text: /***********************************************************************
   * Copyright (c) 2015 Pieter Wuille, Andrew Poelstra                   *
   * Distributed under the MIT software license, see the accompanying    *
   * file COPYING or https://www.opensource.org/licenses/mit-license.php.*
   ***********************************************************************/
  
  #ifndef SECP256K1_ECMULT_CONST_IMPL_H
  #define SECP256K1_ECMULT_CONST_IMPL_H
  
  #include "scalar.h"
  #include "group.h"
  #include "ecmult_const.h"
  #include "ecmult_impl.h"
  
  /** Fill a table 'pre' with precomputed odd multiples of a.
   *
   *  The resulting point set is brought to a single constant Z denominator, stores the X and Y
   *  coordinates as ge_storage points in pre, and stores the global Z in globalz.
   *  It only operates on tables sized for WINDOW_A wnaf multiples.
   */
  static void secp256k1_ecmult_odd_multiples_table_globalz_windowa(secp256k1_ge *pre, secp256k1_fe *globalz, const secp256k1_gej *a) {
      secp256k1_gej prej[ECMULT_TABLE_SIZE(WINDOW_A)];
      secp256k1_fe zr[ECMULT_TABLE_SIZE(WINDOW_A)];
  
      /* Compute the odd multiples in Jacobian form. */
      secp256k1_ecmult_odd_multiples_table(ECMULT_TABLE_SIZE(WINDOW_A), prej, zr, a);
      /* Bring them to the same Z denominator. */
      secp256k1_ge_globalz_set_table_gej(ECMULT_TABLE_SIZE(WINDOW_A), pre, globalz, prej, zr);
  }
  
  /* This is like `ECMULT_TABLE_GET_GE` but is constant time */
  #define ECMULT_CONST_TABLE_GET_GE(r,pre,n,w) do { \
      int m = 0; \
      /* Extract the sign-bit for a constant time absolute-value. */ \
      int mask = (n) >> (sizeof(n) * CHAR_BIT - 1); \
      int abs_n = ((n) + mask) ^ mask; \
      int idx_n = abs_n >> 1; \
      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)); \
      /* Unconditionally set r->x = (pre)[m].x. r->y = (pre)[m].y. because it's either the correct one \
       * or will get replaced in the later iterations, this is needed to make sure `r` is initialized. */ \
      (r)->x = (pre)[m].x; \
      (r)->y = (pre)[m].y; \
      for (m = 1; 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..WNAF_SIZE(w)+1) - return_val.
   *  It has 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 always WNAF_SIZE(w) + 1
   *
   *  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-Verlag 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, const secp256k1_scalar *scalar, int w, int size) {
      int global_sign;
      int skew = 0;
      int word = 0;
  
      /* 1 2 3 */
      int u_last;
      int u;
  
      int flip;
      int bit;
      secp256k1_scalar s;
      int not_neg_one;
  
      VERIFY_CHECK(w > 0);
      VERIFY_CHECK(size > 0);
  
      /* 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.
       *
       * In fact, we _do_ want to negate numbers to minimize their bit-lengths (and in
       * particular, to ensure that the outputs from the endomorphism-split fit into
       * 128 bits). If we negate, the parity of our number flips, inverting which of
       * {1, 2} we want to add to the scalar when ensuring that it's odd. Further
       * complicating things, -1 interacts badly with `secp256k1_scalar_cadd_bit` and
       * we need to special-case it in this logic. */
      flip = secp256k1_scalar_is_high(scalar);
      /* We add 1 to even numbers, 2 to odd ones, noting that negation flips parity */
      bit = flip ^ !secp256k1_scalar_is_even(scalar);
      /* We check for negative one, since adding 2 to it will cause an overflow */
      secp256k1_scalar_negate(&s, scalar);
      not_neg_one = !secp256k1_scalar_is_one(&s);
      s = *scalar;
      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);
      do {
          int even;
  
