trezor-firmware/crypto/bignum.c

/**
 * Copyright (c) 2013-2014 Tomas Dzetkulic
 * Copyright (c) 2013-2014 Pavol Rusnak
 * Copyright (c)      2015 Jochen Hoenicke
 * Copyright (c)      2016 Alex Beregszaszi
 *
 * Permission is hereby granted, free of charge, to any person obtaining
 * a copy of this software and associated documentation files (the "Software"),
 * to deal in the Software without restriction, including without limitation
 * the rights to use, copy, modify, merge, publish, distribute, sublicense,
 * and/or sell copies of the Software, and to permit persons to whom the
 * Software is furnished to do so, subject to the following conditions:
 *
 * The above copyright notice and this permission notice shall be included
 * in all copies or substantial portions of the Software.
 *
 * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
 * OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
 * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
 * THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES
 * OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
 * ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
 * OTHER DEALINGS IN THE SOFTWARE.
 */

#include "bignum.h"
#include <assert.h>
#include <stdio.h>
#include <string.h>
#include "memzero.h"

/* big number library */

/* The structure bignum256 is an array of nine 32-bit values, which
 * are digits in base 2^30 representation.  I.e. the number
 *   bignum256 a;
 * represents the value
 *   sum_{i=0}^8 a.val[i] * 2^{30 i}.
 *
 * The number is *normalized* iff every digit is < 2^30.
 *
 * As the name suggests, a bignum256 is intended to represent a 256
 * bit number, but it can represent 270 bits.  Numbers are usually
 * reduced using a prime, either the group order or the field prime.
 * The reduction is often partly done by bn_fast_mod, and similarly
 * implicitly in bn_multiply.  A *partly reduced number* is a
 * normalized number between 0 (inclusive) and 2*prime (exclusive).
 *
 * A partly reduced number can be fully reduced by calling bn_mod.
 * Only a fully reduced number is guaranteed to fit in 256 bit.
 *
 * All functions assume that the prime in question is slightly smaller
 * than 2^256.  In particular it must be between 2^256-2^224 and
 * 2^256 and it must be a prime number.
 */

inline uint32_t read_be(const uint8_t *data) {
  return (((uint32_t)data[0]) << 24) | (((uint32_t)data[1]) << 16) |
         (((uint32_t)data[2]) << 8) | (((uint32_t)data[3]));
}

inline void write_be(uint8_t *data, uint32_t x) {
  data[0] = x >> 24;
  data[1] = x >> 16;
  data[2] = x >> 8;
  data[3] = x;
}

inline uint32_t read_le(const uint8_t *data) {
  return (((uint32_t)data[3]) << 24) | (((uint32_t)data[2]) << 16) |
         (((uint32_t)data[1]) << 8) | (((uint32_t)data[0]));
}

inline void write_le(uint8_t *data, uint32_t x) {
  data[3] = x >> 24;
  data[2] = x >> 16;
  data[1] = x >> 8;
  data[0] = x;
}

// convert a raw bigendian 256 bit value into a normalized bignum.
// out_number is partly reduced (since it fits in 256 bit).
void bn_read_be(const uint8_t *in_number, bignum256 *out_number) {
  int i;
  uint32_t temp = 0;
  for (i = 0; i < 8; i++) {
    // invariant: temp = (in_number % 2^(32i)) >> 30i
    // get next limb = (in_number % 2^(32(i+1))) >> 32i
    uint32_t limb = read_be(in_number + (7 - i) * 4);
    // temp = (in_number % 2^(32(i+1))) << 30i
    temp |= limb << (2 * i);
    // store 30 bits into val[i]
    out_number->val[i] = temp & 0x3FFFFFFF;
    // prepare temp for next round
    temp = limb >> (30 - 2 * i);
  }
  out_number->val[8] = temp;
}

// convert a normalized bignum to a raw bigendian 256 bit number.
// in_number must be fully reduced.
void bn_write_be(const bignum256 *in_number, uint8_t *out_number) {
  int i;
  uint32_t temp = in_number->val[8];
  for (i = 0; i < 8; i++) {
    // invariant: temp = (in_number >> 30*(8-i))
    uint32_t limb = in_number->val[7 - i];
    temp = (temp << (16 + 2 * i)) | (limb >> (14 - 2 * i));
    write_be(out_number + i * 4, temp);
    temp = limb;
  }
}

// convert a raw little endian 256 bit value into a normalized bignum.
// out_number is partly reduced (since it fits in 256 bit).
void bn_read_le(const uint8_t *in_number, bignum256 *out_number) {
  int i;
  uint32_t temp = 0;
  for (i = 0; i < 8; i++) {
    // invariant: temp = (in_number % 2^(32i)) >> 30i
    // get next limb = (in_number % 2^(32(i+1))) >> 32i
    uint32_t limb = read_le(in_number + i * 4);
    // temp = (in_number % 2^(32(i+1))) << 30i
    temp |= limb << (2 * i);
    // store 30 bits into val[i]
    out_number->val[i] = temp & 0x3FFFFFFF;
    // prepare temp for next round
    temp = limb >> (30 - 2 * i);
  }
  out_number->val[8] = temp;
}

