src/ecmult_const_impl.h

   1 /**********************************************************************
   2  * Copyright (c) 2015 Pieter Wuille, Andrew Poelstra                  *
   3  * Distributed under the MIT software license, see the accompanying   *
   4  * file COPYING or http://www.opensource.org/licenses/mit-license.php.*
   5  **********************************************************************/
   6
   7 #ifndef SECP256K1_ECMULT_CONST_IMPL_H
   8 #define SECP256K1_ECMULT_CONST_IMPL_H
   9
  10 #include "scalar.h"
  11 #include "group.h"
  12 #include "ecmult_const.h"
  13 #include "ecmult_impl.h"
  14
  15 #ifdef USE_ENDOMORPHISM
  16     #define WNAF_BITS 128
  17 #else
  18     #define WNAF_BITS 256
  19 #endif
  20 #define WNAF_SIZE(w) ((WNAF_BITS + (w) - 1) / (w))
  21
  22 /* This is like `ECMULT_TABLE_GET_GE` but is constant time */
  23 #define ECMULT_CONST_TABLE_GET_GE(r,pre,n,w) do { \
  24     int m; \
  25     int abs_n = (n) * (((n) > 0) * 2 - 1); \
  26     int idx_n = abs_n / 2; \
  27     secp256k1_fe neg_y; \
  28     VERIFY_CHECK(((n) & 1) == 1); \
  29     VERIFY_CHECK((n) >= -((1 << ((w)-1)) - 1)); \
  30     VERIFY_CHECK((n) <=  ((1 << ((w)-1)) - 1)); \
  31     VERIFY_SETUP(secp256k1_fe_clear(&(r)->x)); \
  32     VERIFY_SETUP(secp256k1_fe_clear(&(r)->y)); \
  33     for (m = 0; m < ECMULT_TABLE_SIZE(w); m++) { \
  34         /* This loop is used to avoid secret data in array indices. See
  35          * the comment in ecmult_gen_impl.h for rationale. */ \
  36         secp256k1_fe_cmov(&(r)->x, &(pre)[m].x, m == idx_n); \
  37         secp256k1_fe_cmov(&(r)->y, &(pre)[m].y, m == idx_n); \
  38     } \
  39     (r)->infinity = 0; \
  40     secp256k1_fe_negate(&neg_y, &(r)->y, 1); \
  41     secp256k1_fe_cmov(&(r)->y, &neg_y, (n) != abs_n); \
  42 } while(0)
  43
  44
  45 /** Convert a number to WNAF notation.
  46  *  The number becomes represented by sum(2^{wi} * wnaf[i], i=0..WNAF_SIZE(w)+1) - return_val.
  47  *  It has the following guarantees:
  48  *  - each wnaf[i] an odd integer between -(1 << w) and (1 << w)
  49  *  - each wnaf[i] is nonzero
  50  *  - the number of words set is always WNAF_SIZE(w) + 1
  51  *
  52  *  Adapted from `The Width-w NAF Method Provides Small Memory and Fast Elliptic Scalar
  53  *  Multiplications Secure against Side Channel Attacks`, Okeya and Tagaki. M. Joye (Ed.)