          /* 4.1 4.4 */
          u = secp256k1_scalar_shr_int(&s, w);
          /* 4.2 */
          even = ((u & 1) == 0);
          /* In contrast to the original algorithm, u_last is always > 0 and
           * therefore we do not need to check its sign. In particular, it's easy
           * to see that u_last is never < 0 because u is never < 0. Moreover,
           * u_last is never = 0 because u is never even after a loop
           * iteration. The same holds analogously for the initial value of
           * u_last (in the first loop iteration). */
          VERIFY_CHECK(u_last > 0);
          VERIFY_CHECK((u_last & 1) == 1);
          u += even;
          u_last -= even * (1 << w);
  
          /* 4.3, adapted for global sign change */
          wnaf[word++] = u_last * global_sign;
  
          u_last = u;
      } while (word * w < size);
      wnaf[word] = u * global_sign;
  
      VERIFY_CHECK(secp256k1_scalar_is_zero(&s));
      VERIFY_CHECK(word == WNAF_SIZE_BITS(size, w));
      return skew;
  }
  
  static void secp256k1_ecmult_const(secp256k1_gej *r, const secp256k1_ge *a, const secp256k1_scalar *scalar, int size) {
      secp256k1_ge pre_a[ECMULT_TABLE_SIZE(WINDOW_A)];
      secp256k1_ge tmpa;
      secp256k1_fe Z;
  
      int skew_1;
      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;
      int wnaf_1[1 + WNAF_SIZE(WINDOW_A - 1)];
  
      int i;
  
      /* build wnaf representation for q. */
      int rsize = size;
      if (size > 128) {
          rsize = 128;
          /* 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, scalar);
          skew_1   = secp256k1_wnaf_const(wnaf_1,   &q_1,   WINDOW_A - 1, 128);
          skew_lam = secp256k1_wnaf_const(wnaf_lam, &q_lam, WINDOW_A - 1, 128);
      } else
      {
          skew_1   = secp256k1_wnaf_const(wnaf_1, scalar, WINDOW_A - 1, size);
          skew_lam = 0;
      }
  
      /* 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.
       */
      VERIFY_CHECK(!a->infinity);
      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);
      }
      if (size > 128) {
          for (i = 0; i < ECMULT_TABLE_SIZE(WINDOW_A); i++) {
              secp256k1_ge_mul_lambda(&pre_a_lam[i], &pre_a[i]);
          }
  
      }
  
      /* 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_BITS(rsize, WINDOW_A - 1)];
      VERIFY_CHECK(i != 0);
      ECMULT_CONST_TABLE_GET_GE(&tmpa, pre_a, i, WINDOW_A);
      secp256k1_gej_set_ge(r, &tmpa);
      if (size > 128) {
          i = wnaf_lam[WNAF_SIZE_BITS(rsize, 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);
      }
      /* remaining loop iterations */
      for (i = WNAF_SIZE_BITS(rsize, WINDOW_A - 1) - 1; i >= 0; i--) {
          int n;
          int j;
          for (j = 0; j < WINDOW_A - 1; ++j) {
              secp256k1_gej_double(r, r);
          }
  
          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);
          if (size > 128) {
              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);
          }
      }
  
      secp256k1_fe_mul(&r->z, &r->z, &Z);
  
      {
          /* Correct for wNAF skew */
          secp256k1_ge correction = *a;
          secp256k1_ge_storage correction_1_stor;
          secp256k1_ge_storage correction_lam_stor;
          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);
          if (size > 128) {
              secp256k1_ge_to_storage(&correction_lam_stor, a);
          }
          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);
          if (size > 128) {
              secp256k1_ge_storage_cmov(&correction_lam_stor, &a2_stor, skew_lam == 2);
          }
  
          /* Apply the correction */
          secp256k1_ge_from_storage(&correction, &correction_1_stor);
          secp256k1_ge_neg(&correction, &correction);
          secp256k1_gej_add_ge(r, r, &correction);
  
          if (size > 128) {
              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 /* SECP256K1_ECMULT_CONST_IMPL_H */
  