// convert a normalized bignum to a raw little endian 256 bit number.
// in_number must be fully reduced.
void bn_write_le(const bignum256 *in_number, uint8_t *out_number) {
  int i;
  uint32_t temp = in_number->val[8];
  for (i = 0; i < 8; i++) {
    // invariant: temp = (in_number >> 30*(8-i))
    uint32_t limb = in_number->val[7 - i];
    temp = (temp << (16 + 2 * i)) | (limb >> (14 - 2 * i));
    write_le(out_number + (7 - i) * 4, temp);
    temp = limb;
  }
}

void bn_read_uint32(uint32_t in_number, bignum256 *out_number) {
  out_number->val[0] = in_number & 0x3FFFFFFF;
  out_number->val[1] = in_number >> 30;
  out_number->val[2] = 0;
  out_number->val[3] = 0;
  out_number->val[4] = 0;
  out_number->val[5] = 0;
  out_number->val[6] = 0;
  out_number->val[7] = 0;
  out_number->val[8] = 0;
}

void bn_read_uint64(uint64_t in_number, bignum256 *out_number) {
  out_number->val[0] = in_number & 0x3FFFFFFF;
  out_number->val[1] = (in_number >>= 30) & 0x3FFFFFFF;
  out_number->val[2] = in_number >>= 30;
  out_number->val[3] = 0;
  out_number->val[4] = 0;
  out_number->val[5] = 0;
  out_number->val[6] = 0;
  out_number->val[7] = 0;
  out_number->val[8] = 0;
}

// a must be normalized
int bn_bitcount(const bignum256 *a) {
  int i;
  for (i = 8; i >= 0; i--) {
    int tmp = a->val[i];
    if (tmp != 0) {
      return i * 30 + (32 - __builtin_clz(tmp));
    }
  }
  return 0;
}

#define DIGITS 78  // log10(2 ^ 256)

unsigned int bn_digitcount(const bignum256 *a) {
  bignum256 val;
  memcpy(&val, a, sizeof(bignum256));

  unsigned int digits = 1;

  for (unsigned int i = 0; i < DIGITS; i += 3) {
    uint32_t limb;
    bn_divmod1000(&val, &limb);

    if (limb >= 100) {
      digits = i + 3;
    } else if (limb >= 10) {
      digits = i + 2;
    } else if (limb >= 1) {
      digits = i + 1;
    }
  }

  return digits;
}

// sets a bignum to zero.
void bn_zero(bignum256 *a) {
  int i;
  for (i = 0; i < 9; i++) {
    a->val[i] = 0;
  }
}

// sets a bignum to one.
void bn_one(bignum256 *a) {
  a->val[0] = 1;
  a->val[1] = 0;
  a->val[2] = 0;
  a->val[3] = 0;
  a->val[4] = 0;
  a->val[5] = 0;
  a->val[6] = 0;
  a->val[7] = 0;
  a->val[8] = 0;
}

// checks that a bignum is zero.
// a must be normalized
// function is constant time (on some architectures, in particular ARM).
int bn_is_zero(const bignum256 *a) {
  int i;
  uint32_t result = 0;
  for (i = 0; i < 9; i++) {
    result |= a->val[i];
  }
  return !result;
}

// Check whether a < b
// a and b must be normalized
// function is constant time (on some architectures, in particular ARM).
int bn_is_less(const bignum256 *a, const bignum256 *b) {
  int i;
  uint32_t res1 = 0;
  uint32_t res2 = 0;
  for (i = 8; i >= 0; i--) {
    res1 = (res1 << 1) | (a->val[i] < b->val[i]);
    res2 = (res2 << 1) | (a->val[i] > b->val[i]);
  }
  return res1 > res2;
}

// Check whether a == b
// a and b must be normalized
// function is constant time (on some architectures, in particular ARM).
int bn_is_equal(const bignum256 *a, const bignum256 *b) {
  int i;
  uint32_t result = 0;
  for (i = 0; i < 9; i++) {
    result |= (a->val[i] ^ b->val[i]);
  }
  return !result;
}

// Assigns res = cond ? truecase : falsecase
// assumes that cond is either 0 or 1.
// function is constant time.
void bn_cmov(bignum256 *res, int cond, const bignum256 *truecase,
             const bignum256 *falsecase) {
  int i;
  uint32_t tmask = (uint32_t)-cond;
  uint32_t fmask = ~tmask;

  assert(cond == 1 || cond == 0);
  for (i = 0; i < 9; i++) {
    res->val[i] = (truecase->val[i] & tmask) | (falsecase->val[i] & fmask);
  }
}

// shift number to the left, i.e multiply it by 2.
// a must be normalized.  The result is normalized but not reduced.
void bn_lshift(bignum256 *a) {
  int i;
  for (i = 8; i > 0; i--) {
    a->val[i] =
        ((a->val[i] << 1) & 0x3FFFFFFF) | ((a->val[i - 1] & 0x20000000) >> 29);
  }
  a->val[0] = (a->val[0] << 1) & 0x3FFFFFFF;
}

