text: /**********************************************************************
   * Copyright (c) 2015 Pieter Wuille, Andrew Poelstra                  *
   * Distributed under the MIT software license, see the accompanying   *
   * file COPYING or http://www.opensource.org/licenses/mit-license.php.*
   **********************************************************************/
  
  #ifndef _SECP256K1_ECMULT_CONST_IMPL_
  #define _SECP256K1_ECMULT_CONST_IMPL_
  
  #include "scalar.h"
  #include "group.h"
  #include "ecmult_const.h"
  #include "ecmult_impl.h"
  
  #ifdef USE_ENDOMORPHISM
  #define WNAF_BITS 128
  #else
  #define WNAF_BITS 256
  #endif
  #define WNAF_SIZE(w) ((WNAF_BITS + (w) - 1) / (w))
  
  /* This is like `ECMULT_TABLE_GET_GE` but is constant time */
  #define ECMULT_CONST_TABLE_GET_GE(r,pre,n,w) do { \
      int m; \
      int abs_n = (n) * (((n) > 0) * 2 - 1); \
      int idx_n = abs_n / 2; \
      secp256k1_fe neg_y; \
      VERIFY_CHECK(((n) & 1) == 1); \
      VERIFY_CHECK((n) >= -((1 << ((w)-1)) - 1)); \
      VERIFY_CHECK((n) <=  ((1 << ((w)-1)) - 1)); \
      VERIFY_SETUP(secp256k1_fe_clear(&(r)->x)); \
      VERIFY_SETUP(secp256k1_fe_clear(&(r)->y)); \
      for (m = 0; m < ECMULT_TABLE_SIZE(w); m++) { \
          /* This loop is used to avoid secret data in array indices. See
           * the comment in ecmult_gen_impl.h for rationale. */ \
          secp256k1_fe_cmov(&(r)->x, &(pre)[m].x, m == idx_n); \
          secp256k1_fe_cmov(&(r)->y, &(pre)[m].y, m == idx_n); \
      } \
      (r)->infinity = 0; \
      secp256k1_fe_negate(&neg_y, &(r)->y, 1); \
      secp256k1_fe_cmov(&(r)->y, &neg_y, (n) != abs_n); \
  } while(0)
  
  
  /** Convert a number to WNAF notation. The number becomes represented by sum(2^{wi} * wnaf[i], i=0..return_val)
   *  with the following guarantees:
   *  - each wnaf[i] an odd integer between -(1 << w) and (1 << w)
   *  - each wnaf[i] is nonzero
   *  - the number of words set is returned; this is always (WNAF_BITS + w - 1) / w
   *
   *  Adapted from `The Width-w NAF Method Provides Small Memory and Fast Elliptic Scalar
   *  Multiplications Secure against Side Channel Attacks`, Okeya and Tagaki. M. Joye (Ed.)
   *  CT-RSA 2003, LNCS 2612, pp. 328-443, 2003. Springer-Verlagy Berlin Heidelberg 2003
   *
   *  Numbers reference steps of `Algorithm SPA-resistant Width-w NAF with Odd Scalar` on pp. 335
   */
  static int secp256k1_wnaf_const(int *wnaf, secp256k1_scalar s, int w) {
      int global_sign;
      int skew = 0;
      int word = 0;
  
      /* 1 2 3 */
      int u_last;
      int u;
  
      int flip;
      int bit;
      secp256k1_scalar neg_s;
      int not_neg_one;
      /* Note that we cannot handle even numbers by negating them to be odd, as is
       * done in other implementations, since if our scalars were specified to have
       * width < 256 for performance reasons, their negations would have width 256
       * and we'd lose any performance benefit. Instead, we use a technique from
       * Section 4.2 of the Okeya/Tagaki paper, which is to add either 1 (for even)
       * or 2 (for odd) to the number we are encoding, returning a skew value indicating
       * this, and having the caller compensate after doing the multiplication. */
  
