1/* 2 * Copyright (c) 2007, 2017, Oracle and/or its affiliates. All rights reserved. 3 * Use is subject to license terms. 4 * 5 * This library is free software; you can redistribute it and/or 6 * modify it under the terms of the GNU Lesser General Public 7 * License as published by the Free Software Foundation; either 8 * version 2.1 of the License, or (at your option) any later version. 9 * 10 * This library is distributed in the hope that it will be useful, 11 * but WITHOUT ANY WARRANTY; without even the implied warranty of 12 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 13 * Lesser General Public License for more details. 14 * 15 * You should have received a copy of the GNU Lesser General Public License 16 * along with this library; if not, write to the Free Software Foundation, 17 * Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA. 18 * 19 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA 20 * or visit www.oracle.com if you need additional information or have any 21 * questions. 22 */ 23 24/* ********************************************************************* 25 * 26 * The Original Code is the elliptic curve math library for binary polynomial field curves. 27 * 28 * The Initial Developer of the Original Code is 29 * Sun Microsystems, Inc. 30 * Portions created by the Initial Developer are Copyright (C) 2003 31 * the Initial Developer. All Rights Reserved. 32 * 33 * Contributor(s): 34 * Sheueling Chang-Shantz <sheueling.chang@sun.com>, 35 * Stephen Fung <fungstep@hotmail.com>, and 36 * Douglas Stebila <douglas@stebila.ca>, Sun Microsystems Laboratories. 37 * 38 * Last Modified Date from the Original Code: May 2017 39 *********************************************************************** */ 40 41#include "ec2.h" 42#include "mplogic.h" 43#include "mp_gf2m.h" 44#ifndef _KERNEL 45#include <stdlib.h> 46#endif 47 48/* Compute the x-coordinate x/z for the point 2*(x/z) in Montgomery 49 * projective coordinates. Uses algorithm Mdouble in appendix of Lopez, J. 50 * and Dahab, R. "Fast multiplication on elliptic curves over GF(2^m) 51 * without precomputation". modified to not require precomputation of 52 * c=b^{2^{m-1}}. */ 53static mp_err 54gf2m_Mdouble(mp_int *x, mp_int *z, const ECGroup *group, int kmflag) 55{ 56 mp_err res = MP_OKAY; 57 mp_int t1; 58 59 MP_DIGITS(&t1) = 0; 60 MP_CHECKOK(mp_init(&t1, kmflag)); 61 62 MP_CHECKOK(group->meth->field_sqr(x, x, group->meth)); 63 MP_CHECKOK(group->meth->field_sqr(z, &t1, group->meth)); 64 MP_CHECKOK(group->meth->field_mul(x, &t1, z, group->meth)); 65 MP_CHECKOK(group->meth->field_sqr(x, x, group->meth)); 66 MP_CHECKOK(group->meth->field_sqr(&t1, &t1, group->meth)); 67 MP_CHECKOK(group->meth-> 68 field_mul(&group->curveb, &t1, &t1, group->meth)); 69 MP_CHECKOK(group->meth->field_add(x, &t1, x, group->meth)); 70 71 CLEANUP: 72 mp_clear(&t1); 73 return res; 74} 75 76/* Compute the x-coordinate x1/z1 for the point (x1/z1)+(x2/x2) in 77 * Montgomery projective coordinates. Uses algorithm Madd in appendix of 78 * Lopex, J. and Dahab, R. "Fast multiplication on elliptic curves