// shift number to the right, i.e divide by 2 while rounding down.
// a must be normalized.  The result is normalized.
void bn_rshift(bignum256 *a) {
  int i;
  for (i = 0; i < 8; i++) {
    a->val[i] = (a->val[i] >> 1) | ((a->val[i + 1] & 1) << 29);
  }
  a->val[8] >>= 1;
}

// sets bit in bignum
void bn_setbit(bignum256 *a, uint8_t bit) {
  a->val[bit / 30] |= (1u << (bit % 30));
}

// clears bit in bignum
void bn_clearbit(bignum256 *a, uint8_t bit) {
  a->val[bit / 30] &= ~(1u << (bit % 30));
}

// tests bit in bignum
uint32_t bn_testbit(bignum256 *a, uint8_t bit) {
  return a->val[bit / 30] & (1u << (bit % 30));
}

// a = b ^ c
void bn_xor(bignum256 *a, const bignum256 *b, const bignum256 *c) {
  int i;
  for (i = 0; i < 9; i++) {
    a->val[i] = b->val[i] ^ c->val[i];
  }
}

// multiply x by 1/2 modulo prime.
// it computes x = (x & 1) ? (x + prime) >> 1 : x >> 1.
// assumes x is normalized.
// if x was partly reduced, it is also partly reduced on exit.
// function is constant time.
void bn_mult_half(bignum256 *x, const bignum256 *prime) {
  int j;
  uint32_t xodd = -(x->val[0] & 1);
  // compute x = x/2 mod prime
  // if x is odd compute (x+prime)/2
  uint32_t tmp1 = (x->val[0] + (prime->val[0] & xodd)) >> 1;
  for (j = 0; j < 8; j++) {
    uint32_t tmp2 = (x->val[j + 1] + (prime->val[j + 1] & xodd));
    tmp1 += (tmp2 & 1) << 29;
    x->val[j] = tmp1 & 0x3fffffff;
    tmp1 >>= 30;
    tmp1 += tmp2 >> 1;
  }
  x->val[8] = tmp1;
}

// multiply x by k modulo prime.
// assumes x is normalized, 0 <= k <= 4.
// guarantees x is partly reduced.
void bn_mult_k(bignum256 *x, uint8_t k, const bignum256 *prime) {
  int j;
  for (j = 0; j < 9; j++) {
    x->val[j] = k * x->val[j];
  }
  bn_fast_mod(x, prime);
}

// compute x = x mod prime  by computing  x >= prime ? x - prime : x.
// assumes x partly reduced, guarantees x fully reduced.
void bn_mod(bignum256 *x, const bignum256 *prime) {
  const int flag = bn_is_less(x, prime);  // x < prime
  bignum256 temp;
  bn_subtract(x, prime, &temp);  // temp = x - prime
  bn_cmov(x, flag, x, &temp);
}

// auxiliary function for multiplication.
// compute k * x as a 540 bit number in base 2^30 (normalized).
// assumes that k and x are normalized.
void bn_multiply_long(const bignum256 *k, const bignum256 *x,
                      uint32_t res[18]) {
  int i, j;
  uint64_t temp = 0;

  // compute lower half of long multiplication
  for (i = 0; i < 9; i++) {
    for (j = 0; j <= i; j++) {
      // no overflow, since 9*2^60 < 2^64
      temp += k->val[j] * (uint64_t)x->val[i - j];
    }
    res[i] = temp & 0x3FFFFFFFu;
    temp >>= 30;
  }
  // compute upper half
  for (; i < 17; i++) {
    for (j = i - 8; j < 9; j++) {
      // no overflow, since 9*2^60 < 2^64
      temp += k->val[j] * (uint64_t)x->val[i - j];
    }
    res[i] = temp & 0x3FFFFFFFu;
    temp >>= 30;
  }
  res[17] = temp;
}

// auxiliary function for multiplication.
// reduces res modulo prime.
// assumes i >= 8 and i <= 16
// assumes    res normalized, res < 2^(30(i-7)) * 2 * prime
// guarantees res normalized, res < 2^(30(i-8)) * 2 * prime
void bn_multiply_reduce_step(uint32_t res[18], const bignum256 *prime,
                             uint32_t i) {
  // let k = i-8.
  // on entry:
  //   0 <= res < 2^(30k + 31) * prime
  // estimate coef = (res / prime / 2^30k)
  // by coef = res / 2^(30k + 256)  rounded down
  // 0 <= coef < 2^31
  // subtract (coef * 2^(30k) * prime) from res
  // note that we unrolled the first iteration
  assert(i >= 8 && i <= 16);
  uint32_t j;
  uint32_t coef = (res[i] >> 16) + (res[i + 1] << 14);
  uint64_t temp =
      0x2000000000000000ull + res[i - 8] - prime->val[0] * (uint64_t)coef;
  assert(coef < 0x80000000u);
  res[i - 8] = temp & 0x3FFFFFFF;
  for (j = 1; j < 9; j++) {
    temp >>= 30;
    // Note: coeff * prime->val[j] <= (2^31-1) * (2^30-1)
    // Hence, this addition will not underflow.
    temp +=
        0x1FFFFFFF80000000ull + res[i - 8 + j] - prime->val[j] * (uint64_t)coef;
    res[i - 8 + j] = temp & 0x3FFFFFFF;
    // 0 <= temp < 2^61 + 2^30
  }
  temp >>= 30;
  temp += 0x1FFFFFFF80000000ull + res[i - 8 + j];
  res[i - 8 + j] = temp & 0x3FFFFFFF;
  // we rely on the fact that prime > 2^256 - 2^224
  //   res = oldres - coef*2^(30k) * prime;
  // and
  //   coef * 2^(30k + 256) <= oldres < (coef+1) * 2^(30k + 256)
  // Hence, 0 <= res < 2^30k (2^256 + coef * (2^256 - prime))
  //                 < 2^30k (2^256 + 2^31 * 2^224)
  //                 < 2^30k (2 * prime)
}

