curve25519.c

/*
 * Copyright 2016 The OpenSSL Project Authors. All Rights Reserved.
 *
 * Licensed under the OpenSSL license (the "License").  You may not use
 * this file except in compliance with the License.  You can obtain a copy
 * in the file LICENSE in the source distribution or at
 * https://www.openssl.org/source/license.html
 */

/* This code is mostly taken from the ref10 version of Ed25519 in SUPERCOP
 * 20141124 (http://bench.cr.yp.to/supercop.html).
 *
 * The field functions are shared by Ed25519 and X25519 where possible. */

#include <string.h>
#include "ec_lcl.h"


/* fe means field element. Here the field is \Z/(2^255-19). An element t,
 * entries t[0]...t[9], represents the integer t[0]+2^26 t[1]+2^51 t[2]+2^77
 * t[3]+2^102 t[4]+...+2^230 t[9]. Bounds on each t[i] vary depending on
 * context.  */
typedef int32_t fe[10];

static const int64_t kBottom25Bits = 0x1ffffffLL;
static const int64_t kBottom26Bits = 0x3ffffffLL;
static const int64_t kTop39Bits = 0xfffffffffe000000LL;
static const int64_t kTop38Bits = 0xfffffffffc000000LL;

static uint64_t load_3(const uint8_t *in) {
  uint64_t result;
  result = (uint64_t)in[0];
  result |= ((uint64_t)in[1]) << 8;
  result |= ((uint64_t)in[2]) << 16;
  return result;
}

static uint64_t load_4(const uint8_t *in) {
  uint64_t result;
  result = (uint64_t)in[0];
  result |= ((uint64_t)in[1]) << 8;
  result |= ((uint64_t)in[2]) << 16;
  result |= ((uint64_t)in[3]) << 24;
  return result;
}

static void fe_frombytes(fe h, const uint8_t *s) {
  /* Ignores top bit of h. */
  int64_t h0 = load_4(s);
  int64_t h1 = load_3(s + 4) << 6;
  int64_t h2 = load_3(s + 7) << 5;
  int64_t h3 = load_3(s + 10) << 3;
  int64_t h4 = load_3(s + 13) << 2;
  int64_t h5 = load_4(s + 16);
  int64_t h6 = load_3(s + 20) << 7;
  int64_t h7 = load_3(s + 23) << 5;
  int64_t h8 = load_3(s + 26) << 4;
  int64_t h9 = (load_3(s + 29) & 8388607) << 2;
  int64_t carry0;
  int64_t carry1;
  int64_t carry2;
  int64_t carry3;
  int64_t carry4;
  int64_t carry5;
  int64_t carry6;
  int64_t carry7;
  int64_t carry8;
  int64_t carry9;

  carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;
  carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
  carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
  carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;
  carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;

  carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
  carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
  carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
  carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;
  carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;

  h[0] = h0;
  h[1] = h1;
  h[2] = h2;
  h[3] = h3;
  h[4] = h4;
  h[5] = h5;
  h[6] = h6;
  h[7] = h7;
  h[8] = h8;
  h[9] = h9;
}

/* Preconditions:
 *  |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
 *
 * Write p=2^255-19; q=floor(h/p).
 * Basic claim: q = floor(2^(-255)(h + 19 2^(-25)h9 + 2^(-1))).
 *
 * Proof:
 *   Have |h|<=p so |q|<=1 so |19^2 2^(-255) q|<1/4.
 *   Also have |h-2^230 h9|<2^231 so |19 2^(-255)(h-2^230 h9)|<1/4.
 *
 *   Write y=2^(-1)-19^2 2^(-255)q-19 2^(-255)(h-2^230 h9).
 *   Then 0<y<1.
 *
 *   Write r=h-pq.
 *   Have 0<=r<=p-1=2^255-20.
 *   Thus 0<=r+19(2^-255)r<r+19(2^-255)2^255<=2^255-1.
 *
 *   Write x=r+19(2^-255)r+y.
 *   Then 0<x<2^255 so floor(2^(-255)x) = 0 so floor(q+2^(-255)x) = q.
 *
 *   Have q+2^(-255)x = 2^(-255)(h + 19 2^(-25) h9 + 2^(-1))
 *   so floor(2^(-255)(h + 19 2^(-25) h9 + 2^(-1))) = q. */
static void fe_tobytes(uint8_t *s, const fe h) {
  int32_t h0 = h[0];
  int32_t h1 = h[1];
  int32_t h2 = h[2];
  int32_t h3 = h[3];
  int32_t h4 = h[4];
  int32_t h5 = h[5];
  int32_t h6 = h[6];
  int32_t h7 = h[7];
  int32_t h8 = h[8];
  int32_t h9 = h[9];
  int32_t q;

  q = (19 * h9 + (((int32_t) 1) << 24)) >> 25;
  q = (h0 + q) >> 26;
  q = (h1 + q) >> 25;
  q = (h2 + q) >> 26;
  q = (h3 + q) >> 25;
  q = (h4 + q) >> 26;
  q = (h5 + q) >> 25;
  q = (h6 + q) >> 26;
  q = (h7 + q) >> 25;
  q = (h8 + q) >> 26;
  q = (h9 + q) >> 25;

  /* Goal: Output h-(2^255-19)q, which is between 0 and 2^255-20. */
  h0 += 19 * q;
  /* Goal: Output h-2^255 q, which is between 0 and 2^255-20. */

  h1 += h0 >> 26; h0 &= kBottom26Bits;
  h2 += h1 >> 25; h1 &= kBottom25Bits;
  h3 += h2 >> 26; h2 &= kBottom26Bits;
  h4 += h3 >> 25; h3 &= kBottom25Bits;
  h5 += h4 >> 26; h4 &= kBottom26Bits;
  h6 += h5 >> 25; h5 &= kBottom25Bits;
  h7 += h6 >> 26; h6 &= kBottom26Bits;
  h8 += h7 >> 25; h7 &= kBottom25Bits;
  h9 += h8 >> 26; h8 &= kBottom26Bits;
                  h9 &= kBottom25Bits;
                  /* h10 = carry9 */

  /* Goal: Output h0+...+2^255 h10-2^255 q, which is between 0 and 2^255-20.
   * Have h0+...+2^230 h9 between 0 and 2^255-1;
   * evidently 2^255 h10-2^255 q = 0.
   * Goal: Output h0+...+2^230 h9.  */

  s[0] = h0 >> 0;
  s[1] = h0 >> 8;
  s[2] = h0 >> 16;
  s[3] = (h0 >> 24) | ((uint32_t)(h1) << 2);
  s[4] = h1 >> 6;
  s[5] = h1 >> 14;
  s[6] = (h1 >> 22) | ((uint32_t)(h2) << 3);
  s[7] = h2 >> 5;
  s[8] = h2 >> 13;
  s[9] = (h2 >> 21) | ((uint32_t)(h3) << 5);
  s[10] = h3 >> 3;
  s[11] = h3 >> 11;
  s[12] = (h3 >> 19) | ((uint32_t)(h4) << 6);
  s[13] = h4 >> 2;
  s[14] = h4 >> 10;
  s[15] = h4 >> 18;
  s[16] = h5 >> 0;
  s[17] = h5 >> 8;
  s[18] = h5 >> 16;
  s[19] = (h5 >> 24) | ((uint32_t)(h6) << 1);
  s[20] = h6 >> 7;
  s[21] = h6 >> 15;
  s[22] = (h6 >> 23) | ((uint32_t)(h7) << 3);
  s[23] = h7 >> 5;
  s[24] = h7 >> 13;
  s[25] = (h7 >> 21) | ((uint32_t)(h8) << 4);
  s[26] = h8 >> 4;
  s[27] = h8 >> 12;
  s[28] = (h8 >> 20) | ((uint32_t)(h9) << 6);
  s[29] = h9 >> 2;
  s[30] = h9 >> 10;
  s[31] = h9 >> 18;
}

/* h = f */
static void fe_copy(fe h, const fe f) {
  memmove(h, f, sizeof(int32_t) * 10);
}

/* h = 0 */
static void fe_0(fe h) { memset(h, 0, sizeof(int32_t) * 10); }

/* h = 1 */
static void fe_1(fe h) {
  memset(h, 0, sizeof(int32_t) * 10);
  h[0] = 1;
}

/* h = f + g
 * Can overlap h with f or g.
 *
 * Preconditions:
 *    |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
 *    |g| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
 *
 * Postconditions:
 *    |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. */
static void fe_add(fe h, const fe f, const fe g) {
  unsigned i;
  for (i = 0; i < 10; i++) {
    h[i] = f[i] + g[i];
  }
}

/* h = f - g
 * Can overlap h with f or g.
 *
 * Preconditions:
 *    |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
 *    |g| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
 *
 * Postconditions:
 *    |h| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. */
static void fe_sub(fe h, const fe f, const fe g) {
  unsigned i;
  for (i = 0; i < 10; i++) {
    h[i] = f[i] - g[i];
  }
}

