(2^m) without precomputation" (CHES '99, LNCS 1717).
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| 38 | * modified to not require precomputation of c=b^{2^{m-1}}.
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| 39 | */
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| 40 | static int gf2m_Mdouble(const EC_GROUP *group, BIGNUM *x, BIGNUM *z,
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| 41 | BN_CTX *ctx)
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| 42 | {
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| 43 | BIGNUM *t1;
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| 44 | int ret = 0;
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| 45 |
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| 46 | /* Since Mdouble is static we can guarantee that ctx != NULL. */
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| 47 | BN_CTX_start(ctx);
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| 48 | t1 = BN_CTX_get(ctx);
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| 49 | if (t1 == NULL)
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| 50 | goto err;
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| 51 |
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| 52 | if (!group->meth->field_sqr(group, x, x, ctx))
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| 53 | goto err;
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| 54 | if (!group->meth->field_sqr(group, t1, z, ctx))
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| 55 | goto err;
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| 56 | if (!group->meth->field_mul(group, z, x, t1, ctx))
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| 57 | goto err;
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| 58 | if (!group->meth->field_sqr(group, x, x, ctx))
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| 59 | goto err;
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| 60 | if (!group->meth->field_sqr(group, t1, t1, ctx))
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| 61 | goto err;
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| 62 | if (!group->meth->field_mul(group, t1, group->b, t1, ctx))
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| 63 | goto err;
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| 64 | if (!BN_GF2m_add(x, x, t1))
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| 65 | goto err;
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| 66 |
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| 67 | ret = 1;
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| 68 |
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| 69 | err:
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| 70 | BN_CTX_end(ctx);
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| 71 | return ret;
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| 72 | }
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| 73 |
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| 74 | /*-
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| 75 | * Compute the x-coordinate x1/z1 for the point (x1/z1)+(x2/x2) in Montgomery
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| 76 | * projective coordinates.
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| 77 | * Uses algorithm Madd in appendix of
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| 78 | * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over
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| 79 | * GF(2^m) without precomputation" (CHES '99, LNCS 1717).
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| 80 | */
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| 81 | static int gf2m_Madd(const EC_GROUP *group, const BIGNUM *x, BIGNUM *x1,
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| 82 | BIGNUM *z1, const BIGNUM *x2, const BIGNUM *z2,
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| 83 | BN_CTX *ctx)
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| 84 | {
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| 85 | BIGNUM *t1, *t2;
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| 86 | int ret = 0;
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| 87 |
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| 88 | /* Since Madd is static we can guarantee that ctx != NULL. */
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| 89 | BN_CTX_start(ctx);
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| 90 | t1 = BN_CTX_get(ctx);
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| 91 | t2 = BN_CTX_get(ctx);
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| 92 | if (t2 == NULL)
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| 93 | goto err;
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| 94 |
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| 95 | if (!BN_copy(t1, x))
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| 96 | goto err;
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| 97 | if (!group->meth->field_mul(group, x1, x1, z2, ctx))
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| 98 | goto err;
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| 99 | if (!group->meth->field_mul(group, z1, z1, x2, ctx))
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| 100 | goto err;
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| 101 | if (!group->meth->field_mul(group, t2, x1, z1, ctx))
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| 102 | goto err;
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| 103 | if (!BN_GF2m_add(z1, z1, x1))
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| 104 | goto err;
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| 105 | if (!group->meth->field_sqr(group, z1, z1, ctx))
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| 106 | goto err;
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| 107 | if (!group->meth->field_mul(group, x1, z1, t1, ctx))
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| 108 | goto err;
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| 109 | if (!BN_GF2m_add(x1, x1, t2))
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| 110 | goto err;
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| 111 |
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| 112 | ret = 1;
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| 113 |
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| 114 | err:
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| 115 | BN_CTX_end(ctx);
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| 116 | return ret;
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| 117 | }
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| 118 |
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| 119 | /*-
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| 120 | * Compute the x, y affine coordinates from the point (x1, z1) (x2, z2)
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| 121 | * using Montgomery point multiplication algorithm Mxy() in appendix of
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| 122 | * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over
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| 123 | * GF(2^m) without precomputation" (CHES '99, LNCS 1717).