  54  *  CT-RSA 2003, LNCS 2612, pp. 328-443, 2003. Springer-Verlagy Berlin Heidelberg 2003
  55  *
  56  *  Numbers reference steps of `Algorithm SPA-resistant Width-w NAF with Odd Scalar` on pp. 335
  57  */
  58 static int secp256k1_wnaf_const(int *wnaf, secp256k1_scalar s, int w) {
  59     int global_sign;
  60     int skew = 0;
  61     int word = 0;
  62
  63     /* 1 2 3 */
  64     int u_last;
  65     int u;
  66
  67     int flip;
  68     int bit;
  69     secp256k1_scalar neg_s;
  70     int not_neg_one;
  71     /* Note that we cannot handle even numbers by negating them to be odd, as is
  72      * done in other implementations, since if our scalars were specified to have
  73      * width < 256 for performance reasons, their negations would have width 256
  74      * and we'd lose any performance benefit. Instead, we use a technique from
  75      * Section 4.2 of the Okeya/Tagaki paper, which is to add either 1 (for even)
  76      * or 2 (for odd) to the number we are encoding, returning a skew value indicating
  77      * this, and having the caller compensate after doing the multiplication. */
  78
  79     /* Negative numbers will be negated to keep their bit representation below the maximum width */
  80     flip = secp256k1_scalar_is_high(&s);
  81     /* We add 1 to even numbers, 2 to odd ones, noting that negation flips parity */
  82     bit = flip ^ !secp256k1_scalar_is_even(&s);
  83     /* We check for negative one, since adding 2 to it will cause an overflow */
  84     secp256k1_scalar_negate(&neg_s, &s);
  85     not_neg_one = !secp256k1_scalar_is_one(&neg_s);
  86     secp256k1_scalar_cadd_bit(&s, bit, not_neg_one);
  87     /* If we had negative one, flip == 1, s.d[0] == 0, bit == 1, so caller expects
  88      * that we added two to it and flipped it. In fact for -1 these operations are
  89      * identical. We only flipped, but since skewing is required (in the sense that
  90      * the skew must be 1 or 2, never zero) and flipping is not, we need to change
  91      * our flags to claim that we only skewed. */
  92     global_sign = secp256k1_scalar_cond_negate(&s, flip);
  93     global_sign *= not_neg_one * 2 - 1;
  94     skew = 1 << bit;
  95
  96     /* 4 */
  97     u_last = secp256k1_scalar_shr_int(&s, w);
  98     while (word * w < WNAF_BITS) {
  99         int sign;
 100         int even;
 101
 102         /* 4.1 4.4 */
 103         u = secp256k1_scalar_shr_int(&s, w);
 104         /* 4.2 */
 105         even = ((u & 1) == 0);
 106         sign = 2 * (u_last > 0) - 1;
 107         u += sign * even;
 108         u_last -= sign * even * (1 << w);
 109
 110         /* 4.3, adapted for global sign change */
 111         wnaf[word++] = u_last * global_sign;
 112
 113         u_last = u;
 114     }
 115     wnaf[word] = u * global_sign;
 116
 117     VERIFY_CHECK(secp256k1_scalar_is_zero(&s));
 118     VERIFY_CHECK(word == WNAF_SIZE(w));
 119     return skew;
 120 }
 121
 122
 123 static void secp256k1_ecmult_const(secp256k1_gej *r, const secp256k1_ge *a, const secp256k1_scalar *scalar) {
 124     secp256k1_ge pre_a[ECMULT_TABLE_SIZE(WINDOW_A)];
 125     secp256k1_ge tmpa;
 126     secp256k1_fe Z;
 127
 128     int skew_1;
 129     int wnaf_1[1 + WNAF_SIZE(WINDOW_A - 1)];
 130 #ifdef USE_ENDOMORPHISM
 131     secp256k1_ge pre_a_lam[ECMULT_TABLE_SIZE(WINDOW_A)];
 132     int wnaf_lam[1 + WNAF_SIZE(WINDOW_A - 1)];
 133     int skew_lam;
 134     secp256k1_scalar q_1, q_lam;
 135 #endif
 136
 137     int i;
 138     secp256k1_scalar sc = *scalar;
 139
 140     /* build wnaf representation for q. */
 141 #ifdef USE_ENDOMORPHISM
 142     /* split q into q_1 and q_lam (where q = q_1 + q_lam*lambda, and q_1 and q_lam are ~128 bit) */
 143     secp256k1_scalar_split_lambda(&q_1, &q_lam, &sc);
 144     skew_1   = secp256k1_wnaf_const(wnaf_1,   q_1,   WINDOW_A - 1);
 145     skew_lam = secp256k1_wnaf_const(wnaf_lam, q_lam, WINDOW_A - 1);
 146 #else
 147     skew_1   = secp256k1_wnaf_const(wnaf_1, sc, WINDOW_A - 1);
 148 #endif
 149
 150     /* Calculate odd multiples of a.
 151      * All multiples are brought to the same Z 'denominator', which is stored
 152      * in Z. Due to secp256k1' isomorphism we can do all operations pretending
 153      * that the Z coordinate was 1, use affine addition formulae, and correct
 154      * the Z coordinate of the result once at the end.