/* h = f * g
 * Can overlap h with f or g.
 *
 * Preconditions:
 *    |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
 *    |g| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
 *
 * Postconditions:
 *    |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
 *
 * Notes on implementation strategy:
 *
 * Using schoolbook multiplication.
 * Karatsuba would save a little in some cost models.
 *
 * Most multiplications by 2 and 19 are 32-bit precomputations;
 * cheaper than 64-bit postcomputations.
 *
 * There is one remaining multiplication by 19 in the carry chain;
 * one *19 precomputation can be merged into this,
 * but the resulting data flow is considerably less clean.
 *
 * There are 12 carries below.
 * 10 of them are 2-way parallelizable and vectorizable.
 * Can get away with 11 carries, but then data flow is much deeper.
 *
 * With tighter constraints on inputs can squeeze carries into int32. */
static void fe_mul(fe h, const fe f, const fe g) {
  int32_t f0 = f[0];
  int32_t f1 = f[1];
  int32_t f2 = f[2];
  int32_t f3 = f[3];
  int32_t f4 = f[4];
  int32_t f5 = f[5];
  int32_t f6 = f[6];
  int32_t f7 = f[7];
  int32_t f8 = f[8];
  int32_t f9 = f[9];
  int32_t g0 = g[0];
  int32_t g1 = g[1];
  int32_t g2 = g[2];
  int32_t g3 = g[3];
  int32_t g4 = g[4];
  int32_t g5 = g[5];
  int32_t g6 = g[6];
  int32_t g7 = g[7];
  int32_t g8 = g[8];
  int32_t g9 = g[9];
  int32_t g1_19 = 19 * g1; /* 1.959375*2^29 */
  int32_t g2_19 = 19 * g2; /* 1.959375*2^30; still ok */
  int32_t g3_19 = 19 * g3;
  int32_t g4_19 = 19 * g4;
  int32_t g5_19 = 19 * g5;
  int32_t g6_19 = 19 * g6;
  int32_t g7_19 = 19 * g7;
  int32_t g8_19 = 19 * g8;
  int32_t g9_19 = 19 * g9;
  int32_t f1_2 = 2 * f1;
  int32_t f3_2 = 2 * f3;
  int32_t f5_2 = 2 * f5;
  int32_t f7_2 = 2 * f7;
  int32_t f9_2 = 2 * f9;
  int64_t f0g0    = f0   * (int64_t) g0;
  int64_t f0g1    = f0   * (int64_t) g1;
  int64_t f0g2    = f0   * (int64_t) g2;
  int64_t f0g3    = f0   * (int64_t) g3;
  int64_t f0g4    = f0   * (int64_t) g4;
  int64_t f0g5    = f0   * (int64_t) g5;
  int64_t f0g6    = f0   * (int64_t) g6;
  int64_t f0g7    = f0   * (int64_t) g7;
  int64_t f0g8    = f0   * (int64_t) g8;
  int64_t f0g9    = f0   * (int64_t) g9;
  int64_t f1g0    = f1   * (int64_t) g0;
  int64_t f1g1_2  = f1_2 * (int64_t) g1;
  int64_t f1g2    = f1   * (int64_t) g2;
  int64_t f1g3_2  = f1_2 * (int64_t) g3;
  int64_t f1g4    = f1   * (int64_t) g4;
  int64_t f1g5_2  = f1_2 * (int64_t) g5;
  int64_t f1g6    = f1   * (int64_t) g6;
  int64_t f1g7_2  = f1_2 * (int64_t) g7;
  int64_t f1g8    = f1   * (int64_t) g8;
  int64_t f1g9_38 = f1_2 * (int64_t) g9_19;
  int64_t f2g0    = f2   * (int64_t) g0;
  int64_t f2g1    = f2   * (int64_t) g1;
  int64_t f2g2    = f2   * (int64_t) g2;
  int64_t f2g3    = f2   * (int64_t) g3;
  int64_t f2g4    = f2   * (int64_t) g4;
  int64_t f2g5    = f2   * (int64_t) g5;
  int64_t f2g6    = f2   * (int64_t) g6;
  int64_t f2g7    = f2   * (int64_t) g7;
  int64_t f2g8_19 = f2   * (int64_t) g8_19;
  int64_t f2g9_19 = f2   * (int64_t) g9_19;
  int64_t f3g0    = f3   * (int64_t) g0;
  int64_t f3g1_2  = f3_2 * (int64_t) g1;
  int64_t f3g2    = f3   * (int64_t) g2;
  int64_t f3g3_2  = f3_2 * (int64_t) g3;
  int64_t f3g4    = f3   * (int64_t) g4;
  int64_t f3g5_2  = f3_2 * (int64_t) g5;
  int64_t f3g6    = f3   * (int64_t) g6;
  int64_t f3g7_38 = f3_2 * (int64_t) g7_19;
  int64_t f3g8_19 = f3   * (int64_t) g8_19;
  int64_t f3g9_38 = f3_2 * (int64_t) g9_19;
  int64_t f4g0    = f4   * (int64_t) g0;
  int64_t f4g1    = f4   * (int64_t) g1;
  int64_t f4g2    = f4   * (int64_t) g2;
  int64_t f4g3    = f4   * (int64_t) g3;
  int64_t f4g4    = f4   * (int64_t) g4;
  int64_t f4g5    = f4   * (int64_t) g5;
  int64_t f4g6_19 = f4   * (int64_t) g6_19;
  int64_t f4g7_19 = f4   * (int64_t) g7_19;
  int64_t f4g8_19 = f4   * (int64_t) g8_19;
  int64_t f4g9_19 = f4   * (int64_t) g9_19;
  int64_t f5g0    = f5   * (int64_t) g0;
  int64_t f5g1_2  = f5_2 * (int64_t) g1;
  int64_t f5g2    = f5   * (int64_t) g2;
  int64_t f5g3_2  = f5_2 * (int64_t) g3;
  int64_t f5g4    = f5   * (int64_t) g4;
  int64_t f5g5_38 = f5_2 * (int64_t) g5_19;
  int64_t f5g6_19 = f5   * (int64_t) g6_19;
  int64_t f5g7_38 = f5_2 * (int64_t) g7_19;
  int64_t f5g8_19 = f5   * (int64_t) g8_19;
  int64_t f5g9_38 = f5_2 * (int64_t) g9_19;
  int64_t f6g0    = f6   * (int64_t) g0;
  int64_t f6g1    = f6   * (int64_t) g1;
  int64_t f6g2    = f6   * (int64_t) g2;
  int64_t f6g3    = f6   * (int64_t) g3;
  int64_t f6g4_19 = f6   * (int64_t) g4_19;
  int64_t f6g5_19 = f6   * (int64_t) g5_19;
  int64_t f6g6_19 = f6   * (int64_t) g6_19;
  int64_t f6g7_19 = f6   * (int64_t) g7_19;
  int64_t f6g8_19 = f6   * (int64_t) g8_19;
  int64_t f6g9_19 = f6   * (int64_t) g9_19;
  int64_t f7g0    = f7   * (int64_t) g0;
  int64_t f7g1_2  = f7_2 * (int64_t) g1;
  int64_t f7g2    = f7   * (int64_t) g2;
  int64_t f7g3_38 = f7_2 * (int64_t) g3_19;
  int64_t f7g4_19 = f7   * (int64_t) g4_19;
  int64_t f7g5_38 = f7_2 * (int64_t) g5_19;
  int64_t f7g6_19 = f7   * (int64_t) g6_19;
  int64_t f7g7_38 = f7_2 * (int64_t) g7_19;
  int64_t f7g8_19 = f7   * (int64_t) g8_19;
  int64_t f7g9_38 = f7_2 * (int64_t) g9_19;
  int64_t f8g0    = f8   * (int64_t) g0;
  int64_t f8g1    = f8   * (int64_t) g1;
  int64_t f8g2_19 = f8   * (int64_t) g2_19;
  int64_t f8g3_19 = f8   * (int64_t) g3_19;
  int64_t f8g4_19 = f8   * (int64_t) g4_19;
  int64_t f8g5_19 = f8   * (int64_t) g5_19;
  int64_t f8g6_19 = f8   * (int64_t) g6_19;
  int64_t f8g7_19 = f8   * (int64_t) g7_19;
  int64_t f8g8_19 = f8   * (int64_t) g8_19;
  int64_t f8g9_19 = f8   * (int64_t) g9_19;
  int64_t f9g0    = f9   * (int64_t) g0;
  int64_t f9g1_38 = f9_2 * (int64_t) g1_19;
  int64_t f9g2_19 = f9   * (int64_t) g2_19;
  int64_t f9g3_38 = f9_2 * (int64_t) g3_19;
  int64_t f9g4_19 = f9   * (int64_t) g4_19;
  int64_t f9g5_38 = f9_2 * (int64_t) g5_19;
  int64_t f9g6_19 = f9   * (int64_t) g6_19;
  int64_t f9g7_38 = f9_2 * (int64_t) g7_19;
  int64_t f9g8_19 = f9   * (int64_t) g8_19;
  int64_t f9g9_38 = f9_2 * (int64_t) g9_19;
  int64_t h0 = f0g0+f1g9_38+f2g8_19+f3g7_38+f4g6_19+f5g5_38+f6g4_19+f7g3_38+f8g2_19+f9g1_38;
  int64_t h1 = f0g1+f1g0   +f2g9_19+f3g8_19+f4g7_19+f5g6_19+f6g5_19+f7g4_19+f8g3_19+f9g2_19;
  int64_t h2 = f0g2+f1g1_2 +f2g0   +f3g9_38+f4g8_19+f5g7_38+f6g6_19+f7g5_38+f8g4_19+f9g3_38;
  int64_t h3 = f0g3+f1g2   +f2g1   +f3g0   +f4g9_19+f5g8_19+f6g7_19+f7g6_19+f8g5_19+f9g4_19;
  int64_t h4 = f0g4+f1g3_2 +f2g2   +f3g1_2 +f4g0   +f5g9_38+f6g8_19+f7g7_38+f8g6_19+f9g5_38;
  int64_t h5 = f0g5+f1g4   +f2g3   +f3g2   +f4g1   +f5g0   +f6g9_19+f7g8_19+f8g7_19+f9g6_19;
  int64_t h6 = f0g6+f1g5_2 +f2g4   +f3g3_2 +f4g2   +f5g1_2 +f6g0   +f7g9_38+f8g8_19+f9g7_38;
  int64_t h7 = f0g7+f1g6   +f2g5   +f3g4   +f4g3   +f5g2   +f6g1   +f7g0   +f8g9_19+f9g8_19;
  int64_t h8 = f0g8+f1g7_2 +f2g6   +f3g5_2 +f4g4   +f5g3_2 +f6g2   +f7g1_2 +f8g0   +f9g9_38;
  int64_t h9 = f0g9+f1g8   +f2g7   +f3g6   +f4g5   +f5g4   +f6g3   +f7g2   +f8g1   +f9g0   ;
  int64_t carry0;
  int64_t carry1;
  int64_t carry2;
  int64_t carry3;
  int64_t carry4;
  int64_t carry5;
  int64_t carry6;
  int64_t carry7;
  int64_t carry8;
  int64_t carry9;