      /* Negative numbers will be negated to keep their bit representation below the maximum width */
      flip = secp256k1_scalar_is_high(&s);
      /* We add 1 to even numbers, 2 to odd ones, noting that negation flips parity */
      bit = flip ^ !secp256k1_scalar_is_even(&s);
      /* We check for negative one, since adding 2 to it will cause an overflow */
      secp256k1_scalar_negate(&neg_s, &s);
      not_neg_one = !secp256k1_scalar_is_one(&neg_s);
      secp256k1_scalar_cadd_bit(&s, bit, not_neg_one);
      /* If we had negative one, flip == 1, s.d[0] == 0, bit == 1, so caller expects
       * that we added two to it and flipped it. In fact for -1 these operations are
       * identical. We only flipped, but since skewing is required (in the sense that
       * the skew must be 1 or 2, never zero) and flipping is not, we need to change
       * our flags to claim that we only skewed. */
      global_sign = secp256k1_scalar_cond_negate(&s, flip);
      global_sign *= not_neg_one * 2 - 1;
      skew = 1 << bit;
  
      /* 4 */
      u_last = secp256k1_scalar_shr_int(&s, w);
      while (word * w < WNAF_BITS) {
          int sign;
          int even;
  
          /* 4.1 4.4 */
          u = secp256k1_scalar_shr_int(&s, w);
          /* 4.2 */
          even = ((u & 1) == 0);
          sign = 2 * (u_last > 0) - 1;
          u += sign * even;
          u_last -= sign * even * (1 << w);
  
          /* 4.3, adapted for global sign change */
          wnaf[word++] = u_last * global_sign;
  
          u_last = u;
      }
      wnaf[word] = u * global_sign;
  
      VERIFY_CHECK(secp256k1_scalar_is_zero(&s));
      VERIFY_CHECK(word == WNAF_SIZE(w));
      return skew;
  }
  
  
  static void secp256k1_ecmult_const(secp256k1_gej *r, const secp256k1_ge *a, const secp256k1_scalar *scalar) {
      secp256k1_ge pre_a[ECMULT_TABLE_SIZE(WINDOW_A)];
      secp256k1_ge tmpa;
      secp256k1_fe Z;
  
      int skew_1;
      int wnaf_1[1 + WNAF_SIZE(WINDOW_A - 1)];
  #ifdef USE_ENDOMORPHISM
      secp256k1_ge pre_a_lam[ECMULT_TABLE_SIZE(WINDOW_A)];
      int wnaf_lam[1 + WNAF_SIZE(WINDOW_A - 1)];
      int skew_lam;
      secp256k1_scalar q_1, q_lam;
  #endif
  
      int i;
      secp256k1_scalar sc = *scalar;
  
      /* build wnaf representation for q. */
  #ifdef USE_ENDOMORPHISM
      /* split q into q_1 and q_lam (where q = q_1 + q_lam*lambda, and q_1 and q_lam are ~128 bit) */
      secp256k1_scalar_split_lambda(&q_1, &q_lam, &sc);
      skew_1   = secp256k1_wnaf_const(wnaf_1,   q_1,   WINDOW_A - 1);
      skew_lam = secp256k1_wnaf_const(wnaf_lam, q_lam, WINDOW_A - 1);
  #else
      skew_1   = secp256k1_wnaf_const(wnaf_1, sc, WINDOW_A - 1);
  #endif
  
      /* Calculate odd multiples of a.
       * All multiples are brought to the same Z 'denominator', which is stored
       * in Z. Due to secp256k1' isomorphism we can do all operations pretending
       * that the Z coordinate was 1, use affine addition formulae, and correct
       * the Z coordinate of the result once at the end.
       */
      secp256k1_gej_set_ge(r, a);
      secp256k1_ecmult_odd_multiples_table_globalz_windowa(pre_a, &Z, r);
      for (i = 0; i < ECMULT_TABLE_SIZE(WINDOW_A); i++) {
          secp256k1_fe_normalize_weak(&pre_a[i].y);
      }
  #ifdef USE_ENDOMORPHISM
      for (i = 0; i < ECMULT_TABLE_SIZE(WINDOW_A); i++) {
          secp256k1_ge_mul_lambda(&pre_a_lam[i], &pre_a[i]);
      }
  #endif
  