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| 124 | * Returns:
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| 125 | * 0 on error
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| 126 | * 1 if return value should be the point at infinity
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| 127 | * 2 otherwise
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| 128 | */
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| 129 | static int gf2m_Mxy(const EC_GROUP *group, const BIGNUM *x, const BIGNUM *y,
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| 130 | BIGNUM *x1, BIGNUM *z1, BIGNUM *x2, BIGNUM *z2,
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| 131 | BN_CTX *ctx)
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| 132 | {
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| 133 | BIGNUM *t3, *t4, *t5;
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| 134 | int ret = 0;
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| 135 |
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| 136 | if (BN_is_zero(z1)) {
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| 137 | BN_zero(x2);
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| 138 | BN_zero(z2);
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| 139 | return 1;
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| 140 | }
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| 141 |
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| 142 | if (BN_is_zero(z2)) {
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| 143 | if (!BN_copy(x2, x))
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| 144 | return 0;
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| 145 | if (!BN_GF2m_add(z2, x, y))
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| 146 | return 0;
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| 147 | return 2;
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| 148 | }
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| 149 |
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| 150 | /* Since Mxy is static we can guarantee that ctx != NULL. */
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| 151 | BN_CTX_start(ctx);
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| 152 | t3 = BN_CTX_get(ctx);
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| 153 | t4 = BN_CTX_get(ctx);
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| 154 | t5 = BN_CTX_get(ctx);
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| 155 | if (t5 == NULL)
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| 156 | goto err;
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| 157 |
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| 158 | if (!BN_one(t5))
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| 159 | goto err;
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| 160 |
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| 161 | if (!group->meth->field_mul(group, t3, z1, z2, ctx))
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| 162 | goto err;
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| 163 |
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| 164 | if (!group->meth->field_mul(group, z1, z1, x, ctx))
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| 165 | goto err;
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| 166 | if (!BN_GF2m_add(z1, z1, x1))
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| 167 | goto err;
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| 168 | if (!group->meth->field_mul(group, z2, z2, x, ctx))
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| 169 | goto err;
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| 170 | if (!group->meth->field_mul(group, x1, z2, x1, ctx))
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| 171 | goto err;
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| 172 | if (!BN_GF2m_add(z2, z2, x2))
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| 173 | goto err;
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| 174 |
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| 175 | if (!group->meth->field_mul(group, z2, z2, z1, ctx))
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| 176 | goto err;
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| 177 | if (!group->meth->field_sqr(group, t4, x, ctx))
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| 178 | goto err;
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| 179 | if (!BN_GF2m_add(t4, t4, y))
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| 180 | goto err;
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| 181 | if (!group->meth->field_mul(group, t4, t4, t3, ctx))
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| 182 | goto err;
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| 183 | if (!BN_GF2m_add(t4, t4, z2))
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| 184 | goto err;
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| 185 |
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| 186 | if (!group->meth->field_mul(group, t3, t3, x, ctx))
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| 187 | goto err;
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| 188 | if (!group->meth->field_div(group, t3, t5, t3, ctx))
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| 189 | goto err;
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| 190 | if (!group->meth->field_mul(group, t4, t3, t4, ctx))
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| 191 | goto err;
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| 192 | if (!group->meth->field_mul(group, x2, x1, t3, ctx))
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| 193 | goto err;
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| 194 | if (!BN_GF2m_add(z2, x2, x))
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| 195 | goto err;
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| 196 |
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| 197 | if (!group->meth->field_mul(group, z2, z2, t4, ctx))
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| 198 | goto err;
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| 199 | if (!BN_GF2m_add(z2, z2, y))
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| 200 | goto err;
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| 201 |
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| 202 | ret = 2;
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| 203 |
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| 204 | err:
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| 205 | BN_CTX_end(ctx);
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| 206 | return ret;
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| 207 | }
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| 208 |
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| 209 | /*-
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| 210 | * Computes scalar*point and stores the result in r.
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| 211 | * point can not equal r.
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| 212 | * Uses a modified algorithm 2P of
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| 213 | * Lopez, J. and Dahab, R. "Fast multiplication on elliptic curves over
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| 214 | * GF(2^m) without precomputation" (CHES '99, LNCS 1717).
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| 215 | *
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| 216 | * To protect against side-channel attack the function uses constant time swap,
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| 217 | * avoiding conditional branches.
