root/crypto/p224.cc

DEFINITIONS

1. IsZero
2. Add
3. Subtract
4. ReduceLarge
5. Mul
6. Square
7. Reduce
8. Invert
9. Contract
10. AddJacobian
11. DoubleJacobian
12. CopyConditional
13. ScalarMult
14. Get224Bits
15. Put224Bits
16. SetFromString
17. ToString
18. ScalarMult
19. ScalarBaseMult
20. Add
21. Negate

// Copyright (c) 2012 The Chromium Authors. All rights reserved.
// Use of this source code is governed by a BSD-style license that can be
// found in the LICENSE file.

// This is an implementation of the P224 elliptic curve group. It's written to
// be short and simple rather than fast, although it's still constant-time.
//
// See http://www.imperialviolet.org/2010/12/04/ecc.html ([1]) for background.

#include "crypto/p224.h"

#include <string.h>

#include "base/sys_byteorder.h"

namespace {

using base::HostToNet32;
using base::NetToHost32;

// Field element functions.
//
// The field that we're dealing with is ℤ/pℤ where p = 2**224 - 2**96 + 1.
//
// Field elements are represented by a FieldElement, which is a typedef to an
// array of 8 uint32's. The value of a FieldElement, a, is:
//   a[0] + 2**28·a[1] + 2**56·a[1] + ... + 2**196·a[7]
//
// Using 28-bit limbs means that there's only 4 bits of headroom, which is less
// than we would really like. But it has the useful feature that we hit 2**224
// exactly, making the reflections during a reduce much nicer.

using crypto::p224::FieldElement;

// kP is the P224 prime.
const FieldElement kP = {
  1, 0, 0, 268431360,
  268435455, 268435455, 268435455, 268435455,
};

void Contract(FieldElement* inout);

// IsZero returns 0xffffffff if a == 0 mod p and 0 otherwise.
uint32 IsZero(const FieldElement& a) {
  FieldElement minimal;
  memcpy(&minimal, &a, sizeof(minimal));
  Contract(&minimal);

  uint32 is_zero = 0, is_p = 0;
  for (unsigned i = 0; i < 8; i++) {
    is_zero |= minimal[i];
    is_p |= minimal[i] - kP[i];
  }

  // If either is_zero or is_p is 0, then we should return 1.
  is_zero |= is_zero >> 16;
  is_zero |= is_zero >> 8;
  is_zero |= is_zero >> 4;
  is_zero |= is_zero >> 2;
  is_zero |= is_zero >> 1;

  is_p |= is_p >> 16;
  is_p |= is_p >> 8;
  is_p |= is_p >> 4;
  is_p |= is_p >> 2;
  is_p |= is_p >> 1;

  // For is_zero and is_p, the LSB is 0 iff all the bits are zero.
  is_zero &= is_p & 1;
  is_zero = (~is_zero) << 31;
  is_zero = static_cast<int32>(is_zero) >> 31;
  return is_zero;
}

// Add computes *out = a+b
//
// a[i] + b[i] < 2**32
void Add(FieldElement* out, const FieldElement& a, const FieldElement& b) {
  for (int i = 0; i < 8; i++) {
    (*out)[i] = a[i] + b[i];
  }
}

static const uint32 kTwo31p3 = (1u<<31) + (1u<<3);
static const uint32 kTwo31m3 = (1u<<31) - (1u<<3);
static const uint32 kTwo31m15m3 = (1u<<31) - (1u<<15) - (1u<<3);
// kZero31ModP is 0 mod p where bit 31 is set in all limbs so that we can
// subtract smaller amounts without underflow. See the section "Subtraction" in
// [1] for why.
static const FieldElement kZero31ModP = {
  kTwo31p3, kTwo31m3, kTwo31m3, kTwo31m15m3,
  kTwo31m3, kTwo31m3, kTwo31m3, kTwo31m3
};