      /* first loop iteration (separated out so we can directly set r, rather
       * than having it start at infinity, get doubled several times, then have
       * its new value added to it) */
      i = wnaf_1[WNAF_SIZE(WINDOW_A - 1)];
      VERIFY_CHECK(i != 0);
      ECMULT_CONST_TABLE_GET_GE(&tmpa, pre_a, i, WINDOW_A);
      secp256k1_gej_set_ge(r, &tmpa);
  #ifdef USE_ENDOMORPHISM
      i = wnaf_lam[WNAF_SIZE(WINDOW_A - 1)];
      VERIFY_CHECK(i != 0);
      ECMULT_CONST_TABLE_GET_GE(&tmpa, pre_a_lam, i, WINDOW_A);
      secp256k1_gej_add_ge(r, r, &tmpa);
  #endif
      /* remaining loop iterations */
      for (i = WNAF_SIZE(WINDOW_A - 1) - 1; i >= 0; i--) {
          int n;
          int j;
          for (j = 0; j < WINDOW_A - 1; ++j) {
              secp256k1_gej_double_nonzero(r, r, NULL);
          }
  
          n = wnaf_1[i];
          ECMULT_CONST_TABLE_GET_GE(&tmpa, pre_a, n, WINDOW_A);
          VERIFY_CHECK(n != 0);
          secp256k1_gej_add_ge(r, r, &tmpa);
  #ifdef USE_ENDOMORPHISM
          n = wnaf_lam[i];
          ECMULT_CONST_TABLE_GET_GE(&tmpa, pre_a_lam, n, WINDOW_A);
          VERIFY_CHECK(n != 0);
          secp256k1_gej_add_ge(r, r, &tmpa);
  #endif
      }
  
      secp256k1_fe_mul(&r->z, &r->z, &Z);
  
      {
          /* Correct for wNAF skew */
          secp256k1_ge correction = *a;
          secp256k1_ge_storage correction_1_stor;
  #ifdef USE_ENDOMORPHISM
          secp256k1_ge_storage correction_lam_stor;
  #endif
          secp256k1_ge_storage a2_stor;
          secp256k1_gej tmpj;
          secp256k1_gej_set_ge(&tmpj, &correction);
          secp256k1_gej_double_var(&tmpj, &tmpj, NULL);
          secp256k1_ge_set_gej(&correction, &tmpj);
          secp256k1_ge_to_storage(&correction_1_stor, a);
  #ifdef USE_ENDOMORPHISM
          secp256k1_ge_to_storage(&correction_lam_stor, a);
  #endif
          secp256k1_ge_to_storage(&a2_stor, &correction);
  
          /* For odd numbers this is 2a (so replace it), for even ones a (so no-op) */
          secp256k1_ge_storage_cmov(&correction_1_stor, &a2_stor, skew_1 == 2);
  #ifdef USE_ENDOMORPHISM
          secp256k1_ge_storage_cmov(&correction_lam_stor, &a2_stor, skew_lam == 2);
  #endif
  
          /* Apply the correction */
          secp256k1_ge_from_storage(&correction, &correction_1_stor);
          secp256k1_ge_neg(&correction, &correction);
          secp256k1_gej_add_ge(r, r, &correction);
  
  #ifdef USE_ENDOMORPHISM
          secp256k1_ge_from_storage(&correction, &correction_lam_stor);
          secp256k1_ge_neg(&correction, &correction);
          secp256k1_ge_mul_lambda(&correction, &correction);
          secp256k1_gej_add_ge(r, r, &correction);
  #endif
      }
  }
  
  #endif
  