text: /**********************************************************************
   * Copyright (c) 2013, 2014 Pieter Wuille                             *
   * Distributed under the MIT software license, see the accompanying   *
   * file COPYING or http://www.opensource.org/licenses/mit-license.php.*
   **********************************************************************/
  
  #ifndef _SECP256K1_FIELD_IMPL_H_
  #define _SECP256K1_FIELD_IMPL_H_
  
  #if defined HAVE_CONFIG_H
  #include "libsecp256k1-config.h"
  #endif
  
  #include "util.h"
  
  #if defined(USE_FIELD_10X26)
  #include "field_10x26_impl.h"
  #elif defined(USE_FIELD_5X52)
  #include "field_5x52_impl.h"
  #else
  #error "Please select field implementation"
  #endif
  
  SECP256K1_INLINE static int secp256k1_fe_equal(const secp256k1_fe *a, const secp256k1_fe *b) {
      secp256k1_fe na;
      secp256k1_fe_negate(&na, a, 1);
      secp256k1_fe_add(&na, b);
      return secp256k1_fe_normalizes_to_zero(&na);
  }
  
  SECP256K1_INLINE static int secp256k1_fe_equal_var(const secp256k1_fe *a, const secp256k1_fe *b) {
      secp256k1_fe na;
      secp256k1_fe_negate(&na, a, 1);
      secp256k1_fe_add(&na, b);
      return secp256k1_fe_normalizes_to_zero_var(&na);
  }
  
  static int secp256k1_fe_sqrt(secp256k1_fe *r, const secp256k1_fe *a) {
      /** Given that p is congruent to 3 mod 4, we can compute the square root of
       *  a mod p as the (p+1)/4'th power of a.
       *
       *  As (p+1)/4 is an even number, it will have the same result for a and for
       *  (-a). Only one of these two numbers actually has a square root however,
       *  so we test at the end by squaring and comparing to the input.
       *  Also because (p+1)/4 is an even number, the computed square root is
       *  itself always a square (a ** ((p+1)/4) is the square of a ** ((p+1)/8)).
       */
      secp256k1_fe x2, x3, x6, x9, x11, x22, x44, x88, x176, x220, x223, t1;
      int j;
  
      /** The binary representation of (p + 1)/4 has 3 blocks of 1s, with lengths in
       *  { 2, 22, 223 }. Use an addition chain to calculate 2^n - 1 for each block:
       *  1, [2], 3, 6, 9, 11, [22], 44, 88, 176, 220, [223]
       */
  
      secp256k1_fe_sqr(&x2, a);
      secp256k1_fe_mul(&x2, &x2, a);
  
      secp256k1_fe_sqr(&x3, &x2);
      secp256k1_fe_mul(&x3, &x3, a);
  
      x6 = x3;
      for (j=0; j<3; j++) {
          secp256k1_fe_sqr(&x6, &x6);
      }
      secp256k1_fe_mul(&x6, &x6, &x3);
  
      x9 = x6;
      for (j=0; j<3; j++) {
          secp256k1_fe_sqr(&x9, &x9);
      }
      secp256k1_fe_mul(&x9, &x9, &x3);
  
      x11 = x9;
      for (j=0; j<2; j++) {
          secp256k1_fe_sqr(&x11, &x11);
      }
      secp256k1_fe_mul(&x11, &x11, &x2);
  
      x22 = x11;
      for (j=0; j<11; j++) {
          secp256k1_fe_sqr(&x22, &x22);
      }
      secp256k1_fe_mul(&x22, &x22, &x11);
  
      x44 = x22;
      for (j=0; j<22; j++) {
          secp256k1_fe_sqr(&x44, &x44);
      }
      secp256k1_fe_mul(&x44, &x44, &x22);
  
      x88 = x44;
      for (j=0; j<44; j++) {
          secp256k1_fe_sqr(&x88, &x88);
      }
      secp256k1_fe_mul(&x88, &x88, &x44);
  
      x176 = x88;
      for (j=0; j<88; j++) {
          secp256k1_fe_sqr(&x176, &x176);
      }
      secp256k1_fe_mul(&x176, &x176, &x88);
  
      x220 = x176;
      for (j=0; j<44; j++) {
          secp256k1_fe_sqr(&x220, &x220);
      }
      secp256k1_fe_mul(&x220, &x220, &x44);
  
      x223 = x220;
      for (j=0; j<3; j++) {
          secp256k1_fe_sqr(&x223, &x223);
      }
      secp256k1_fe_mul(&x223, &x223, &x3);
  