over 79 * GF(2^m) without precomputation". */ 80static mp_err 81gf2m_Madd(const mp_int *x, mp_int *x1, mp_int *z1, mp_int *x2, mp_int *z2, 82 const ECGroup *group, int kmflag) 83{ 84 mp_err res = MP_OKAY; 85 mp_int t1, t2; 86 87 MP_DIGITS(&t1) = 0; 88 MP_DIGITS(&t2) = 0; 89 MP_CHECKOK(mp_init(&t1, kmflag)); 90 MP_CHECKOK(mp_init(&t2, kmflag)); 91 92 MP_CHECKOK(mp_copy(x, &t1)); 93 MP_CHECKOK(group->meth->field_mul(x1, z2, x1, group->meth)); 94 MP_CHECKOK(group->meth->field_mul(z1, x2, z1, group->meth)); 95 MP_CHECKOK(group->meth->field_mul(x1, z1, &t2, group->meth)); 96 MP_CHECKOK(group->meth->field_add(z1, x1, z1, group->meth)); 97 MP_CHECKOK(group->meth->field_sqr(z1, z1, group->meth)); 98 MP_CHECKOK(group->meth->field_mul(z1, &t1, x1, group->meth)); 99 MP_CHECKOK(group->meth->field_add(x1, &t2, x1, group->meth)); 100 101 CLEANUP: 102 mp_clear(&t1); 103 mp_clear(&t2); 104 return res; 105} 106 107/* Compute the x, y affine coordinates from the point (x1, z1) (x2, z2) 108 * using Montgomery point multiplication algorithm Mxy() in appendix of 109 * Lopex, J. and Dahab, R. "Fast multiplication on elliptic curves over 110 * GF(2^m) without precomputation". Returns: 0 on error 1 if return value 111 * should be the point at infinity 2 otherwise */ 112static int 113gf2m_Mxy(const mp_int *x, const mp_int *y, mp_int *x1, mp_int *z1, 114 mp_int *x2, mp_int *z2, const ECGroup *group) 115{ 116 mp_err res = MP_OKAY; 117 int ret = 0; 118 mp_int t3, t4, t5; 119 120 MP_DIGITS(&t3) = 0; 121 MP_DIGITS(&t4) = 0; 122 MP_DIGITS(&t5) = 0; 123 MP_CHECKOK(mp_init(&t3, FLAG(x2))); 124 MP_CHECKOK(mp_init(&t4, FLAG(x2))); 125 MP_CHECKOK(mp_init(&t5, FLAG(x2))); 126 127 if (mp_cmp_z(z1) == 0) { 128 mp_zero(x2); 129 mp_zero(z2); 130 ret = 1; 131 goto CLEANUP; 132 } 133 134 if (mp_cmp_z(z2) == 0) { 135 MP_CHECKOK(mp_copy(x, x2)); 136 MP_CHECKOK(group->meth->field_add(x, y, z2, group->meth)); 137 ret = 2; 138 goto CLEANUP; 139 } 140 141 MP_CHECKOK(mp_set_int(&t5, 1)); 142 if (group->meth->field_enc) { 143 MP_CHECKOK(group->meth->field_enc(&t5, &t5, group->meth)); 144 } 145 146 MP_CHECKOK(group->meth->field_mul(z1, z2, &t3, group->meth)); 147 148 MP_CHECKOK(group->meth->field_mul(z1, x, z1, group->meth)); 149 MP_CHECKOK(group->meth->field_add(z1, x1, z1, group->meth)); 150 MP_CHECKOK(group->meth->field_mul(z2, x, z2, group->meth)); 151 MP_CHECKOK(group->meth->field_mul(z2, x1, x1, group->meth)); 152 MP_CHECKOK(group->meth->field_add(z2, x2, z2, group->meth)); 153 154 MP_CHECKOK(group->meth->field_mul(z2, z1, z2, group->meth)); 155 MP_CHECKOK(group->meth->field_sqr(x, &t4, group->meth)); 156 MP_CHECKOK(group->meth->field_add(&t4, y, &t4, group->meth)); 157 MP_CHECKOK(group->meth->field_mul(&t4, &t3, &t4, group->meth)); 158 MP_CHECKOK(group->meth->field_add(&t4, z2, &t4, group->meth)); 159 160 MP_CHECKOK(group->meth->field_mul(&t3, x, &t3, group->meth)); 161 MP_CHECKOK(group->meth->field_div(&t5, &t3, &t3, group->meth)); 162 MP_CHECKOK(group->meth->field_mul(&t3, &t4, &t4, group->meth)); 163 MP_CHECKOK(group->meth->field_mul(x1, &t3, x2, group->meth)); 164 