  /* |h0| <= (1.65*1.65*2^52*(1+19+19+19+19)+1.65*1.65*2^50*(38+38+38+38+38))
   *   i.e. |h0| <= 1.4*2^60; narrower ranges for h2, h4, h6, h8
   * |h1| <= (1.65*1.65*2^51*(1+1+19+19+19+19+19+19+19+19))
   *   i.e. |h1| <= 1.7*2^59; narrower ranges for h3, h5, h7, h9 */

  carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
  carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
  /* |h0| <= 2^25 */
  /* |h4| <= 2^25 */
  /* |h1| <= 1.71*2^59 */
  /* |h5| <= 1.71*2^59 */

  carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
  carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;
  /* |h1| <= 2^24; from now on fits into int32 */
  /* |h5| <= 2^24; from now on fits into int32 */
  /* |h2| <= 1.41*2^60 */
  /* |h6| <= 1.41*2^60 */

  carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
  carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;
  /* |h2| <= 2^25; from now on fits into int32 unchanged */
  /* |h6| <= 2^25; from now on fits into int32 unchanged */
  /* |h3| <= 1.71*2^59 */
  /* |h7| <= 1.71*2^59 */

  carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
  carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;
  /* |h3| <= 2^24; from now on fits into int32 unchanged */
  /* |h7| <= 2^24; from now on fits into int32 unchanged */
  /* |h4| <= 1.72*2^34 */
  /* |h8| <= 1.41*2^60 */

  carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
  carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;
  /* |h4| <= 2^25; from now on fits into int32 unchanged */
  /* |h8| <= 2^25; from now on fits into int32 unchanged */
  /* |h5| <= 1.01*2^24 */
  /* |h9| <= 1.71*2^59 */

  carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;
  /* |h9| <= 2^24; from now on fits into int32 unchanged */
  /* |h0| <= 1.1*2^39 */

  carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
  /* |h0| <= 2^25; from now on fits into int32 unchanged */
  /* |h1| <= 1.01*2^24 */

  h[0] = h0;
  h[1] = h1;
  h[2] = h2;
  h[3] = h3;
  h[4] = h4;
  h[5] = h5;
  h[6] = h6;
  h[7] = h7;
  h[8] = h8;
  h[9] = h9;
}

/* h = f * f
 * Can overlap h with f.
 *
 * Preconditions:
 *    |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
 *
 * Postconditions:
 *    |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
 *
 * See fe_mul.c for discussion of implementation strategy. */
static void fe_sq(fe h, const fe f) {
  int32_t f0 = f[0];
  int32_t f1 = f[1];
  int32_t f2 = f[2];
  int32_t f3 = f[3];
  int32_t f4 = f[4];
  int32_t f5 = f[5];
  int32_t f6 = f[6];
  int32_t f7 = f[7];
  int32_t f8 = f[8];
  int32_t f9 = f[9];
  int32_t f0_2 = 2 * f0;
  int32_t f1_2 = 2 * f1;
  int32_t f2_2 = 2 * f2;
  int32_t f3_2 = 2 * f3;
  int32_t f4_2 = 2 * f4;
  int32_t f5_2 = 2 * f5;
  int32_t f6_2 = 2 * f6;
  int32_t f7_2 = 2 * f7;
  int32_t f5_38 = 38 * f5; /* 1.959375*2^30 */
  int32_t f6_19 = 19 * f6; /* 1.959375*2^30 */
  int32_t f7_38 = 38 * f7; /* 1.959375*2^30 */
  int32_t f8_19 = 19 * f8; /* 1.959375*2^30 */
  int32_t f9_38 = 38 * f9; /* 1.959375*2^30 */
  int64_t f0f0    = f0   * (int64_t) f0;
  int64_t f0f1_2  = f0_2 * (int64_t) f1;
  int64_t f0f2_2  = f0_2 * (int64_t) f2;
  int64_t f0f3_2  = f0_2 * (int64_t) f3;
  int64_t f0f4_2  = f0_2 * (int64_t) f4;
  int64_t f0f5_2  = f0_2 * (int64_t) f5;
  int64_t f0f6_2  = f0_2 * (int64_t) f6;
  int64_t f0f7_2  = f0_2 * (int64_t) f7;
  int64_t f0f8_2  = f0_2 * (int64_t) f8;
  int64_t f0f9_2  = f0_2 * (int64_t) f9;
  int64_t f1f1_2  = f1_2 * (int64_t) f1;
  int64_t f1f2_2  = f1_2 * (int64_t) f2;
  int64_t f1f3_4  = f1_2 * (int64_t) f3_2;
  int64_t f1f4_2  = f1_2 * (int64_t) f4;
  int64_t f1f5_4  = f1_2 * (int64_t) f5_2;
  int64_t f1f6_2  = f1_2 * (int64_t) f6;
  int64_t f1f7_4  = f1_2 * (int64_t) f7_2;
  int64_t f1f8_2  = f1_2 * (int64_t) f8;
  int64_t f1f9_76 = f1_2 * (int64_t) f9_38;
  int64_t f2f2    = f2   * (int64_t) f2;
  int64_t f2f3_2  = f2_2 * (int64_t) f3;
  int64_t f2f4_2  = f2_2 * (int64_t) f4;
  int64_t f2f5_2  = f2_2 * (int64_t) f5;
  int64_t f2f6_2  = f2_2 * (int64_t) f6;
  int64_t f2f7_2  = f2_2 * (int64_t) f7;
  int64_t f2f8_38 = f2_2 * (int64_t) f8_19;
  int64_t f2f9_38 = f2   * (int64_t) f9_38;
  int64_t f3f3_2  = f3_2 * (int64_t) f3;
  int64_t f3f4_2  = f3_2 * (int64_t) f4;
  int64_t f3f5_4  = f3_2 * (int64_t) f5_2;
  int64_t f3f6_2  = f3_2 * (int64_t) f6;
  int64_t f3f7_76 = f3_2 * (int64_t) f7_38;
  int64_t f3f8_38 = f3_2 * (int64_t) f8_19;
  int64_t f3f9_76 = f3_2 * (int64_t) f9_38;
  int64_t f4f4    = f4   * (int64_t) f4;
  int64_t f4f5_2  = f4_2 * (int64_t) f5;
  int64_t f4f6_38 = f4_2 * (int64_t) f6_19;
  int64_t f4f7_38 = f4   * (int64_t) f7_38;
  int64_t f4f8_38 = f4_2 * (int64_t) f8_19;
  int64_t f4f9_38 = f4   * (int64_t) f9_38;
  int64_t f5f5_38 = f5   * (int64_t) f5_38;
  int64_t f5f6_38 = f5_2 * (int64_t) f6_19;
  int64_t f5f7_76 = f5_2 * (int64_t) f7_38;
  int64_t f5f8_38 = f5_2 * (int64_t) f8_19;
  int64_t f5f9_76 = f5_2 * (int64_t) f9_38;
  int64_t f6f6_19 = f6   * (int64_t) f6_19;
  int64_t f6f7_38 = f6   * (int64_t) f7_38;
  int64_t f6f8_38 = f6_2 * (int64_t) f8_19;
  int64_t f6f9_38 = f6   * (int64_t) f9_38;
  int64_t f7f7_38 = f7   * (int64_t) f7_38;
  int64_t f7f8_38 = f7_2 * (int64_t) f8_19;
  int64_t f7f9_76 = f7_2 * (int64_t) f9_38;
  int64_t f8f8_19 = f8   * (int64_t) f8_19;
  int64_t f8f9_38 = f8   * (int64_t) f9_38;
  int64_t f9f9_38 = f9   * (int64_t) f9_38;
  int64_t h0 = f0f0  +f1f9_76+f2f8_38+f3f7_76+f4f6_38+f5f5_38;
  int64_t h1 = f0f1_2+f2f9_38+f3f8_38+f4f7_38+f5f6_38;
  int64_t h2 = f0f2_2+f1f1_2 +f3f9_76+f4f8_38+f5f7_76+f6f6_19;
  int64_t h3 = f0f3_2+f1f2_2 +f4f9_38+f5f8_38+f6f7_38;
  int64_t h4 = f0f4_2+f1f3_4 +f2f2   +f5f9_76+f6f8_38+f7f7_38;
  int64_t h5 = f0f5_2+f1f4_2 +f2f3_2 +f6f9_38+f7f8_38;
  int64_t h6 = f0f6_2+f1f5_4 +f2f4_2 +f3f3_2 +f7f9_76+f8f8_19;
  int64_t h7 = f0f7_2+f1f6_2 +f2f5_2 +f3f4_2 +f8f9_38;
  int64_t h8 = f0f8_2+f1f7_4 +f2f6_2 +f3f5_4 +f4f4   +f9f9_38;
  int64_t h9 = f0f9_2+f1f8_2 +f2f7_2 +f3f6_2 +f4f5_2;
  int64_t carry0;
  int64_t carry1;
  int64_t carry2;
  int64_t carry3;
  int64_t carry4;
  int64_t carry5;
  int64_t carry6;
  int64_t carry7;
  int64_t carry8;
  int64_t carry9;

  carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
  carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;

  carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
  carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;

  carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
  carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;

  carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
  carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;

  carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
  carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;

  carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;

  carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;

  h[0] = h0;
  h[1] = h1;
  h[2] = h2;
  h[3] = h3;
  h[4] = h4;
  h[5] = h5;
  h[6] = h6;
  h[7] = h7;
  h[8] = h8;
  h[9] = h9;
}

static void fe_invert(fe out, const fe z) {
  fe t0;
  fe t1;
  fe t2;
  fe t3;
  int i;

  /*
   * Compute z ** -1 = z ** (2 ** 255 - 19 - 2) with the exponent as
   * 2 ** 255 - 21 = (2 ** 5) * (2 ** 250 - 1) + 11.
   */

  /* t0 = z ** 2 */
  fe_sq(t0, z);

  /* t1 = t0 ** (2 ** 2) = z ** 8 */
  fe_sq(t1, t0);
  fe_sq(t1, t1);

  /* t1 = z * t1 = z ** 9 */
  fe_mul(t1, z, t1);
  /* t0 = t0 * t1 = z ** 11 -- stash t0 away for the end. */
  fe_mul(t0, t0, t1);

  /* t2 = t0 ** 2 = z ** 22 */
  fe_sq(t2, t0);

  /* t1 = t1 * t2 = z ** (2 ** 5 - 1) */
  fe_mul(t1, t1, t2);