// auxiliary function for multiplication.
// reduces x = res modulo prime.
// assumes    res normalized, res < 2^270 * 2 * prime
// guarantees x partly reduced, i.e., x < 2 * prime
void bn_multiply_reduce(bignum256 *x, uint32_t res[18],
                        const bignum256 *prime) {
  int i;
  // res = k * x is a normalized number (every limb < 2^30)
  // 0 <= res < 2^270 * 2 * prime.
  for (i = 16; i >= 8; i--) {
    bn_multiply_reduce_step(res, prime, i);
    assert(res[i + 1] == 0);
  }
  // store the result
  for (i = 0; i < 9; i++) {
    x->val[i] = res[i];
  }
}

// Compute x := k * x  (mod prime)
// both inputs must be smaller than 180 * prime.
// result is partly reduced (0 <= x < 2 * prime)
// This only works for primes between 2^256-2^224 and 2^256.
void bn_multiply(const bignum256 *k, bignum256 *x, const bignum256 *prime) {
  uint32_t res[18] = {0};
  bn_multiply_long(k, x, res);
  bn_multiply_reduce(x, res, prime);
  memzero(res, sizeof(res));
}

// partly reduce x modulo prime
// input x does not have to be normalized.
// x can be any number that fits.
// prime must be between (2^256 - 2^224) and 2^256
// result is partly reduced, smaller than 2*prime
void bn_fast_mod(bignum256 *x, const bignum256 *prime) {
  int j;
  uint32_t coef;
  uint64_t temp;

  coef = x->val[8] >> 16;
  // substract (coef * prime) from x
  // note that we unrolled the first iteration
  temp = 0x2000000000000000ull + x->val[0] - prime->val[0] * (uint64_t)coef;
  x->val[0] = temp & 0x3FFFFFFF;
  for (j = 1; j < 9; j++) {
    temp >>= 30;
    temp += 0x1FFFFFFF80000000ull + x->val[j] - prime->val[j] * (uint64_t)coef;
    x->val[j] = temp & 0x3FFFFFFF;
  }
}

// square root of x = x^((p+1)/4)
// http://en.wikipedia.org/wiki/Quadratic_residue#Prime_or_prime_power_modulus
// assumes    x is normalized but not necessarily reduced.
// guarantees x is reduced
void bn_sqrt(bignum256 *x, const bignum256 *prime) {
  // this method compute x^1/2 = x^(prime+1)/4
  uint32_t i, j, limb;
  bignum256 res, p;
  bn_one(&res);
  // compute p = (prime+1)/4
  memcpy(&p, prime, sizeof(bignum256));
  bn_addi(&p, 1);
  bn_rshift(&p);
  bn_rshift(&p);
  for (i = 0; i < 9; i++) {
    // invariants:
    //    x   = old(x)^(2^(i*30))
    //    res = old(x)^(p % 2^(i*30))
    // get the i-th limb of prime - 2
    limb = p.val[i];
    for (j = 0; j < 30; j++) {
      // invariants:
      //    x    = old(x)^(2^(i*30+j))
      //    res  = old(x)^(p % 2^(i*30+j))
      //    limb = (p % 2^(i*30+30)) / 2^(i*30+j)
      if (i == 8 && limb == 0) break;
      if (limb & 1) {
        bn_multiply(x, &res, prime);
      }
      limb >>= 1;
      bn_multiply(x, x, prime);
    }
  }
  bn_mod(&res, prime);
  memcpy(x, &res, sizeof(bignum256));
  memzero(&res, sizeof(res));
  memzero(&p, sizeof(p));
}