MP_CHECKOK(group->meth->field_add(x2, x, z2, group->meth)); 165 166 MP_CHECKOK(group->meth->field_mul(z2, &t4, z2, group->meth)); 167 MP_CHECKOK(group->meth->field_add(z2, y, z2, group->meth)); 168 169 ret = 2; 170 171 CLEANUP: 172 mp_clear(&t3); 173 mp_clear(&t4); 174 mp_clear(&t5); 175 if (res == MP_OKAY) { 176 return ret; 177 } else { 178 return 0; 179 } 180} 181 182/* Computes R = nP based on algorithm 2P of Lopex, J. and Dahab, R. "Fast 183 * multiplication on elliptic curves over GF(2^m) without 184 * precomputation". Elliptic curve points P and R can be identical. Uses 185 * Montgomery projective coordinates. The timing parameter is ignored 186 * because this algorithm resists timing attacks by default. */ 187mp_err 188ec_GF2m_pt_mul_mont(const mp_int *n, const mp_int *px, const mp_int *py, 189 mp_int *rx, mp_int *ry, const ECGroup *group, 190 int timing) 191{ 192 mp_err res = MP_OKAY; 193 mp_int x1, x2, z1, z2; 194 int i, j; 195 mp_digit top_bit, mask; 196 197 MP_DIGITS(&x1) = 0; 198 MP_DIGITS(&x2) = 0; 199 MP_DIGITS(&z1) = 0; 200 MP_DIGITS(&z2) = 0; 201 MP_CHECKOK(mp_init(&x1, FLAG(n))); 202 MP_CHECKOK(mp_init(&x2, FLAG(n))); 203 MP_CHECKOK(mp_init(&z1, FLAG(n))); 204 MP_CHECKOK(mp_init(&z2, FLAG(n))); 205 206 /* if result should be point at infinity */ 207 if ((mp_cmp_z(n) == 0) || (ec_GF2m_pt_is_inf_aff(px, py) == MP_YES)) { 208 MP_CHECKOK(ec_GF2m_pt_set_inf_aff(rx, ry)); 209 goto CLEANUP; 210 } 211 212 MP_CHECKOK(mp_copy(px, &x1)); /* x1 = px */ 213 MP_CHECKOK(mp_set_int(&z1, 1)); /* z1 = 1 */ 214 MP_CHECKOK(group->meth->field_sqr(&x1, &z2, group->meth)); /* z2 = 215 * x1^2 = 216 * px^2 */ 217 MP_CHECKOK(group->meth->field_sqr(&z2, &x2, group->meth)); 218 MP_CHECKOK(group->meth->field_add(&x2, &group->curveb, &x2, group->meth)); /* x2 219 * = 220 * px^4 221 * + 222 * b 223 */ 224 225 /* find top-most bit and go one past it */ 226 i = MP_USED(n) - 1; 227 j = MP_DIGIT_BIT - 1; 228 top_bit = 1; 229 top_bit <<= MP_DIGIT_BIT - 1; 230 mask = top_bit; 231 while (!(MP_DIGITS(n)[i] & mask)) { 232 mask >>= 1; 233 j--; 234 } 235 mask >>= 1; 236 j--; 237 238 /* if top most bit was at word break, go to next word */ 239 if (!mask) { 240 i--; 241 j = MP_DIGIT_BIT - 1; 242 mask = top_bit; 243 } 244 245 for (; i >= 0; i--) { 246 for (; j >= 0; j--) { 247 if (MP_DIGITS(n)[i] & mask) { 248 MP_CHECKOK(gf2m_Madd(px, &x1, &z1, &x2, &z2, group, FLAG(n))); 249 MP_CHECKOK(gf2m_Mdouble(&x2, &z2, group, FLAG(n))); 250 } else { 251 MP_CHECKOK(gf2m_Madd(px, &x2, &z2, &x1, &z1, group, FLAG(n))); 252 MP_CHECKOK(gf2m_Mdouble(&x1, &z1, group, FLAG(n))); 253 } 254 mask >>= 1; 255 } 256 j = MP_DIGIT_BIT - 1; 257 mask = top_bit; 258 } 259 260 /* convert out of "projective" coordinates */ 261 i = gf2m_Mxy(px, py, &x1, &z1, &x2, &z2, group); 262 if (i == 0) { 263 res = MP_BADARG; 264 goto CLEANUP; 265 } else if (i == 1) { 266 MP_CHECKOK(ec_GF2m_pt_set_inf_aff(rx, ry)); 267 } else { 268 MP_CHECKOK(mp_copy(&x2, rx)); 269 MP_CHECKOK(mp_copy(&z2, ry)); 270 } 271 272 CLEANUP: 273 mp_clear(&x1); 274 mp_clear(&x2); 275 mp_clear(&z1); 276 mp_clear(&z2); 277 return res; 278} 279