  /* t2 = t1 ** (2 ** 5) = z ** ((2 ** 5) * (2 ** 5 - 1)) */
  fe_sq(t2, t1);
  for (i = 1; i < 5; ++i) {
    fe_sq(t2, t2);
  }

  /* t1 = t1 * t2 = z ** ((2 ** 5 + 1) * (2 ** 5 - 1)) = z ** (2 ** 10 - 1) */
  fe_mul(t1, t2, t1);

  /* Continuing similarly... */

  /* t2 = z ** (2 ** 20 - 1) */
  fe_sq(t2, t1);
  for (i = 1; i < 10; ++i) {
    fe_sq(t2, t2);
  }
  fe_mul(t2, t2, t1);

  /* t2 = z ** (2 ** 40 - 1) */
  fe_sq(t3, t2);
  for (i = 1; i < 20; ++i) {
    fe_sq(t3, t3);
  }
  fe_mul(t2, t3, t2);

  /* t2 = z ** (2 ** 10) * (2 ** 40 - 1) */
  for (i = 0; i < 10; ++i) {
    fe_sq(t2, t2);
  }
  /* t1 = z ** (2 ** 50 - 1) */
  fe_mul(t1, t2, t1);

  /* t2 = z ** (2 ** 100 - 1) */
  fe_sq(t2, t1);
  for (i = 1; i < 50; ++i) {
    fe_sq(t2, t2);
  }
  fe_mul(t2, t2, t1);

  /* t2 = z ** (2 ** 200 - 1) */
  fe_sq(t3, t2);
  for (i = 1; i < 100; ++i) {
    fe_sq(t3, t3);
  }
  fe_mul(t2, t3, t2);

  /* t2 = z ** ((2 ** 50) * (2 ** 200 - 1) */
  fe_sq(t2, t2);
  for (i = 1; i < 50; ++i) {
    fe_sq(t2, t2);
  }

  /* t1 = z ** (2 ** 250 - 1) */
  fe_mul(t1, t2, t1);

  /* t1 = z ** ((2 ** 5) * (2 ** 250 - 1)) */
  fe_sq(t1, t1);
  for (i = 1; i < 5; ++i) {
    fe_sq(t1, t1);
  }

  /* Recall t0 = z ** 11; out = z ** (2 ** 255 - 21) */
  fe_mul(out, t1, t0);
}

/* h = -f
 *
 * Preconditions:
 *    |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
 *
 * Postconditions:
 *    |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. */
static void fe_neg(fe h, const fe f) {
  unsigned i;
  for (i = 0; i < 10; i++) {
    h[i] = -f[i];
  }
}

/* Replace (f,g) with (g,g) if b == 1;
 * replace (f,g) with (f,g) if b == 0.
 *
 * Preconditions: b in {0,1}. */
static void fe_cmov(fe f, const fe g, unsigned b) {
  size_t i;
  b = 0-b;
  for (i = 0; i < 10; i++) {
    int32_t x = f[i] ^ g[i];
    x &= b;
    f[i] ^= x;
  }
}

/* h = 2 * f * f
 * Can overlap h with f.
 *
 * Preconditions:
 *    |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
 *
 * Postconditions:
 *    |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
 *
 * See fe_mul.c for discussion of implementation strategy. */
static void fe_sq2(fe h, const fe f) {
  int32_t f0 = f[0];
  int32_t f1 = f[1];
  int32_t f2 = f[2];
  int32_t f3 = f[3];
  int32_t f4 = f[4];
  int32_t f5 = f[5];
  int32_t f6 = f[6];
  int32_t f7 = f[7];
  int32_t f8 = f[8];
  int32_t f9 = f[9];
  int32_t f0_2 = 2 * f0;
  int32_t f1_2 = 2 * f1;
  int32_t f2_2 = 2 * f2;
  int32_t f3_2 = 2 * f3;
  int32_t f4_2 = 2 * f4;
  int32_t f5_2 = 2 * f5;
  int32_t f6_2 = 2 * f6;
  int32_t f7_2 = 2 * f7;
  int32_t f5_38 = 38 * f5; /* 1.959375*2^30 */
  int32_t f6_19 = 19 * f6; /* 1.959375*2^30 */
  int32_t f7_38 = 38 * f7; /* 1.959375*2^30 */
  int32_t f8_19 = 19 * f8; /* 1.959375*2^30 */
  int32_t f9_38 = 38 * f9; /* 1.959375*2^30 */
  int64_t f0f0    = f0   * (int64_t) f0;
  int64_t f0f1_2  = f0_2 * (int64_t) f1;
  int64_t f0f2_2  = f0_2 * (int64_t) f2;
  int64_t f0f3_2  = f0_2 * (int64_t) f3;
  int64_t f0f4_2  = f0_2 * (int64_t) f4;
  int64_t f0f5_2  = f0_2 * (int64_t) f5;
  int64_t f0f6_2  = f0_2 * (int64_t) f6;
  int64_t f0f7_2  = f0_2 * (int64_t) f7;
  int64_t f0f8_2  = f0_2 * (int64_t) f8;
  int64_t f0f9_2  = f0_2 * (int64_t) f9;
  int64_t f1f1_2  = f1_2 * (int64_t) f1;
  int64_t f1f2_2  = f1_2 * (int64_t) f2;
  int64_t f1f3_4  = f1_2 * (int64_t) f3_2;
  int64_t f1f4_2  = f1_2 * (int64_t) f4;
  int64_t f1f5_4  = f1_2 * (int64_t) f5_2;
  int64_t f1f6_2  = f1_2 * (int64_t) f6;
  int64_t f1f7_4  = f1_2 * (int64_t) f7_2;
  int64_t f1f8_2  = f1_2 * (int64_t) f8;
  int64_t f1f9_76 = f1_2 * (int64_t) f9_38;
  int64_t f2f2    = f2   * (int64_t) f2;
  int64_t f2f3_2  = f2_2 * (int64_t) f3;
  int64_t f2f4_2  = f2_2 * (int64_t) f4;
  int64_t f2f5_2  = f2_2 * (int64_t) f5;
  int64_t f2f6_2  = f2_2 * (int64_t) f6;
  int64_t f2f7_2  = f2_2 * (int64_t) f7;
  int64_t f2f8_38 = f2_2 * (int64_t) f8_19;
  int64_t f2f9_38 = f2   * (int64_t) f9_38;
  int64_t f3f3_2  = f3_2 * (int64_t) f3;
  int64_t f3f4_2  = f3_2 * (int64_t) f4;
  int64_t f3f5_4  = f3_2 * (int64_t) f5_2;
  int64_t f3f6_2  = f3_2 * (int64_t) f6;
  int64_t f3f7_76 = f3_2 * (int64_t) f7_38;
  int64_t f3f8_38 = f3_2 * (int64_t) f8_19;
  int64_t f3f9_76 = f3_2 * (int64_t) f9_38;
  int64_t f4f4    = f4   * (int64_t) f4;
  int64_t f4f5_2  = f4_2 * (int64_t) f5;
  int64_t f4f6_38 = f4_2 * (int64_t) f6_19;
  int64_t f4f7_38 = f4   * (int64_t) f7_38;
  int64_t f4f8_38 = f4_2 * (int64_t) f8_19;
  int64_t f4f9_38 = f4   * (int64_t) f9_38;
  int64_t f5f5_38 = f5   * (int64_t) f5_38;
  int64_t f5f6_38 = f5_2 * (int64_t) f6_19;
  int64_t f5f7_76 = f5_2 * (int64_t) f7_38;
  int64_t f5f8_38 = f5_2 * (int64_t) f8_19;
  int64_t f5f9_76 = f5_2 * (int64_t) f9_38;
  int64_t f6f6_19 = f6   * (int64_t) f6_19;
  int64_t f6f7_38 = f6   * (int64_t) f7_38;
  int64_t f6f8_38 = f6_2 * (int64_t) f8_19;
  int64_t f6f9_38 = f6   * (int64_t) f9_38;
  int64_t f7f7_38 = f7   * (int64_t) f7_38;
  int64_t f7f8_38 = f7_2 * (int64_t) f8_19;
  int64_t f7f9_76 = f7_2 * (int64_t) f9_38;
  int64_t f8f8_19 = f8   * (int64_t) f8_19;
  int64_t f8f9_38 = f8   * (int64_t) f9_38;
  int64_t f9f9_38 = f9   * (int64_t) f9_38;
  int64_t h0 = f0f0  +f1f9_76+f2f8_38+f3f7_76+f4f6_38+f5f5_38;
  int64_t h1 = f0f1_2+f2f9_38+f3f8_38+f4f7_38+f5f6_38;
  int64_t h2 = f0f2_2+f1f1_2 +f3f9_76+f4f8_38+f5f7_76+f6f6_19;
  int64_t h3 = f0f3_2+f1f2_2 +f4f9_38+f5f8_38+f6f7_38;
  int64_t h4 = f0f4_2+f1f3_4 +f2f2   +f5f9_76+f6f8_38+f7f7_38;
  int64_t h5 = f0f5_2+f1f4_2 +f2f3_2 +f6f9_38+f7f8_38;
  int64_t h6 = f0f6_2+f1f5_4 +f2f4_2 +f3f3_2 +f7f9_76+f8f8_19;
  int64_t h7 = f0f7_2+f1f6_2 +f2f5_2 +f3f4_2 +f8f9_38;
  int64_t h8 = f0f8_2+f1f7_4 +f2f6_2 +f3f5_4 +f4f4   +f9f9_38;
  int64_t h9 = f0f9_2+f1f8_2 +f2f7_2 +f3f6_2 +f4f5_2;
  int64_t carry0;
  int64_t carry1;
  int64_t carry2;
  int64_t carry3;
  int64_t carry4;
  int64_t carry5;
  int64_t carry6;
  int64_t carry7;
  int64_t carry8;
  int64_t carry9;

  h0 += h0;
  h1 += h1;
  h2 += h2;
  h3 += h3;
  h4 += h4;
  h5 += h5;
  h6 += h6;
  h7 += h7;
  h8 += h8;
  h9 += h9;

  carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
  carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;

  carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
  carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;

  carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
  carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;

  carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
  carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;

  carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
  carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;

  carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;

  carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;

  h[0] = h0;
  h[1] = h1;
  h[2] = h2;
  h[3] = h3;
  h[4] = h4;
  h[5] = h5;
  h[6] = h6;
  h[7] = h7;
  h[8] = h8;
  h[9] = h9;
}

/* ge means group element.