// Subtract computes *out = a-b
//
// a[i], b[i] < 2**30
// out[i] < 2**32
void Subtract(FieldElement* out, const FieldElement& a, const FieldElement& b) {
  for (int i = 0; i < 8; i++) {
    // See the section on "Subtraction" in [1] for details.
    (*out)[i] = a[i] + kZero31ModP[i] - b[i];
  }
}

static const uint64 kTwo63p35 = (1ull<<63) + (1ull<<35);
static const uint64 kTwo63m35 = (1ull<<63) - (1ull<<35);
static const uint64 kTwo63m35m19 = (1ull<<63) - (1ull<<35) - (1ull<<19);
// kZero63ModP is 0 mod p where bit 63 is set in all limbs. See the section
// "Subtraction" in [1] for why.
static const uint64 kZero63ModP[8] = {
  kTwo63p35, kTwo63m35, kTwo63m35, kTwo63m35,
  kTwo63m35m19, kTwo63m35, kTwo63m35, kTwo63m35,
};

static const uint32 kBottom28Bits = 0xfffffff;

// LargeFieldElement also represents an element of the field. The limbs are
// still spaced 28-bits apart and in little-endian order. So the limbs are at
// 0, 28, 56, ..., 392 bits, each 64-bits wide.
typedef uint64 LargeFieldElement[15];

// ReduceLarge converts a LargeFieldElement to a FieldElement.
//
// in[i] < 2**62
void ReduceLarge(FieldElement* out, LargeFieldElement* inptr) {
  LargeFieldElement& in(*inptr);

  for (int i = 0; i < 8; i++) {
    in[i] += kZero63ModP[i];
  }

  // Eliminate the coefficients at 2**224 and greater while maintaining the
  // same value mod p.
  for (int i = 14; i >= 8; i--) {
    in[i-8] -= in[i];  // reflection off the "+1" term of p.
    in[i-5] += (in[i] & 0xffff) << 12;  // part of the "-2**96" reflection.
    in[i-4] += in[i] >> 16;  // the rest of the "-2**96" reflection.
  }
  in[8] = 0;
  // in[0..8] < 2**64

  // As the values become small enough, we start to store them in |out| and use
  // 32-bit operations.
  for (int i = 1; i < 8; i++) {
    in[i+1] += in[i] >> 28;
    (*out)[i] = static_cast<uint32>(in[i] & kBottom28Bits);
  }
  // Eliminate the term at 2*224 that we introduced while keeping the same
  // value mod p.
  in[0] -= in[8];  // reflection off the "+1" term of p.
  (*out)[3] += static_cast<uint32>(in[8] & 0xffff) << 12; // "-2**96" term
  (*out)[4] += static_cast<uint32>(in[8] >> 16);  // rest of "-2**96" term
  // in[0] < 2**64
  // out[3] < 2**29
  // out[4] < 2**29
  // out[1,2,5..7] < 2**28

  (*out)[0] = static_cast<uint32>(in[0] & kBottom28Bits);
  (*out)[1] += static_cast<uint32>((in[0] >> 28) & kBottom28Bits);
  (*out)[2] += static_cast<uint32>(in[0] >> 56);
  // out[0] < 2**28
  // out[1..4] < 2**29
  // out[5..7] < 2**28
}

// Mul computes *out = a*b
//
// a[i] < 2**29, b[i] < 2**30 (or vice versa)
// out[i] < 2**29
void Mul(FieldElement* out, const FieldElement& a, const FieldElement& b) {
  LargeFieldElement tmp;
  memset(&tmp, 0, sizeof(tmp));

  for (int i = 0; i < 8; i++) {
    for (int j = 0; j < 8; j++) {
      tmp[i+j] += static_cast<uint64>(a[i]) * static_cast<uint64>(b[j]);
    }
  }