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| 218 | */
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| 219 | static int ec_GF2m_montgomery_point_multiply(const EC_GROUP *group,
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| 220 | EC_POINT *r,
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| 221 | const BIGNUM *scalar,
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| 222 | const EC_POINT *point,
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| 223 | BN_CTX *ctx)
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| 224 | {
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| 225 | BIGNUM *x1, *x2, *z1, *z2;
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| 226 | int ret = 0, i, group_top;
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| 227 | BN_ULONG mask, word;
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| 228 |
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| 229 | if (r == point) {
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| 230 | ECerr(EC_F_EC_GF2M_MONTGOMERY_POINT_MULTIPLY, EC_R_INVALID_ARGUMENT);
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| 231 | return 0;
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| 232 | }
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| 233 |
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| 234 | /* if result should be point at infinity */
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| 235 | if ((scalar == NULL) || BN_is_zero(scalar) || (point == NULL) ||
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| 236 | EC_POINT_is_at_infinity(group, point)) {
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| 237 | return EC_POINT_set_to_infinity(group, r);
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| 238 | }
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| 239 |
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| 240 | /* only support affine coordinates */
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| 241 | if (!point->Z_is_one)
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| 242 | return 0;
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| 243 |
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| 244 | /*
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| 245 | * Since point_multiply is static we can guarantee that ctx != NULL.
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| 246 | */
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| 247 | BN_CTX_start(ctx);
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| 248 | x1 = BN_CTX_get(ctx);
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| 249 | z1 = BN_CTX_get(ctx);
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| 250 | if (z1 == NULL)
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| 251 | goto err;
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| 252 |
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| 253 | x2 = r->X;
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| 254 | z2 = r->Y;
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| 255 |
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| 256 | group_top = bn_get_top(group->field);
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| 257 | if (bn_wexpand(x1, group_top) == NULL
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| 258 | || bn_wexpand(z1, group_top) == NULL
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| 259 | || bn_wexpand(x2, group_top) == NULL
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| 260 | || bn_wexpand(z2, group_top) == NULL)
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| 261 | goto err;
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| 262 |
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| 263 | if (!BN_GF2m_mod_arr(x1, point->X, group->poly))
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| 264 | goto err; /* x1 = x */
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| 265 | if (!BN_one(z1))
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| 266 | goto err; /* z1 = 1 */
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| 267 | if (!group->meth->field_sqr(group, z2, x1, ctx))
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| 268 | goto err; /* z2 = x1^2 = x^2 */
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| 269 | if (!group->meth->field_sqr(group, x2, z2, ctx))
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| 270 | goto err;
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| 271 | if (!BN_GF2m_add(x2, x2, group->b))
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| 272 | goto err; /* x2 = x^4 + b */
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| 273 |
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| 274 | /* find top most bit and go one past it */
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| 275 | i = bn_get_top(scalar) - 1;
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| 276 | mask = BN_TBIT;
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| 277 | word = bn_get_words(scalar)[i];
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| 278 | while (!(word & mask))
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| 279 | mask >>= 1;
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| 280 | mask >>= 1;
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| 281 | /* if top most bit was at word break, go to next word */
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| 282 | if (!mask) {
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| 283 | i--;
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| 284 | mask = BN_TBIT;
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| 285 | }
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| 286 |
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| 287 | for (; i >= 0; i--) {
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| 288 | word = bn_get_words(scalar)[i];
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| 289 | while (mask) {
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| 290 | BN_consttime_swap(word & mask, x1, x2, group_top);
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| 291 | BN_consttime_swap(word & mask, z1, z2, group_top);
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| 292 | if (!gf2m_Madd(group, point->X, x2, z2, x1, z1, ctx))
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| 293 | goto err;
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| 294 | if (!gf2m_Mdouble(group, x1, z1, ctx))
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| 295 | goto err;
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| 296 | BN_consttime_swap(word & mask, x1, x2, group_top);
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| 297 | BN_consttime_swap(word & mask, z1, z2, group_top);
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| 298 | mask >>= 1;
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| 299 | }
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| 300 | mask = BN_TBIT;
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| 301 | }
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| 302 |
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| 303 | /* convert out of "projective" coordinates */
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| 304 | i = gf2m_Mxy(group, point->X, point->Y, x1, z1, x2, z2, ctx);
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| 305 | if (i == 0)
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| 306 | goto err;
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| 307 | else if (i == 1) {
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| 308 | if (!EC_POINT_set_to_infinity(group, r))
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| 309 | goto err;
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| 310 | } else {
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| 311 | if (!BN_one(r->Z))
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| 312 | goto err;
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| 313 | r->Z_is_one = 1;
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| 314 | }
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| 315 |
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| 316 | /* GF(2^m) field elements should always have BIGNUM::neg = 0 */
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| 317 | BN_set_negative(r->X, 0);
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| 318 | BN_set_negative(r->Y, 0);
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| 319 |
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| 320 | ret = 1;
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| 321 |
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| 322 | err:
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| 323 | BN_CTX_end(ctx);
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| 324 | return ret;
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| 325 | }
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| 326 |
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| 327 | /*-
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| 328 | * Computes the sum
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| 329 | * scalar*group->generator + scalars[0]*points[0] + ... + scalars[num-1]*points[num-1]
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| 330 | * gracefully ignoring NULL scalar values.