 * Here the group is the set of pairs (x,y) of field elements (see fe.h)
 * satisfying -x^2 + y^2 = 1 + d x^2y^2
 * where d = -121665/121666.
 *
 * Representations:
 *   ge_p2 (projective): (X:Y:Z) satisfying x=X/Z, y=Y/Z
 *   ge_p3 (extended): (X:Y:Z:T) satisfying x=X/Z, y=Y/Z, XY=ZT
 *   ge_p1p1 (completed): ((X:Z),(Y:T)) satisfying x=X/Z, y=Y/T
 *   ge_precomp (Duif): (y+x,y-x,2dxy) */

typedef struct {
  fe X;
  fe Y;
  fe Z;
} ge_p2;

typedef struct {
  fe X;
  fe Y;
  fe Z;
  fe T;
} ge_p3;

typedef struct {
  fe X;
  fe Y;
  fe Z;
  fe T;
} ge_p1p1;

typedef struct {
  fe yplusx;
  fe yminusx;
  fe xy2d;
} ge_precomp;

typedef struct {
  fe YplusX;
  fe YminusX;
  fe Z;
  fe T2d;
} ge_cached;

static void ge_p3_0(ge_p3 *h) {
  fe_0(h->X);
  fe_1(h->Y);
  fe_1(h->Z);
  fe_0(h->T);
}

static void ge_precomp_0(ge_precomp *h) {
  fe_1(h->yplusx);
  fe_1(h->yminusx);
  fe_0(h->xy2d);
}

/* r = p */
static void ge_p3_to_p2(ge_p2 *r, const ge_p3 *p) {
  fe_copy(r->X, p->X);
  fe_copy(r->Y, p->Y);
  fe_copy(r->Z, p->Z);
}

/* r = p */
static void ge_p1p1_to_p2(ge_p2 *r, const ge_p1p1 *p) {
  fe_mul(r->X, p->X, p->T);
  fe_mul(r->Y, p->Y, p->Z);
  fe_mul(r->Z, p->Z, p->T);
}

/* r = p */
static void ge_p1p1_to_p3(ge_p3 *r, const ge_p1p1 *p) {
  fe_mul(r->X, p->X, p->T);
  fe_mul(r->Y, p->Y, p->Z);
  fe_mul(r->Z, p->Z, p->T);
  fe_mul(r->T, p->X, p->Y);
}

/* r = 2 * p */
static void ge_p2_dbl(ge_p1p1 *r, const ge_p2 *p) {
  fe t0;

  fe_sq(r->X, p->X);
  fe_sq(r->Z, p->Y);
  fe_sq2(r->T, p->Z);
  fe_add(r->Y, p->X, p->Y);
  fe_sq(t0, r->Y);
  fe_add(r->Y, r->Z, r->X);
  fe_sub(r->Z, r->Z, r->X);
  fe_sub(r->X, t0, r->Y);
  fe_sub(r->T, r->T, r->Z);
}

/* r = 2 * p */
static void ge_p3_dbl(ge_p1p1 *r, const ge_p3 *p) {
  ge_p2 q;
  ge_p3_to_p2(&q, p);
  ge_p2_dbl(r, &q);
}

/* r = p + q */
static void ge_madd(ge_p1p1 *r, const ge_p3 *p, const ge_precomp *q) {
  fe t0;

  fe_add(r->X, p->Y, p->X);
  fe_sub(r->Y, p->Y, p->X);
  fe_mul(r->Z, r->X, q->yplusx);
  fe_mul(r->Y, r->Y, q->yminusx);
  fe_mul(r->T, q->xy2d, p->T);
  fe_add(t0, p->Z, p->Z);
  fe_sub(r->X, r->Z, r->Y);
  fe_add(r->Y, r->Z, r->Y);
  fe_add(r->Z, t0, r->T);
  fe_sub(r->T, t0, r->T);
}

static uint8_t equal(signed char b, signed char c) {
  uint8_t ub = b;
  uint8_t uc = c;
  uint8_t x = ub ^ uc; /* 0: yes; 1..255: no */
  uint32_t y = x;      /* 0: yes; 1..255: no */
  y -= 1;              /* 4294967295: yes; 0..254: no */
  y >>= 31;            /* 1: yes; 0: no */
  return y;
}

static void cmov(ge_precomp *t, const ge_precomp *u, uint8_t b) {
  fe_cmov(t->yplusx, u->yplusx, b);
  fe_cmov(t->yminusx, u->yminusx, b);
  fe_cmov(t->xy2d, u->xy2d, b);
}