  ReduceLarge(out, &tmp);
}

// Square computes *out = a*a
//
// a[i] < 2**29
// out[i] < 2**29
void Square(FieldElement* out, const FieldElement& a) {
  LargeFieldElement tmp;
  memset(&tmp, 0, sizeof(tmp));

  for (int i = 0; i < 8; i++) {
    for (int j = 0; j <= i; j++) {
      uint64 r = static_cast<uint64>(a[i]) * static_cast<uint64>(a[j]);
      if (i == j) {
        tmp[i+j] += r;
      } else {
        tmp[i+j] += r << 1;
      }
    }
  }

  ReduceLarge(out, &tmp);
}

// Reduce reduces the coefficients of in_out to smaller bounds.
//
// On entry: a[i] < 2**31 + 2**30
// On exit: a[i] < 2**29
void Reduce(FieldElement* in_out) {
  FieldElement& a = *in_out;

  for (int i = 0; i < 7; i++) {
    a[i+1] += a[i] >> 28;
    a[i] &= kBottom28Bits;
  }
  uint32 top = a[7] >> 28;
  a[7] &= kBottom28Bits;

  // top < 2**4
  // Constant-time: mask = (top != 0) ? 0xffffffff : 0
  uint32 mask = top;
  mask |= mask >> 2;
  mask |= mask >> 1;
  mask <<= 31;
  mask = static_cast<uint32>(static_cast<int32>(mask) >> 31);

  // Eliminate top while maintaining the same value mod p.
  a[0] -= top;
  a[3] += top << 12;

  // We may have just made a[0] negative but, if we did, then we must
  // have added something to a[3], thus it's > 2**12. Therefore we can
  // carry down to a[0].
  a[3] -= 1 & mask;
  a[2] += mask & ((1<<28) - 1);
  a[1] += mask & ((1<<28) - 1);
  a[0] += mask & (1<<28);
}

// Invert calcuates *out = in**-1 by computing in**(2**224 - 2**96 - 1), i.e.
// Fermat's little theorem.
void Invert(FieldElement* out, const FieldElement& in) {
  FieldElement f1, f2, f3, f4;

  Square(&f1, in);                        // 2
  Mul(&f1, f1, in);                       // 2**2 - 1
  Square(&f1, f1);                        // 2**3 - 2
  Mul(&f1, f1, in);                       // 2**3 - 1
  Square(&f2, f1);                        // 2**4 - 2
  Square(&f2, f2);                        // 2**5 - 4
  Square(&f2, f2);                        // 2**6 - 8
  Mul(&f1, f1, f2);                       // 2**6 - 1
  Square(&f2, f1);                        // 2**7 - 2
  for (int i = 0; i < 5; i++) {           // 2**12 - 2**6
    Square(&f2, f2);
  }
  Mul(&f2, f2, f1);                       // 2**12 - 1
  Square(&f3, f2);                        // 2**13 - 2
  for (int i = 0; i < 11; i++) {          // 2**24 - 2**12
    Square(&f3, f3);
  }
  Mul(&f2, f3, f2);                       // 2**24 - 1
  Square(&f3, f2);                        // 2**25 - 2
  for (int i = 0; i < 23; i++) {          // 2**48 - 2**24
    Square(&f3, f3);
  }
  Mul(&f3, f3, f2);                       // 2**48 - 1
  Square(&f4, f3);                        // 2**49 - 2
  for (int i = 0; i < 47; i++) {          // 2**96 - 2**48
    Square(&f4, f4);
  }
  Mul(&f3, f3, f4);                       // 2**96 - 1
  Square(&f4, f3);                        // 2**97 - 2
  for (int i = 0; i < 23; i++) {          // 2**120 - 2**24
    Square(&f4, f4);
  }
  Mul(&f2, f4, f2);                       // 2**120 - 1
  for (int i = 0; i < 6; i++) {           // 2**126 - 2**6
    Square(&f2, f2);
  }
  Mul(&f1, f1, f2);                       // 2**126 - 1
  Square(&f1, f1);                        // 2**127 - 2
  Mul(&f1, f1, in);                       // 2**127 - 1
  for (int i = 0; i < 97; i++) {          // 2**224 - 2**97
    Square(&f1, f1);
  }
  Mul(out, f1, f3);                       // 2**224 - 2**96 - 1
}