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| 331 | */
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| 332 | int ec_GF2m_simple_mul(const EC_GROUP *group, EC_POINT *r,
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| 333 | const BIGNUM *scalar, size_t num,
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| 334 | const EC_POINT *points[], const BIGNUM *scalars[],
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| 335 | BN_CTX *ctx)
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| 336 | {
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| 337 | BN_CTX *new_ctx = NULL;
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| 338 | int ret = 0;
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| 339 | size_t i;
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| 340 | EC_POINT *p = NULL;
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| 341 | EC_POINT *acc = NULL;
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| 342 |
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| 343 | if (ctx == NULL) {
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| 344 | ctx = new_ctx = BN_CTX_new();
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| 345 | if (ctx == NULL)
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| 346 | return 0;
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| 347 | }
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| 348 |
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| 349 | /*
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| 350 | * This implementation is more efficient than the wNAF implementation for
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| 351 | * 2 or fewer points. Use the ec_wNAF_mul implementation for 3 or more
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| 352 | * points, or if we can perform a fast multiplication based on
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| 353 | * precomputation.
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| 354 | */
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| 355 | if ((scalar && (num > 1)) || (num > 2)
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| 356 | || (num == 0 && EC_GROUP_have_precompute_mult(group))) {
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| 357 | ret = ec_wNAF_mul(group, r, scalar, num, points, scalars, ctx);
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| 358 | goto err;
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| 359 | }
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| 360 |
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| 361 | if ((p = EC_POINT_new(group)) == NULL)
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| 362 | goto err;
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| 363 | if ((acc = EC_POINT_new(group)) == NULL)
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| 364 | goto err;
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| 365 |
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| 366 | if (!EC_POINT_set_to_infinity(group, acc))
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| 367 | goto err;
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| 368 |
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| 369 | if (scalar) {
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| 370 | if (!ec_GF2m_montgomery_point_multiply
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| 371 | (group, p, scalar, group->generator, ctx))
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| 372 | goto err;
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| 373 | if (BN_is_negative(scalar))
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| 374 | if (!group->meth->invert(group, p, ctx))
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| 375 | goto err;
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| 376 | if (!group->meth->add(group, acc, acc, p, ctx))
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| 377 | goto err;
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| 378 | }
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| 379 |
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| 380 | for (i = 0; i < num; i++) {
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| 381 | if (!ec_GF2m_montgomery_point_multiply
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| 382 | (group, p, scalars[i], points[i], ctx))
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| 383 | goto err;
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| 384 | if (BN_is_negative(scalars[i]))
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| 385 | if (!group->meth->invert(group, p, ctx))
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| 386 | goto err;
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| 387 | if (!group->meth->add(group, acc, acc, p, ctx))
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| 388 | goto err;
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| 389 | }
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| 390 |
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| 391 | if (!EC_POINT_copy(r, acc))
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| 392 | goto err;
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| 393 |
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| 394 | ret = 1;
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| 395 |
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| 396 | err:
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| 397 | EC_POINT_free(p);
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| 398 | EC_POINT_free(acc);
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| 399 | BN_CTX_free(new_ctx);
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| 400 | return ret;
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| 401 | }
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| 402 |
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| 403 | /*
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| 404 | * Precomputation for point multiplication: fall back to wNAF methods because
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| 405 | * ec_GF2m_simple_mul() uses ec_wNAF_mul() if appropriate
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| 406 | */
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| 407 |
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| 408 | int ec_GF2m_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
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| 409 | {
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| 410 | return ec_wNAF_precompute_mult(group, ctx);
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| 411 | }
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| 412 |
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| 413 | int ec_GF2m_have_precompute_mult(const EC_GROUP *group)
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| 414 | {
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| 415 | return ec_wNAF_have_precompute_mult(group);
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| 416 | }
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| 417 |
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| 418 | #endif
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