/* k25519Precomp[i][j] = (j+1)*256^i*B */
static const ge_precomp k25519Precomp[32][8] = {
    {
        {
            {25967493, -14356035, 29566456, 3660896, -12694345, 4014787,
             27544626, -11754271, -6079156, 2047605},
            {-12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692,
             5043384, 19500929, -15469378},
            {-8738181, 4489570, 9688441, -14785194, 10184609, -12363380,
             29287919, 11864899, -24514362, -4438546},
        },
        {
            {-12815894, -12976347, -21581243, 11784320, -25355658, -2750717,
             -11717903, -3814571, -358445, -10211303},
            {-21703237, 6903825, 27185491, 6451973, -29577724, -9554005,
             -15616551, 11189268, -26829678, -5319081},
            {26966642, 11152617, 32442495, 15396054, 14353839, -12752335,
             -3128826, -9541118, -15472047, -4166697},
        },
        {
            {15636291, -9688557, 24204773, -7912398, 616977, -16685262,
             27787600, -14772189, 28944400, -1550024},
            {16568933, 4717097, -11556148, -1102322, 15682896, -11807043,
             16354577, -11775962, 7689662, 11199574},
            {30464156, -5976125, -11779434, -15670865, 23220365, 15915852,
             7512774, 10017326, -17749093, -9920357},
        },
        {
            {-17036878, 13921892, 10945806, -6033431, 27105052, -16084379,
             -28926210, 15006023, 3284568, -6276540},
            {23599295, -8306047, -11193664, -7687416, 13236774, 10506355,
             7464579, 9656445, 13059162, 10374397},
            {7798556, 16710257, 3033922, 2874086, 28997861, 2835604, 32406664,
             -3839045, -641708, -101325},
        },
        {
            {10861363, 11473154, 27284546, 1981175, -30064349, 12577861,
             32867885, 14515107, -15438304, 10819380},
            {4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668,
             12483688, -12668491, 5581306},
            {19563160, 16186464, -29386857, 4097519, 10237984, -4348115,
             28542350, 13850243, -23678021, -15815942},
        },
        {
            {-15371964, -12862754, 32573250, 4720197, -26436522, 5875511,
             -19188627, -15224819, -9818940, -12085777},
            {-8549212, 109983, 15149363, 2178705, 22900618, 4543417, 3044240,
             -15689887, 1762328, 14866737},
            {-18199695, -15951423, -10473290, 1707278, -17185920, 3916101,
             -28236412, 3959421, 27914454, 4383652},
        },
        {
            {5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852,
             5230134, -23952439, -15175766},
            {-30269007, -3463509, 7665486, 10083793, 28475525, 1649722,
             20654025, 16520125, 30598449, 7715701},
            {28881845, 14381568, 9657904, 3680757, -20181635, 7843316,
             -31400660, 1370708, 29794553, -1409300},
        },
        {
            {14499471, -2729599, -33191113, -4254652, 28494862, 14271267,
             30290735, 10876454, -33154098, 2381726},
            {-7195431, -2655363, -14730155, 462251, -27724326, 3941372,
             -6236617, 3696005, -32300832, 15351955},
            {27431194, 8222322, 16448760, -3907995, -18707002, 11938355,
             -32961401, -2970515, 29551813, 10109425},
        },
    },
    {
        {
            {-13657040, -13155431, -31283750, 11777098, 21447386, 6519384,
             -2378284, -1627556, 10092783, -4764171},
            {27939166, 14210322, 4677035, 16277044, -22964462, -12398139,
             -32508754, 12005538, -17810127, 12803510},
            {17228999, -15661624, -1233527, 300140, -1224870, -11714777,
             30364213, -9038194, 18016357, 4397660},
        },
        {
            {-10958843, -7690207, 4776341, -14954238, 27850028, -15602212,
             -26619106, 14544525, -17477504, 982639},
            {29253598, 15796703, -2863982, -9908884, 10057023, 3163536, 7332899,
             -4120128, -21047696, 9934963},
            {5793303, 16271923, -24131614, -10116404, 29188560, 1206517,
             -14747930, 4559895, -30123922, -10897950},
        },
        {
            {-27643952, -11493006, 16282657, -11036493, 28414021, -15012264,
             24191034, 4541697, -13338309, 5500568},
            {12650548, -1497113, 9052871, 11355358, -17680037, -8400164,
             -17430592, 12264343, 10874051, 13524335},
            {25556948, -3045990, 714651, 2510400, 23394682, -10415330, 33119038,
             5080568, -22528059, 5376628},
        },
        {
            {-26088264, -4011052, -17013699, -3537628, -6726793, 1920897,
             -22321305, -9447443, 4535768, 1569007},
            {-2255422, 14606630, -21692440, -8039818, 28430649, 8775819,
             -30494562, 3044290, 31848280, 12543772},
            {-22028579, 2943893, -31857513, 6777306, 13784462, -4292203,
             -27377195, -2062731, 7718482, 14474653},
        },
        {
            {2385315, 2454213, -22631320, 46603, -4437935, -15680415, 656965,
             -7236665, 24316168, -5253567},
            {13741529, 10911568, -33233417, -8603737, -20177830, -1033297,
             33040651, -13424532, -20729456, 8321686},
            {21060490, -2212744, 15712757, -4336099, 1639040, 10656336,
             23845965, -11874838, -9984458, 608372},
        },
        {
            {-13672732, -15087586, -10889693, -7557059, -6036909, 11305547,
             1123968, -6780577, 27229399, 23887},
            {-23244140, -294205, -11744728, 14712571, -29465699, -2029617,
             12797024, -6440308, -1633405, 16678954},
            {-29500620, 4770662, -16054387, 14001338, 7830047, 9564805,
             -1508144, -4795045, -17169265, 4904953},
        },
        {
            {24059557, 14617003, 19037157, -15039908, 19766093, -14906429,
             5169211, 16191880, 2128236, -4326833},
            {-16981152, 4124966, -8540610, -10653797, 30336522, -14105247,
             -29806336, 916033, -6882542, -2986532},
            {-22630907, 12419372, -7134229, -7473371, -16478904, 16739175,
             285431, 2763829, 15736322, 4143876},
        },
        {
            {2379352, 11839345, -4110402, -5988665, 11274298, 794957, 212801,
             -14594663, 23527084, -16458268},
            {33431127, -11130478, -17838966, -15626900, 8909499, 8376530,
             -32625340, 4087881, -15188911, -14416214},
            {1767683, 7197987, -13205226, -2022635, -13091350, 448826, 5799055,
             4357868, -4774191, -16323038},
        },
    },
    {
        {
            {6721966, 13833823, -23523388, -1551314, 26354293, -11863321,
             23365147, -3949732, 7390890, 2759800},
            {4409041, 2052381, 23373853, 10530217, 7676779, -12885954, 21302353,
             -4264057, 1244380, -12919645},
            {-4421239, 7169619, 4982368, -2957590, 30256825, -2777540, 14086413,
             9208236, 15886429, 16489664},
        },
        {
            {1996075, 10375649, 14346367, 13311202, -6874135, -16438411,
             -13693198, 398369, -30606455, -712933},
            {-25307465, 9795880, -2777414, 14878809, -33531835, 14780363,
             13348553, 12076947, -30836462, 5113182},
            {-17770784, 11797796, 31950843, 13929123, -25888302, 12288344,
             -30341101, -7336386, 13847711, 5387222},
        },
        {
            {-18582163, -3416217, 17824843, -2340966, 22744343, -10442611,
             8763061, 3617786, -19600662, 10370991},
            {20246567, -14369378, 22358229, -543712, 18507283, -10413996,
             14554437, -8746092, 32232924, 16763880},
            {9648505, 10094563, 26416693, 14745928, -30374318, -6472621,
             11094161, 15689506, 3140038, -16510092},
        },
        {
            {-16160072, 5472695, 31895588, 4744994, 8823515, 10365685,
             -27224800, 9448613, -28774454, 366295},
            {19153450, 11523972, -11096490, -6503142, -24647631, 5420647,
             28344573, 8041113, 719605, 11671788},
            {8678025, 2694440, -6808014, 2517372, 4964326, 11152271, -15432916,
             -15266516, 27000813, -10195553},
        },
        {
            {-15157904, 7134312, 8639287, -2814877, -7235688, 10421742, 564065,
             5336097, 6750977, -14521026},
            {11836410, -3979488, 26297894, 16080799, 23455045, 15735944,
             1695823, -8819122, 8169720, 16220347},
            {-18115838, 8653647, 17578566, -6092619, -8025777, -16012763,
             -11144307, -2627664, -5990708, -14166033},
        },
        {
            {-23308498, -10968312, 15213228, -10081214, -30853605, -11050004,
             27884329, 2847284, 2655861, 1738395},
            {-27537433, -14253021, -25336301, -8002780, -9370762, 8129821,
             21651608, -3239336, -19087449, -11005278},
            {1533110, 3437855, 23735889, 459276, 29970501, 11335377, 26030092,
             5821408, 10478196, 8544890},
        },
        {
            {32173121, -16129311, 24896207, 3921497, 22579056, -3410854,
             19270449, 12217473, 17789017, -3395995},
            {-30552961, -2228401, -15578829, -10147201, 13243889, 517024,
             15479401, -3853233, 30460520, 1052596},
            {-11614875, 13323618, 32618793, 8175907, -15230173, 12596687,
             27491595, -4612359, 3179268, -9478891},
        },
        {
            {31947069, -14366651, -4640583, -15339921, -15125977, -6039709,
             -14756777, -16411740, 19072640, -9511060},
            {11685058, 11822410, 3158003, -13952594, 33402194, -4165066,
             5977896, -5215017, 473099, 5040608},
            {-20290863, 8198642, -27410132, 11602123, 1290375, -2799760,
             28326862, 1721092, -19558642, -3131606},
        },
    },
    {
        {
            {7881532, 10687937, 7578723, 7738378, -18951012, -2553952, 21820786,
             8076149, -27868496, 11538389},
            {-19935666, 3899861, 18283497, -6801568, -15728660, -11249211,
             8754525, 7446702, -5676054, 5797016},
            {-11295600, -3793569, -15782110, -7964573, 12708869, -8456199,
             2014099, -9050574, -2369172, -5877341},
        },
        {
            {-22472376, -11568741, -27682020, 1146375, 18956691, 16640559,
             1192730, -3714199, 15123619, 10811505},
            {14352098, -3419715, -18942044, 10822655, 32750596, 4699007, -70363,
             15776356, -28886779, -11974553},
            {-28241164, -8072475, -4978962, -5315317, 29416931, 1847569,
             -20654173, -16484855, 4714547, -9600655},
        },
        {
            {15200332, 8368572, 19679101, 15970074, -31872674, 1959451,
             24611599, -4543832, -11745876, 12340220},
            {12876937, -10480056, 33134381, 6590940, -6307776, 14872440,
             9613953, 8241152, 15370987, 9608631},
            {-4143277, -12014408, 8446281, -391603, 4407738, 13629032, -7724868,
             15866074, -28210621, -8814099},
        },
        {
            {26660628, -15677655, 8393734, 358047, -7401291, 992988, -23904233,
             858697, 20571223, 8420556},
            {14620715, 13067227, -15447274, 8264467, 14106269, 15080814,
             33531827, 12516406, -21574435, -12476749},
            {236881, 10476226, 57258, -14677024, 6472998, 2466984, 17258519,
             7256740, 8791136, 15069930},
        },
        {
            {1276410, -9371918, 22949635, -16322807, -23493039, -5702186,
             14711875, 4874229, -30663140, -2331391},
            {5855666, 4990204, -13711848, 7294284, -7804282, 1924647, -1423175,
             -7912378, -33069337, 9234253},
            {20590503, -9018988, 31529744, -7352666, -2706834, 10650548,
             31559055, -11609587, 18979186, 13396066},
        },
        {
            {24474287, 4968103, 22267082, 4407354, 24063882, -8325180,
             -18816887, 13594782, 33514650, 7021958},
            {-11566906, -6565505, -21365085, 15928892, -26158305, 4315421,
             -25948728, -3916677, -21480480, 12868082},
            {-28635013, 13504661, 19988037, -2132761, 21078225, 6443208,
             -21446107, 2244500, -12455797, -8089383},
        },
        {
            {-30595528, 13793479, -5852820, 319136, -25723172, -6263899,
             33086546, 8957937, -15233648, 5540521},
            {-11630176, -11503902, -8119500, -7643073, 2620056, 1022908,
             -23710744, -1568984, -16128528, -14962807},
            {23152971, 775386, 27395463, 14006635, -9701118, 4649512, 1689819,
             892185, -11513277, -15205948},
        },
        {
            {9770129, 9586738, 26496094, 4324120, 1556511, -3550024, 27453819,
             4763127, -19179614, 5867134},
            {-32765025, 1927590, 31726409, -4753295, 23962434, -16019500,
             27846559, 5931263, -29749703, -16108455},
            {27461885, -2977536, 22380810, 1815854, -23033753, -3031938,
             7283490, -15148073, -19526700, 7734629},
        },
    },
    {