 155      */
 156     secp256k1_gej_set_ge(r, a);
 157     secp256k1_ecmult_odd_multiples_table_globalz_windowa(pre_a, &Z, r);
 158     for (i = 0; i < ECMULT_TABLE_SIZE(WINDOW_A); i++) {
 159         secp256k1_fe_normalize_weak(&pre_a[i].y);
 160     }
 161 #ifdef USE_ENDOMORPHISM
 162     for (i = 0; i < ECMULT_TABLE_SIZE(WINDOW_A); i++) {
 163         secp256k1_ge_mul_lambda(&pre_a_lam[i], &pre_a[i]);
 164     }
 165 #endif
 166
 167     /* first loop iteration (separated out so we can directly set r, rather
 168      * than having it start at infinity, get doubled several times, then have
 169      * its new value added to it) */
 170     i = wnaf_1[WNAF_SIZE(WINDOW_A - 1)];
 171     VERIFY_CHECK(i != 0);
 172     ECMULT_CONST_TABLE_GET_GE(&tmpa, pre_a, i, WINDOW_A);
 173     secp256k1_gej_set_ge(r, &tmpa);
 174 #ifdef USE_ENDOMORPHISM
 175     i = wnaf_lam[WNAF_SIZE(WINDOW_A - 1)];
 176     VERIFY_CHECK(i != 0);
 177     ECMULT_CONST_TABLE_GET_GE(&tmpa, pre_a_lam, i, WINDOW_A);
 178     secp256k1_gej_add_ge(r, r, &tmpa);
 179 #endif
 180     /* remaining loop iterations */
 181     for (i = WNAF_SIZE(WINDOW_A - 1) - 1; i >= 0; i--) {
 182         int n;
 183         int j;
 184         for (j = 0; j < WINDOW_A - 1; ++j) {
 185             secp256k1_gej_double_nonzero(r, r, NULL);
 186         }
 187
 188         n = wnaf_1[i];
 189         ECMULT_CONST_TABLE_GET_GE(&tmpa, pre_a, n, WINDOW_A);
 190         VERIFY_CHECK(n != 0);
 191         secp256k1_gej_add_ge(r, r, &tmpa);
 192 #ifdef USE_ENDOMORPHISM
 193         n = wnaf_lam[i];
 194         ECMULT_CONST_TABLE_GET_GE(&tmpa, pre_a_lam, n, WINDOW_A);
 195         VERIFY_CHECK(n != 0);
 196         secp256k1_gej_add_ge(r, r, &tmpa);
 197 #endif
 198     }
 199
 200     secp256k1_fe_mul(&r->z, &r->z, &Z);
 201
 202     {
 203         /* Correct for wNAF skew */
 204         secp256k1_ge correction = *a;
 205         secp256k1_ge_storage correction_1_stor;
 206 #ifdef USE_ENDOMORPHISM
 207         secp256k1_ge_storage correction_lam_stor;
 208 #endif
 209         secp256k1_ge_storage a2_stor;
 210         secp256k1_gej tmpj;
 211         secp256k1_gej_set_ge(&tmpj, &correction);
 212         secp256k1_gej_double_var(&tmpj, &tmpj, NULL);
 213         secp256k1_ge_set_gej(&correction, &tmpj);
 214         secp256k1_ge_to_storage(&correction_1_stor, a);
 215 #ifdef USE_ENDOMORPHISM
 216         secp256k1_ge_to_storage(&correction_lam_stor, a);
 217 #endif
 218         secp256k1_ge_to_storage(&a2_stor, &correction);
 219
 220         /* For odd numbers this is 2a (so replace it), for even ones a (so no-op) */
 221         secp256k1_ge_storage_cmov(&correction_1_stor, &a2_stor, skew_1 == 2);
 222 #ifdef USE_ENDOMORPHISM
 223         secp256k1_ge_storage_cmov(&correction_lam_stor, &a2_stor, skew_lam == 2);
 224 #endif
 225
 226         /* Apply the correction */
 227         secp256k1_ge_from_storage(&correction, &correction_1_stor);
 228         secp256k1_ge_neg(&correction, &correction);
 229         secp256k1_gej_add_ge(r, r, &correction);
 230
 231 #ifdef USE_ENDOMORPHISM
 232         secp256k1_ge_from_storage(&correction, &correction_lam_stor);
 233         secp256k1_ge_neg(&correction, &correction);
 234         secp256k1_ge_mul_lambda(&correction, &correction);
 235         secp256k1_gej_add_ge(r, r, &correction);
 236 #endif
 237     }
 238 }
 239
 240 #endif /* SECP256K1_ECMULT_CONST_IMPL_H */