#if !USE_INVERSE_FAST

// in field G_prime, small but slow
void bn_inverse(bignum256 *x, const bignum256 *prime) {
  // this method compute x^-1 = x^(prime-2)
  uint32_t i, j, limb;
  bignum256 res;
  bn_one(&res);
  for (i = 0; i < 9; i++) {
    // invariants:
    //    x   = old(x)^(2^(i*30))
    //    res = old(x)^((prime-2) % 2^(i*30))
    // get the i-th limb of prime - 2
    limb = prime->val[i];
    // this is not enough in general but fine for secp256k1 & nist256p1 because
    // prime->val[0] > 1
    if (i == 0) limb -= 2;
    for (j = 0; j < 30; j++) {
      // invariants:
      //    x    = old(x)^(2^(i*30+j))
      //    res  = old(x)^((prime-2) % 2^(i*30+j))
      //    limb = ((prime-2) % 2^(i*30+30)) / 2^(i*30+j)
      // early abort when only zero bits follow
      if (i == 8 && limb == 0) break;
      if (limb & 1) {
        bn_multiply(x, &res, prime);
      }
      limb >>= 1;
      bn_multiply(x, x, prime);
    }
  }
  bn_mod(&res, prime);
  memcpy(x, &res, sizeof(bignum256));
}

#else

// in field G_prime, big and complicated but fast
// the input must not be 0 mod prime.
// the result is smaller than prime
void bn_inverse(bignum256 *x, const bignum256 *prime) {
  int i, j, k, cmp;
  struct combo {
    uint32_t a[9];
    int len1;
  } us, vr, *odd, *even;
  uint32_t pp[8];
  uint32_t temp32;
  uint64_t temp;

  // The algorithm is based on Schroeppel et. al. "Almost Modular Inverse"
  // algorithm.  We keep four values u,v,r,s in the combo registers
  // us and vr.  us stores u in the first len1 limbs (little endian)
  // and s in the last 9-len1 limbs (big endian).  vr stores v and r.
  // This is because both u*s and v*r are guaranteed to fit in 8 limbs, so
  // their components are guaranteed to fit in 9.  During the algorithm,
  // the length of u and v shrinks while r and s grow.
  // u,v,r,s correspond to F,G,B,C in Schroeppel's algorithm.

  // reduce x modulo prime.  This is necessary as it has to fit in 8 limbs.
  bn_fast_mod(x, prime);
  bn_mod(x, prime);
  // convert x and prime to 8x32 bit limb form
  temp32 = prime->val[0];
  for (i = 0; i < 8; i++) {
    temp32 |= prime->val[i + 1] << (30 - 2 * i);
    us.a[i] = pp[i] = temp32;
    temp32 = prime->val[i + 1] >> (2 + 2 * i);
  }
  temp32 = x->val[0];
  for (i = 0; i < 8; i++) {
    temp32 |= x->val[i + 1] << (30 - 2 * i);
    vr.a[i] = temp32;
    temp32 = x->val[i + 1] >> (2 + 2 * i);
  }
  us.len1 = 8;
  vr.len1 = 8;
  // set s = 1 and r = 0
  us.a[8] = 1;
  vr.a[8] = 0;
  // set k = 0.
  k = 0;

  // only one of the numbers u,v can be even at any time.  We
  // let even point to that number and odd to the other.
  // Initially the prime u is guaranteed to be odd.
  odd = &us;
  even = &vr;

  // u = prime, v = x
  // r = 0    , s = 1
  // k = 0
  for (;;) {
    // invariants:
    //   let u = limbs us.a[0..u.len1-1] in little endian,
    //   let s = limbs us.a[u.len..8] in big endian,
    //   let v = limbs vr.a[0..u.len1-1] in little endian,
    //   let r = limbs vr.a[u.len..8] in big endian,
    //   r,s >= 0 ; u,v >= 1
    //   x*-r = u*2^k mod prime
    //   x*s  = v*2^k mod prime
    //   u*s + v*r = prime
    //   floor(log2(u)) + floor(log2(v)) + k <= 510
    //   max(u,v) <= 2^k   (*) see comment at end of loop
    //   gcd(u,v) = 1
    //   {odd,even} = {&us, &vr}
    //   odd->a[0] and odd->a[8] are odd
    //   even->a[0] or even->a[8] is even
    //
    // first u/v are large and r/s small
    // later u/v are small and r/s large
    assert(odd->a[0] & 1);
    assert(odd->a[8] & 1);

    // adjust length of even.
    while (even->a[even->len1 - 1] == 0) {
      even->len1--;
      // if input was 0, return.
      // This simple check prevents crashing with stack underflow
      // or worse undesired behaviour for illegal input.
      if (even->len1 < 0) return;
    }

    // reduce even->a while it is even
    while (even->a[0] == 0) {
      // shift right first part of even by a limb
      // and shift left second part of even by a limb.
      for (i = 0; i < 8; i++) {
        even->a[i] = even->a[i + 1];
      }
      even->a[i] = 0;
      even->len1--;
      k += 32;
    }
    // count up to 32 zero bits of even->a.
    j = 0;
    while ((even->a[0] & (1u << j)) == 0) {
      j++;
    }
    if (j > 0) {
      // shift first part of even right by j bits.
      for (i = 0; i + 1 < even->len1; i++) {
        even->a[i] = (even->a[i] >> j) | (even->a[i + 1] << (32 - j));
      }
      even->a[i] = (even->a[i] >> j);
      if (even->a[i] == 0) {
        even->len1--;
      } else {
        i++;
      }