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    {
        {
            {20678546, -8375738, -32671898, 8849123, -5009758, 14574752,
             31186971, -3973730, 9014762, -8579056},
            {-13644050, -10350239, -15962508, 5075808, -1514661, -11534600,
             -33102500, 9160280, 8473550, -3256838},
            {24900749, 14435722, 17209120, -15292541, -22592275, 9878983,
             -7689309, -16335821, -24568481, 11788948},
        },
        {
            {-3118155, -11395194, -13802089, 14797441, 9652448, -6845904,
             -20037437, 10410733, -24568470, -1458691},
            {-15659161, 16736706, -22467150, 10215878, -9097177, 7563911,
             11871841, -12505194, -18513325, 8464118},
            {-23400612, 8348507, -14585951, -861714, -3950205, -6373419,
             14325289, 8628612, 33313881, -8370517},
        },
        {
            {-20186973, -4967935, 22367356, 5271547, -1097117, -4788838,
             -24805667, -10236854, -8940735, -5818269},
            {-6948785, -1795212, -32625683, -16021179, 32635414, -7374245,
             15989197, -12838188, 28358192, -4253904},
            {-23561781, -2799059, -32351682, -1661963, -9147719, 10429267,
             -16637684, 4072016, -5351664, 5596589},
        },
        {
            {-28236598, -3390048, 12312896, 6213178, 3117142, 16078565,
             29266239, 2557221, 1768301, 15373193},
            {-7243358, -3246960, -4593467, -7553353, -127927, -912245, -1090902,
             -4504991, -24660491, 3442910},
            {-30210571, 5124043, 14181784, 8197961, 18964734, -11939093,
             22597931, 7176455, -18585478, 13365930},
        },
        {
            {-7877390, -1499958, 8324673, 4690079, 6261860, 890446, 24538107,
             -8570186, -9689599, -3031667},
            {25008904, -10771599, -4305031, -9638010, 16265036, 15721635,
             683793, -11823784, 15723479, -15163481},
            {-9660625, 12374379, -27006999, -7026148, -7724114, -12314514,
             11879682, 5400171, 519526, -1235876},
        },
        {
            {22258397, -16332233, -7869817, 14613016, -22520255, -2950923,
             -20353881, 7315967, 16648397, 7605640},
            {-8081308, -8464597, -8223311, 9719710, 19259459, -15348212,
             23994942, -5281555, -9468848, 4763278},
            {-21699244, 9220969, -15730624, 1084137, -25476107, -2852390,
             31088447, -7764523, -11356529, 728112},
        },
        {
            {26047220, -11751471, -6900323, -16521798, 24092068, 9158119,
             -4273545, -12555558, -29365436, -5498272},
            {17510331, -322857, 5854289, 8403524, 17133918, -3112612, -28111007,
             12327945, 10750447, 10014012},
            {-10312768, 3936952, 9156313, -8897683, 16498692, -994647,
             -27481051, -666732, 3424691, 7540221},
        },
        {
            {30322361, -6964110, 11361005, -4143317, 7433304, 4989748, -7071422,
             -16317219, -9244265, 15258046},
            {13054562, -2779497, 19155474, 469045, -12482797, 4566042, 5631406,
             2711395, 1062915, -5136345},
            {-19240248, -11254599, -29509029, -7499965, -5835763, 13005411,
             -6066489, 12194497, 32960380, 1459310},
        },
    },
    {
        {
            {19852034, 7027924, 23669353, 10020366, 8586503, -6657907, 394197,
             -6101885, 18638003, -11174937},
            {31395534, 15098109, 26581030, 8030562, -16527914, -5007134,
             9012486, -7584354, -6643087, -5442636},
            {-9192165, -2347377, -1997099, 4529534, 25766844, 607986, -13222,
             9677543, -32294889, -6456008},
        },
        {
            {-2444496, -149937, 29348902, 8186665, 1873760, 12489863, -30934579,
             -7839692, -7852844, -8138429},
            {-15236356, -15433509, 7766470, 746860, 26346930, -10221762,
             -27333451, 10754588, -9431476, 5203576},
            {31834314, 14135496, -770007, 5159118, 20917671, -16768096,
             -7467973, -7337524, 31809243, 7347066},
        },
        {
            {-9606723, -11874240, 20414459, 13033986, 13716524, -11691881,
             19797970, -12211255, 15192876, -2087490},
            {-12663563, -2181719, 1168162, -3804809, 26747877, -14138091,
             10609330, 12694420, 33473243, -13382104},
            {33184999, 11180355, 15832085, -11385430, -1633671, 225884,
             15089336, -11023903, -6135662, 14480053},
        },
        {
            {31308717, -5619998, 31030840, -1897099, 15674547, -6582883,
             5496208, 13685227, 27595050, 8737275},
            {-20318852, -15150239, 10933843, -16178022, 8335352, -7546022,
             -31008351, -12610604, 26498114, 66511},
            {22644454, -8761729, -16671776, 4884562, -3105614, -13559366,
             30540766, -4286747, -13327787, -7515095},
        },
        {
            {-28017847, 9834845, 18617207, -2681312, -3401956, -13307506,
             8205540, 13585437, -17127465, 15115439},
            {23711543, -672915, 31206561, -8362711, 6164647, -9709987,
             -33535882, -1426096, 8236921, 16492939},
            {-23910559, -13515526, -26299483, -4503841, 25005590, -7687270,
             19574902, 10071562, 6708380, -6222424},
        },
        {
            {2101391, -4930054, 19702731, 2367575, -15427167, 1047675, 5301017,
             9328700, 29955601, -11678310},
            {3096359, 9271816, -21620864, -15521844, -14847996, -7592937,
             -25892142, -12635595, -9917575, 6216608},
            {-32615849, 338663, -25195611, 2510422, -29213566, -13820213,
             24822830, -6146567, -26767480, 7525079},
        },
        {
            {-23066649, -13985623, 16133487, -7896178, -3389565, 778788,
             -910336, -2782495, -19386633, 11994101},
            {21691500, -13624626, -641331, -14367021, 3285881, -3483596,
             -25064666, 9718258, -7477437, 13381418},
            {18445390, -4202236, 14979846, 11622458, -1727110, -3582980,
             23111648, -6375247, 28535282, 15779576},
        },
        {
            {30098053, 3089662, -9234387, 16662135, -21306940, 11308411,
             -14068454, 12021730, 9955285, -16303356},
            {9734894, -14576830, -7473633, -9138735, 2060392, 11313496,
             -18426029, 9924399, 20194861, 13380996},
            {-26378102, -7965207, -22167821, 15789297, -18055342, -6168792,
             -1984914, 15707771, 26342023, 10146099},
        },
    },
    {
        {
            {-26016874, -219943, 21339191, -41388, 19745256, -2878700,
             -29637280, 2227040, 21612326, -545728},
            {-13077387, 1184228, 23562814, -5970442, -20351244, -6348714,
             25764461, 12243797, -20856566, 11649658},
            {-10031494, 11262626, 27384172, 2271902, 26947504, -15997771, 39944,
             6114064, 33514190, 2333242},
        },
        {
            {-21433588, -12421821, 8119782, 7219913, -21830522, -9016134,
             -6679750, -12670638, 24350578, -13450001},
            {-4116307, -11271533, -23886186, 4843615, -30088339, 690623,
             -31536088, -10406836, 8317860, 12352766},
            {18200138, -14475911, -33087759, -2696619, -23702521, -9102511,
             -23552096, -2287550, 20712163, 6719373},
        },
        {
            {26656208, 6075253, -7858556, 1886072, -28344043, 4262326, 11117530,
             -3763210, 26224235, -3297458},
            {-17168938, -14854097, -3395676, -16369877, -19954045, 14050420,
             21728352, 9493610, 18620611, -16428628},
            {-13323321, 13325349, 11432106, 5964811, 18609221, 6062965,
             -5269471, -9725556, -30701573, -16479657},
        },
        {
            {-23860538, -11233159, 26961357, 1640861, -32413112, -16737940,
             12248509, -5240639, 13735342, 1934062},
            {25089769, 6742589, 17081145, -13406266, 21909293, -16067981,
             -15136294, -3765346, -21277997, 5473616},
            {31883677, -7961101, 1083432, -11572403, 22828471, 13290673,
             -7125085, 12469656, 29111212, -5451014},
        },
        {
            {24244947, -15050407, -26262976, 2791540, -14997599, 16666678,
             24367466, 6388839, -10295587, 452383},
            {-25640782, -3417841, 5217916, 16224624, 19987036, -4082269,
             -24236251, -5915248, 15766062, 8407814},
            {-20406999, 13990231, 15495425, 16395525, 5377168, 15166495,
             -8917023, -4388953, -8067909, 2276718},
        },
        {
            {30157918, 12924066, -17712050, 9245753, 19895028, 3368142,
             -23827587, 5096219, 22740376, -7303417},
            {2041139, -14256350, 7783687, 13876377, -25946985, -13352459,
             24051124, 13742383, -15637599, 13295222},
            {33338237, -8505733, 12532113, 7977527, 9106186, -1715251,
             -17720195, -4612972, -4451357, -14669444},
        },
        {
            {-20045281, 5454097, -14346548, 6447146, 28862071, 1883651,
             -2469266, -4141880, 7770569, 9620597},
            {23208068, 7979712, 33071466, 8149229, 1758231, -10834995, 30945528,
             -1694323, -33502340, -14767970},
            {1439958, -16270480, -1079989, -793782, 4625402, 10647766, -5043801,
             1220118, 30494170, -11440799},
        },
        {
            {-5037580, -13028295, -2970559, -3061767, 15640974, -6701666,
             -26739026, 926050, -1684339, -13333647},
            {13908495, -3549272, 30919928, -6273825, -21521863, 7989039,
             9021034, 9078865, 3353509, 4033511},
            {-29663431, -15113610, 32259991, -344482, 24295849, -12912123,
             23161163, 8839127, 27485041, 7356032},
        },
    },
    {
        {
            {9661027, 705443, 11980065, -5370154, -1628543, 14661173, -6346142,
             2625015, 28431036, -16771834},
            {-23839233, -8311415, -25945511, 7480958, -17681669, -8354183,
             -22545972, 14150565, 15970762, 4099461},
            {29262576, 16756590, 26350592, -8793563, 8529671, -11208050,
             13617293, -9937143, 11465739, 8317062},
        },
        {
            {-25493081, -6962928, 32500200, -9419051, -23038724, -2302222,
             14898637, 3848455, 20969334, -5157516},
            {-20384450, -14347713, -18336405, 13884722, -33039454, 2842114,
             -21610826, -3649888, 11177095, 14989547},
            {-24496721, -11716016, 16959896, 2278463, 12066309, 10137771,
             13515641, 2581286, -28487508, 9930240},
        },
        {
            {-17751622, -2097826, 16544300, -13009300, -15914807, -14949081,
             18345767, -13403753, 16291481, -5314038},
            {-33229194, 2553288, 32678213, 9875984, 8534129, 6889387, -9676774,
             6957617, 4368891, 9788741},
            {16660756, 7281060, -10830758, 12911820, 20108584, -8101676,
             -21722536, -8613148, 16250552, -11111103},
        },
        {
            {-19765507, 2390526, -16551031, 14161980, 1905286, 6414907, 4689584,
             10604807, -30190403, 4782747},
            {-1354539, 14736941, -7367442, -13292886, 7710542, -14155590,
             -9981571, 4383045, 22546403, 437323},
            {31665577, -12180464, -16186830, 1491339, -18368625, 3294682,
             27343084, 2786261, -30633590, -14097016},
        },
        {
            {-14467279, -683715, -33374107, 7448552, 19294360, 14334329,
             -19690631, 2355319, -19284671, -6114373},
            {15121312, -15796162, 6377020, -6031361, -10798111, -12957845,
             18952177, 15496498, -29380133, 11754228},
            {-2637277, -13483075, 8488727, -14303896, 12728761, -1622493,
             7141596, 11724556, 22761615, -10134141},
        },
        {
            {16918416, 11729663, -18083579, 3022987, -31015732, -13339659,
             -28741185, -12227393, 32851222, 11717399},
            {11166634, 7338049, -6722523, 4531520, -29468672, -7302055,
             31474879, 3483633, -1193175, -4030831},
            {-185635, 9921305, 31456609, -13536438, -12013818, 13348923,
             33142652, 6546660, -19985279, -3948376},
        },
        {
            {-32460596, 11266712, -11197107, -7899103, 31703694, 3855903,
             -8537131, -12833048, -30772034, -15486313},
            {-18006477, 12709068, 3991746, -6479188, -21491523, -10550425,
             -31135347, -16049879, 10928917, 3011958},
            {-6957757, -15594337, 31696059, 334240, 29576716, 14796075,
             -30831056, -12805180, 18008031, 10258577},
        },
        {
            {-22448644, 15655569, 7018479, -4410003, -30314266, -1201591,
             -1853465, 1367120, 25127874, 6671743},
            {29701166, -14373934, -10878120, 9279288, -17568, 13127210,
             21382910, 11042292, 25838796, 4642684},
            {-20430234, 14955537, -24126347, 8124619, -5369288, -5990470,
             30468147, -13900640, 18423289, 4177476},
        },
    },
};