      /* The final result is then assembled using a sliding window over the blocks. */
  
      t1 = x223;
      for (j=0; j<23; j++) {
          secp256k1_fe_sqr(&t1, &t1);
      }
      secp256k1_fe_mul(&t1, &t1, &x22);
      for (j=0; j<6; j++) {
          secp256k1_fe_sqr(&t1, &t1);
      }
      secp256k1_fe_mul(&t1, &t1, &x2);
      secp256k1_fe_sqr(&t1, &t1);
      secp256k1_fe_sqr(r, &t1);
  
      /* Check that a square root was actually calculated */
  
      secp256k1_fe_sqr(&t1, r);
      return secp256k1_fe_equal(&t1, a);
  }
  
  static void secp256k1_fe_inv(secp256k1_fe *r, const secp256k1_fe *a) {
      secp256k1_fe x2, x3, x6, x9, x11, x22, x44, x88, x176, x220, x223, t1;
      int j;
  
      /** The binary representation of (p - 2) has 5 blocks of 1s, with lengths in
       *  { 1, 2, 22, 223 }. Use an addition chain to calculate 2^n - 1 for each block:
       *  [1], [2], 3, 6, 9, 11, [22], 44, 88, 176, 220, [223]
       */
  
      secp256k1_fe_sqr(&x2, a);
      secp256k1_fe_mul(&x2, &x2, a);
  
      secp256k1_fe_sqr(&x3, &x2);
      secp256k1_fe_mul(&x3, &x3, a);
  
      x6 = x3;
      for (j=0; j<3; j++) {
          secp256k1_fe_sqr(&x6, &x6);
      }
      secp256k1_fe_mul(&x6, &x6, &x3);
  
      x9 = x6;
      for (j=0; j<3; j++) {
          secp256k1_fe_sqr(&x9, &x9);
      }
      secp256k1_fe_mul(&x9, &x9, &x3);
  
      x11 = x9;
      for (j=0; j<2; j++) {
          secp256k1_fe_sqr(&x11, &x11);
      }
      secp256k1_fe_mul(&x11, &x11, &x2);
  
      x22 = x11;
      for (j=0; j<11; j++) {
          secp256k1_fe_sqr(&x22, &x22);
      }
      secp256k1_fe_mul(&x22, &x22, &x11);
  
      x44 = x22;
      for (j=0; j<22; j++) {
          secp256k1_fe_sqr(&x44, &x44);
      }
      secp256k1_fe_mul(&x44, &x44, &x22);
  
      x88 = x44;
      for (j=0; j<44; j++) {
          secp256k1_fe_sqr(&x88, &x88);
      }
      secp256k1_fe_mul(&x88, &x88, &x44);
  
      x176 = x88;
      for (j=0; j<88; j++) {
          secp256k1_fe_sqr(&x176, &x176);
      }
      secp256k1_fe_mul(&x176, &x176, &x88);
  
      x220 = x176;
      for (j=0; j<44; j++) {
          secp256k1_fe_sqr(&x220, &x220);
      }
      secp256k1_fe_mul(&x220, &x220, &x44);
  
      x223 = x220;
      for (j=0; j<3; j++) {
          secp256k1_fe_sqr(&x223, &x223);
      }
      secp256k1_fe_mul(&x223, &x223, &x3);
  