      // shift second part of even left by j bits.
      for (; i < 8; i++) {
        even->a[i] = (even->a[i] << j) | (even->a[i + 1] >> (32 - j));
      }
      even->a[i] = (even->a[i] << j);
      // add j bits to k.
      k += j;
    }
    // invariant is reestablished.
    // now both a[0] are odd.
    assert(odd->a[0] & 1);
    assert(odd->a[8] & 1);
    assert(even->a[0] & 1);
    assert((even->a[8] & 1) == 0);

    // cmp > 0 if us.a[0..len1-1] > vr.a[0..len1-1],
    // cmp = 0 if equal, < 0 if less.
    cmp = us.len1 - vr.len1;
    if (cmp == 0) {
      i = us.len1 - 1;
      while (i >= 0 && us.a[i] == vr.a[i]) i--;
      // both are equal to 1 and we are done.
      if (i == -1) break;
      cmp = us.a[i] > vr.a[i] ? 1 : -1;
    }
    if (cmp > 0) {
      even = &us;
      odd = &vr;
    } else {
      even = &vr;
      odd = &us;
    }

    // now even > odd.

    //  even->a[0..len1-1] = (even->a[0..len1-1] - odd->a[0..len1-1]);
    temp = 1;
    for (i = 0; i < odd->len1; i++) {
      temp += 0xFFFFFFFFull + even->a[i] - odd->a[i];
      even->a[i] = temp & 0xFFFFFFFF;
      temp >>= 32;
    }
    for (; i < even->len1; i++) {
      temp += 0xFFFFFFFFull + even->a[i];
      even->a[i] = temp & 0xFFFFFFFF;
      temp >>= 32;
    }
    //  odd->a[len1..8] = (odd->b[len1..8] + even->b[len1..8]);
    temp = 0;
    for (i = 8; i >= even->len1; i--) {
      temp += (uint64_t)odd->a[i] + even->a[i];
      odd->a[i] = temp & 0xFFFFFFFF;
      temp >>= 32;
    }
    for (; i >= odd->len1; i--) {
      temp += (uint64_t)odd->a[i];
      odd->a[i] = temp & 0xFFFFFFFF;
      temp >>= 32;
    }
    // note that
    //  if u > v:
    //   u'2^k = (u - v) 2^k = x(-r) - xs = x(-(r+s)) = x(-r') mod prime
    //   u's' + v'r' = (u-v)s + v(r+s) = us + vr
    //  if u < v:
    //   v'2^k = (v - u) 2^k = xs - x(-r) = x(s+r) = xs' mod prime
    //   u's' + v'r' = u(s+r) + (v-u)r = us + vr

    // even->a[0] is difference between two odd numbers, hence even.
    // odd->a[8] is sum of even and odd number, hence odd.
    assert(odd->a[0] & 1);
    assert(odd->a[8] & 1);
    assert((even->a[0] & 1) == 0);

    // The invariants are (almost) reestablished.
    // The invariant max(u,v) <= 2^k can be invalidated at this point,
    // because odd->a[len1..8] was changed.  We only have
    //
    //     odd->a[len1..8] <= 2^{k+1}
    //
    // Since even->a[0] is even, k will be incremented at the beginning
    // of the next loop while odd->a[len1..8] remains unchanged.
    // So after that, odd->a[len1..8] <= 2^k will hold again.
  }
  // In the last iteration we had u = v and gcd(u,v) = 1.
  // Hence, u=1, v=1, s+r = prime, k <= 510, 2^k > max(s,r) >= prime/2
  // This implies 0 <= s < prime and 255 <= k <= 510.
  //
  // The invariants also give us x*s = 2^k mod prime,
  // hence s = 2^k * x^-1 mod prime.
  // We need to compute s/2^k mod prime.

  // First we compute inverse = -prime^-1 mod 2^32, which we need later.
  // We use the Explicit Quadratic Modular inverse algorithm.
  //   http://arxiv.org/pdf/1209.6626.pdf
  // a^-1  = (2-a) * PROD_i (1 + (a - 1)^(2^i)) mod 2^32
  // the product will converge quickly, because (a-1)^(2^i) will be
  // zero mod 2^32 after at most five iterations.
  // We want to compute -prime^-1 so we start with (pp[0]-2).
  assert(pp[0] & 1);
  uint32_t amone = pp[0] - 1;
  uint32_t inverse = pp[0] - 2;
  while (amone) {
    amone *= amone;
    inverse *= (amone + 1);
  }