static uint8_t negative(signed char b) {
  uint32_t x = b;
  x >>= 31; /* 1: yes; 0: no */
  return x;
}

static void table_select(ge_precomp *t, int pos, signed char b) {
  ge_precomp minust;
  uint8_t bnegative = negative(b);
  uint8_t babs = b - ((uint8_t)((-bnegative) & b) << 1);

  ge_precomp_0(t);
  cmov(t, &k25519Precomp[pos][0], equal(babs, 1));
  cmov(t, &k25519Precomp[pos][1], equal(babs, 2));
  cmov(t, &k25519Precomp[pos][2], equal(babs, 3));
  cmov(t, &k25519Precomp[pos][3], equal(babs, 4));
  cmov(t, &k25519Precomp[pos][4], equal(babs, 5));
  cmov(t, &k25519Precomp[pos][5], equal(babs, 6));
  cmov(t, &k25519Precomp[pos][6], equal(babs, 7));
  cmov(t, &k25519Precomp[pos][7], equal(babs, 8));
  fe_copy(minust.yplusx, t->yminusx);
  fe_copy(minust.yminusx, t->yplusx);
  fe_neg(minust.xy2d, t->xy2d);
  cmov(t, &minust, bnegative);
}

/* h = a * B
 * where a = a[0]+256*a[1]+...+256^31 a[31]
 * B is the Ed25519 base point (x,4/5) with x positive.
 *
 * Preconditions:
 *   a[31] <= 127 */
static void ge_scalarmult_base(ge_p3 *h, const uint8_t *a) {
  signed char e[64];
  signed char carry;
  ge_p1p1 r;
  ge_p2 s;
  ge_precomp t;
  int i;

  for (i = 0; i < 32; ++i) {
    e[2 * i + 0] = (a[i] >> 0) & 15;
    e[2 * i + 1] = (a[i] >> 4) & 15;
  }
  /* each e[i] is between 0 and 15 */
  /* e[63] is between 0 and 7 */

  carry = 0;
  for (i = 0; i < 63; ++i) {
    e[i] += carry;
    carry = e[i] + 8;
    carry >>= 4;
    e[i] -= carry << 4;
  }
  e[63] += carry;
  /* each e[i] is between -8 and 8 */

  ge_p3_0(h);
  for (i = 1; i < 64; i += 2) {
    table_select(&t, i / 2, e[i]);
    ge_madd(&r, h, &t);
    ge_p1p1_to_p3(h, &r);
  }

  ge_p3_dbl(&r, h);
  ge_p1p1_to_p2(&s, &r);
  ge_p2_dbl(&r, &s);
  ge_p1p1_to_p2(&s, &r);
  ge_p2_dbl(&r, &s);
  ge_p1p1_to_p2(&s, &r);
  ge_p2_dbl(&r, &s);
  ge_p1p1_to_p3(h, &r);

  for (i = 0; i < 64; i += 2) {
    table_select(&t, i / 2, e[i]);
    ge_madd(&r, h, &t);
    ge_p1p1_to_p3(h, &r);
  }
}

/* Replace (f,g) with (g,f) if b == 1;
 * replace (f,g) with (f,g) if b == 0.
 *
 * Preconditions: b in {0,1}. */
static void fe_cswap(fe f, fe g, unsigned int b) {
  size_t i;
  b = 0-b;
  for (i = 0; i < 10; i++) {
    int32_t x = f[i] ^ g[i];
    x &= b;
    f[i] ^= x;
    g[i] ^= x;
  }
}

/* h = f * 121666
 * Can overlap h with f.
 *
 * Preconditions:
 *    |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
 *
 * Postconditions:
 *    |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. */
static void fe_mul121666(fe h, fe f) {
  int32_t f0 = f[0];
  int32_t f1 = f[1];
  int32_t f2 = f[2];
  int32_t f3 = f[3];
  int32_t f4 = f[4];
  int32_t f5 = f[5];
  int32_t f6 = f[6];
  int32_t f7 = f[7];
  int32_t f8 = f[8];
  int32_t f9 = f[9];
  int64_t h0 = f0 * (int64_t) 121666;
  int64_t h1 = f1 * (int64_t) 121666;
  int64_t h2 = f2 * (int64_t) 121666;
  int64_t h3 = f3 * (int64_t) 121666;
  int64_t h4 = f4 * (int64_t) 121666;
  int64_t h5 = f5 * (int64_t) 121666;
  int64_t h6 = f6 * (int64_t) 121666;
  int64_t h7 = f7 * (int64_t) 121666;
  int64_t h8 = f8 * (int64_t) 121666;
  int64_t h9 = f9 * (int64_t) 121666;
  int64_t carry0;
  int64_t carry1;
  int64_t carry2;
  int64_t carry3;
  int64_t carry4;
  int64_t carry5;
  int64_t carry6;
  int64_t carry7;
  int64_t carry8;
  int64_t carry9;

  carry9 = h9 + (1 << 24); h0 += (carry9 >> 25) * 19; h9 -= carry9 & kTop39Bits;
  carry1 = h1 + (1 << 24); h2 += carry1 >> 25; h1 -= carry1 & kTop39Bits;
  carry3 = h3 + (1 << 24); h4 += carry3 >> 25; h3 -= carry3 & kTop39Bits;
  carry5 = h5 + (1 << 24); h6 += carry5 >> 25; h5 -= carry5 & kTop39Bits;
  carry7 = h7 + (1 << 24); h8 += carry7 >> 25; h7 -= carry7 & kTop39Bits;

  carry0 = h0 + (1 << 25); h1 += carry0 >> 26; h0 -= carry0 & kTop38Bits;
  carry2 = h2 + (1 << 25); h3 += carry2 >> 26; h2 -= carry2 & kTop38Bits;
  carry4 = h4 + (1 << 25); h5 += carry4 >> 26; h4 -= carry4 & kTop38Bits;
  carry6 = h6 + (1 << 25); h7 += carry6 >> 26; h6 -= carry6 & kTop38Bits;
  carry8 = h8 + (1 << 25); h9 += carry8 >> 26; h8 -= carry8 & kTop38Bits;

  h[0] = h0;
  h[1] = h1;
  h[2] = h2;
  h[3] = h3;
  h[4] = h4;
  h[5] = h5;
  h[6] = h6;
  h[7] = h7;
  h[8] = h8;
  h[9] = h9;
}

static void x25519_scalar_mult_generic(uint8_t out[32],
                                       const uint8_t scalar[32],
                                       const uint8_t point[32]) {
  fe x1, x2, z2, x3, z3, tmp0, tmp1;
  uint8_t e[32];
  unsigned swap = 0;
  int pos;

  memcpy(e, scalar, 32);
  e[0] &= 248;
  e[31] &= 127;
  e[31] |= 64;
  fe_frombytes(x1, point);
  fe_1(x2);
  fe_0(z2);
  fe_copy(x3, x1);
  fe_1(z3);

  for (pos = 254; pos >= 0; --pos) {
    unsigned b = 1 & (e[pos / 8] >> (pos & 7));
    swap ^= b;
    fe_cswap(x2, x3, swap);
    fe_cswap(z2, z3, swap);
    swap = b;
    fe_sub(tmp0, x3, z3);
    fe_sub(tmp1, x2, z2);
    fe_add(x2, x2, z2);
    fe_add(z2, x3, z3);
    fe_mul(z3, tmp0, x2);
    fe_mul(z2, z2, tmp1);
    fe_sq(tmp0, tmp1);
    fe_sq(tmp1, x2);
    fe_add(x3, z3, z2);
    fe_sub(z2, z3, z2);
    fe_mul(x2, tmp1, tmp0);
    fe_sub(tmp1, tmp1, tmp0);
    fe_sq(z2, z2);
    fe_mul121666(z3, tmp1);
    fe_sq(x3, x3);
    fe_add(tmp0, tmp0, z3);
    fe_mul(z3, x1, z2);
    fe_mul(z2, tmp1, tmp0);
  }
  fe_cswap(x2, x3, swap);
  fe_cswap(z2, z3, swap);

  fe_invert(z2, z2);
  fe_mul(x2, x2, z2);
  fe_tobytes(out, x2);
}

static void x25519_scalar_mult(uint8_t out[32], const uint8_t scalar[32],
                               const uint8_t point[32]) {
  x25519_scalar_mult_generic(out, scalar, point);
}

int X25519(uint8_t out_shared_key[32], const uint8_t private_key[32],
           const uint8_t peer_public_value[32]) {
  static const uint8_t kZeros[32] = {0};
  x25519_scalar_mult(out_shared_key, private_key, peer_public_value);
  /* The all-zero output results when the input is a point of small order. */
  return CRYPTO_memcmp(kZeros, out_shared_key, 32) != 0;
}

void X25519_public_from_private(uint8_t out_public_value[32],
                                const uint8_t private_key[32]) {
  uint8_t e[32];
  ge_p3 A;
  fe zplusy, zminusy, zminusy_inv;

  memcpy(e, private_key, 32);
  e[0] &= 248;
  e[31] &= 127;
  e[31] |= 64;

  ge_scalarmult_base(&A, e);

  /* We only need the u-coordinate of the curve25519 point. The map is
   * u=(y+1)/(1-y). Since y=Y/Z, this gives u=(Z+Y)/(Z-Y). */
  fe_add(zplusy, A.Z, A.Y);
  fe_sub(zminusy, A.Z, A.Y);
  fe_invert(zminusy_inv, zminusy);
  fe_mul(zplusy, zplusy, zminusy_inv);
  fe_tobytes(out_public_value, zplusy);
}