      /* The final result is then assembled using a sliding window over the blocks. */
  
      t1 = x223;
      for (j=0; j<23; j++) {
          secp256k1_fe_sqr(&t1, &t1);
      }
      secp256k1_fe_mul(&t1, &t1, &x22);
      for (j=0; j<5; j++) {
          secp256k1_fe_sqr(&t1, &t1);
      }
      secp256k1_fe_mul(&t1, &t1, a);
      for (j=0; j<3; j++) {
          secp256k1_fe_sqr(&t1, &t1);
      }
      secp256k1_fe_mul(&t1, &t1, &x2);
      for (j=0; j<2; j++) {
          secp256k1_fe_sqr(&t1, &t1);
      }
      secp256k1_fe_mul(r, a, &t1);
  }
  
  static void secp256k1_fe_inv_var(secp256k1_fe *r, const secp256k1_fe *a) {
  #if defined(USE_FIELD_INV_BUILTIN)
      secp256k1_fe_inv(r, a);
  #elif defined(USE_FIELD_INV_NUM)
      secp256k1_num n, m;
      static const secp256k1_fe negone = SECP256K1_FE_CONST(
          0xFFFFFFFFUL, 0xFFFFFFFFUL, 0xFFFFFFFFUL, 0xFFFFFFFFUL,
          0xFFFFFFFFUL, 0xFFFFFFFFUL, 0xFFFFFFFEUL, 0xFFFFFC2EUL
      );
      /* secp256k1 field prime, value p defined in "Standards for Efficient Cryptography" (SEC2) 2.7.1. */
      static const unsigned char prime[32] = {
          0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,
          0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,
          0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,
          0xFF,0xFF,0xFF,0xFE,0xFF,0xFF,0xFC,0x2F
      };
      unsigned char b[32];
      int res;
      secp256k1_fe c = *a;
      secp256k1_fe_normalize_var(&c);
      secp256k1_fe_get_b32(b, &c);
      secp256k1_num_set_bin(&n, b, 32);
      secp256k1_num_set_bin(&m, prime, 32);
      secp256k1_num_mod_inverse(&n, &n, &m);
      secp256k1_num_get_bin(b, 32, &n);
      res = secp256k1_fe_set_b32(r, b);
      (void)res;
      VERIFY_CHECK(res);
      /* Verify the result is the (unique) valid inverse using non-GMP code. */
      secp256k1_fe_mul(&c, &c, r);
      secp256k1_fe_add(&c, &negone);
      CHECK(secp256k1_fe_normalizes_to_zero_var(&c));
  #else
  #error "Please select field inverse implementation"
  #endif
  }
  
  static void secp256k1_fe_inv_all_var(secp256k1_fe *r, const secp256k1_fe *a, size_t len) {
      secp256k1_fe u;
      size_t i;
      if (len < 1) {
          return;
      }
  
      VERIFY_CHECK((r + len <= a) || (a + len <= r));
  
      r[0] = a[0];
  
      i = 0;
      while (++i < len) {
          secp256k1_fe_mul(&r[i], &r[i - 1], &a[i]);
      }
  
      secp256k1_fe_inv_var(&u, &r[--i]);
  
      while (i > 0) {
          size_t j = i--;
          secp256k1_fe_mul(&r[j], &r[i], &u);
          secp256k1_fe_mul(&u, &u, &a[j]);
      }
  
      r[0] = u;
  }
  
  static int secp256k1_fe_is_quad_var(const secp256k1_fe *a) {
  #ifndef USE_NUM_NONE
      unsigned char b[32];
      secp256k1_num n;
      secp256k1_num m;
      /* secp256k1 field prime, value p defined in "Standards for Efficient Cryptography" (SEC2) 2.7.1. */
      static const unsigned char prime[32] = {
          0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,
          0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,
          0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,
          0xFF,0xFF,0xFF,0xFE,0xFF,0xFF,0xFC,0x2F
      };
  
      secp256k1_fe c = *a;
      secp256k1_fe_normalize_var(&c);
      secp256k1_fe_get_b32(b, &c);
      secp256k1_num_set_bin(&n, b, 32);
      secp256k1_num_set_bin(&m, prime, 32);
      return secp256k1_num_jacobi(&n, &m) >= 0;
  #else
      secp256k1_fe r;
      return secp256k1_fe_sqrt(&r, a);
  #endif
  }
  
  #endif
  