  while (k >= 32) {
    // compute s / 2^32 modulo prime.
    // Idea: compute factor, such that
    //   s + factor*prime mod 2^32 == 0
    // i.e. factor = s * -1/prime mod 2^32.
    // Then compute s + factor*prime and shift right by 32 bits.
    uint32_t factor = (inverse * us.a[8]) & 0xffffffff;
    temp = us.a[8] + (uint64_t)pp[0] * factor;
    assert((temp & 0xffffffff) == 0);
    temp >>= 32;
    for (i = 0; i < 7; i++) {
      temp += us.a[8 - (i + 1)] + (uint64_t)pp[i + 1] * factor;
      us.a[8 - i] = temp & 0xffffffff;
      temp >>= 32;
    }
    us.a[8 - i] = temp & 0xffffffff;
    k -= 32;
  }
  if (k > 0) {
    // compute s / 2^k  modulo prime.
    // Same idea: compute factor, such that
    //   s + factor*prime mod 2^k == 0
    // i.e. factor = s * -1/prime mod 2^k.
    // Then compute s + factor*prime and shift right by k bits.
    uint32_t mask = (1u << k) - 1;
    uint32_t factor = (inverse * us.a[8]) & mask;
    temp = (us.a[8] + (uint64_t)pp[0] * factor) >> k;
    assert(((us.a[8] + pp[0] * factor) & mask) == 0);
    for (i = 0; i < 7; i++) {
      temp += (us.a[8 - (i + 1)] + (uint64_t)pp[i + 1] * factor) << (32 - k);
      us.a[8 - i] = temp & 0xffffffff;
      temp >>= 32;
    }
    us.a[8 - i] = temp & 0xffffffff;
  }

  // convert s to bignum style
  temp32 = 0;
  for (i = 0; i < 8; i++) {
    x->val[i] = ((us.a[8 - i] << (2 * i)) & 0x3FFFFFFFu) | temp32;
    temp32 = us.a[8 - i] >> (30 - 2 * i);
  }
  x->val[i] = temp32;

  // let's wipe all temp buffers
  memzero(pp, sizeof(pp));
  memzero(&us, sizeof(us));
  memzero(&vr, sizeof(vr));
}
#endif

void bn_normalize(bignum256 *a) { bn_addi(a, 0); }

// add two numbers a = a + b
// assumes that a, b are normalized
// guarantees that a is normalized
void bn_add(bignum256 *a, const bignum256 *b) {
  int i;
  uint32_t tmp = 0;
  for (i = 0; i < 9; i++) {
    tmp += a->val[i] + b->val[i];
    a->val[i] = tmp & 0x3FFFFFFF;
    tmp >>= 30;
  }
}

void bn_addmod(bignum256 *a, const bignum256 *b, const bignum256 *prime) {
  int i;
  for (i = 0; i < 9; i++) {
    a->val[i] += b->val[i];
  }
  bn_fast_mod(a, prime);
}

void bn_addi(bignum256 *a, uint32_t b) {
  int i;
  uint32_t tmp = b;
  for (i = 0; i < 9; i++) {
    tmp += a->val[i];
    a->val[i] = tmp & 0x3FFFFFFF;
    tmp >>= 30;
  }
}

void bn_subi(bignum256 *a, uint32_t b, const bignum256 *prime) {
  assert(b <= prime->val[0]);
  // the possible underflow will be taken care of when adding the prime
  a->val[0] -= b;
  bn_add(a, prime);
}

// res = a - b mod prime.  More exactly res = a + (2*prime - b).
// b must be a partly reduced number
// result is normalized but not reduced.
void bn_subtractmod(const bignum256 *a, const bignum256 *b, bignum256 *res,
                    const bignum256 *prime) {
  int i;
  uint32_t temp = 1;
  for (i = 0; i < 9; i++) {
    temp += 0x3FFFFFFF + a->val[i] + 2u * prime->val[i] - b->val[i];
    res->val[i] = temp & 0x3FFFFFFF;
    temp >>= 30;
  }
}

// res = a - b ; a > b
void bn_subtract(const bignum256 *a, const bignum256 *b, bignum256 *res) {
  int i;
  uint32_t tmp = 1;
  for (i = 0; i < 9; i++) {
    tmp += 0x3FFFFFFF + a->val[i] - b->val[i];
    res->val[i] = tmp & 0x3FFFFFFF;
    tmp >>= 30;
  }
}

// a / 58 = a (+r)
void bn_divmod58(bignum256 *a, uint32_t *r) {
  int i;
  uint32_t rem, tmp;
  rem = a->val[8] % 58;
  a->val[8] /= 58;
  for (i = 7; i >= 0; i--) {
    // invariants:
    //   rem = old(a) >> 30(i+1) % 58
    //   a[i+1..8] = old(a[i+1..8])/58
    //   a[0..i]   = old(a[0..i])
    // 2^30 == 18512790*58 + 4
    tmp = rem * 4 + a->val[i];
    // set a[i] = (rem * 2^30 + a[i])/58
    //          = rem * 18512790 + (rem * 4 + a[i])/58
    a->val[i] = rem * 18512790 + (tmp / 58);
    // set rem = (rem * 2^30 + a[i]) mod 58
    //         = (rem * 4 + a[i]) mod 58
    rem = tmp % 58;
  }
  *r = rem;
}

// a / 1000 = a (+r)
void bn_divmod1000(bignum256 *a, uint32_t *r) {
  int i;
  uint32_t rem, tmp;
  rem = a->val[8] % 1000;
  a->val[8] /= 1000;
  for (i = 7; i >= 0; i--) {
    // invariants:
    //   rem = old(a) >> 30(i+1) % 1000
    //   a[i+1..8] = old(a[i+1..8])/1000
    //   a[0..i]   = old(a[0..i])
    // 2^30 == 1073741*1000 + 824
    tmp = rem * 824 + a->val[i];
    // set a[i] = (rem * 2^30 + a[i])/1000
    //          = rem * 1073741 + (rem * 824 + a[i])/1000
    a->val[i] = rem * 1073741 + (tmp / 1000);
    // set rem = (rem * 2^30 + a[i]) mod 1000
    //         = (rem * 824 + a[i]) mod 1000
    rem = tmp % 1000;
  }
  *r = rem;
}

size_t bn_format(const bignum256 *amnt, const char *prefix, const char *suffix,
                 unsigned int decimals, int exponent, bool trailing, char *out,
                 size_t outlen) {
  // sanity check, 2**256 ~ 10**77; we should never need decimals/exponent
  // bigger than that
  if (decimals > 80 || exponent < -20 || exponent > 80) {
    memzero(out, outlen);
    return 0;
  }

  size_t prefixlen = prefix ? strlen(prefix) : 0;
  size_t suffixlen = suffix ? strlen(suffix) : 0;

  /* add prefix to beginning of out buffer */
  if (prefixlen) {
    memcpy(out, prefix, prefixlen);
  }
  /* add suffix to end of out buffer */
  if (suffixlen) {
    memcpy(&out[outlen - suffixlen - 1], suffix, suffixlen);
  }
  /* nul terminate (even if suffix = NULL) */
  out[outlen - 1] = '\0';

  /* fill number between prefix and suffix (between start and end) */
  char *start = &out[prefixlen], *end = &out[outlen - suffixlen - 1];
  char *str = end;

#define BN_FORMAT_PUSH_CHECKED(c) \
  do {                            \
    if (str == start) return 0;   \
    *--str = (c);                 \
  } while (0)

#define BN_FORMAT_PUSH(n)                                       \
  do {                                                          \
    if (exponent < 0) {                                         \
      exponent++;                                               \
    } else {                                                    \
      if ((n) > 0 || trailing || str != end || decimals <= 1) { \
        BN_FORMAT_PUSH_CHECKED('0' + (n));                      \
      }                                                         \
      if (decimals > 0 && decimals-- == 1) {                    \
        BN_FORMAT_PUSH_CHECKED('.');                            \
      }                                                         \
    }                                                           \
  } while (0)

  bignum256 val;
  memcpy(&val, amnt, sizeof(bignum256));

  if (bn_is_zero(&val)) {
    exponent = 0;
  }

  for (; exponent > 0; exponent--) {
    BN_FORMAT_PUSH(0);
  }

  unsigned int digits = bn_digitcount(&val);
  for (unsigned int i = 0; i < digits / 3; i++) {
    uint32_t limb;
    bn_divmod1000(&val, &limb);

    BN_FORMAT_PUSH(limb % 10);
    limb /= 10;
    BN_FORMAT_PUSH(limb % 10);
    limb /= 10;
    BN_FORMAT_PUSH(limb % 10);
  }

  if (digits % 3 != 0) {
    uint32_t limb;
    bn_divmod1000(&val, &limb);

    switch (digits % 3) {
      case 2:
        BN_FORMAT_PUSH(limb % 10);
        limb /= 10;
        //-fallthrough

      case 1:
        BN_FORMAT_PUSH(limb % 10);
        break;
    }
  }

  while (decimals > 0 || str[0] == '\0' || str[0] == '.') {
    BN_FORMAT_PUSH(0);
  }

  /* finally move number to &out[prefixlen] to close the gap between
   * prefix and str.  len is length of number + suffix + traling 0
   */
  size_t len = &out[outlen] - str;
  memmove(&out[prefixlen], str, len);

  /* return length of number including prefix and suffix without trailing 0 */
  return prefixlen + len - 1;
}

#if USE_BN_PRINT
void bn_print(const bignum256 *a) {
  printf("%04x", a->val[8] & 0x0000FFFF);
  printf("%08x", (a->val[7] << 2) | ((a->val[6] & 0x30000000) >> 28));
  printf("%07x", a->val[6] & 0x0FFFFFFF);
  printf("%08x", (a->val[5] << 2) | ((a->val[4] & 0x30000000) >> 28));
  printf("%07x", a->val[4] & 0x0FFFFFFF);
  printf("%08x", (a->val[3] << 2) | ((a->val[2] & 0x30000000) >> 28));
  printf("%07x", a->val[2] & 0x0FFFFFFF);
  printf("%08x", (a->val[1] << 2) | ((a->val[0] & 0x30000000) >> 28));
  printf("%07x", a->val[0] & 0x0FFFFFFF);
}

void bn_print_raw(const bignum256 *a) {
  int i;
  for (i = 0; i <= 8; i++) {
    printf("0x%08x, ", a->val[i]);
  }
}
#endif