// Contract converts a FieldElement to its minimal, distinguished form.
//
// On entry, in[i] < 2**29
// On exit, in[i] < 2**28
void Contract(FieldElement* inout) {
  FieldElement& out = *inout;

  // Reduce the coefficients to < 2**28.
  for (int i = 0; i < 7; i++) {
    out[i+1] += out[i] >> 28;
    out[i] &= kBottom28Bits;
  }
  uint32 top = out[7] >> 28;
  out[7] &= kBottom28Bits;

  // Eliminate top while maintaining the same value mod p.
  out[0] -= top;
  out[3] += top << 12;

  // We may just have made out[0] negative. So we carry down. If we made
  // out[0] negative then we know that out[3] is sufficiently positive
  // because we just added to it.
  for (int i = 0; i < 3; i++) {
    uint32 mask = static_cast<uint32>(static_cast<int32>(out[i]) >> 31);
    out[i] += (1 << 28) & mask;
    out[i+1] -= 1 & mask;
  }

  // We might have pushed out[3] over 2**28 so we perform another, partial
  // carry chain.
  for (int i = 3; i < 7; i++) {
    out[i+1] += out[i] >> 28;
    out[i] &= kBottom28Bits;
  }
  top = out[7] >> 28;
  out[7] &= kBottom28Bits;

  // Eliminate top while maintaining the same value mod p.
  out[0] -= top;
  out[3] += top << 12;

  // There are two cases to consider for out[3]:
  //   1) The first time that we eliminated top, we didn't push out[3] over
  //      2**28. In this case, the partial carry chain didn't change any values
  //      and top is zero.
  //   2) We did push out[3] over 2**28 the first time that we eliminated top.
  //      The first value of top was in [0..16), therefore, prior to eliminating
  //      the first top, 0xfff1000 <= out[3] <= 0xfffffff. Therefore, after
  //      overflowing and being reduced by the second carry chain, out[3] <=
  //      0xf000. Thus it cannot have overflowed when we eliminated top for the
  //      second time.

  // Again, we may just have made out[0] negative, so do the same carry down.
  // As before, if we made out[0] negative then we know that out[3] is
  // sufficiently positive.
  for (int i = 0; i < 3; i++) {
    uint32 mask = static_cast<uint32>(static_cast<int32>(out[i]) >> 31);
    out[i] += (1 << 28) & mask;
    out[i+1] -= 1 & mask;
  }

  // The value is < 2**224, but maybe greater than p. In order to reduce to a
  // unique, minimal value we see if the value is >= p and, if so, subtract p.

  // First we build a mask from the top four limbs, which must all be
  // equal to bottom28Bits if the whole value is >= p. If top_4_all_ones
  // ends up with any zero bits in the bottom 28 bits, then this wasn't
  // true.
  uint32 top_4_all_ones = 0xffffffffu;
  for (int i = 4; i < 8; i++) {
    top_4_all_ones &= out[i];
  }
  top_4_all_ones |= 0xf0000000;
  // Now we replicate any zero bits to all the bits in top_4_all_ones.
  top_4_all_ones &= top_4_all_ones >> 16;
  top_4_all_ones &= top_4_all_ones >> 8;
  top_4_all_ones &= top_4_all_ones >> 4;
  top_4_all_ones &= top_4_all_ones >> 2;
  top_4_all_ones &= top_4_all_ones >> 1;
  top_4_all_ones =
      static_cast<uint32>(static_cast<int32>(top_4_all_ones << 31) >> 31);

  // Now we test whether the bottom three limbs are non-zero.
  uint32 bottom_3_non_zero = out[0] | out[1] | out[2];
  bottom_3_non_zero |= bottom_3_non_zero >> 16;
  bottom_3_non_zero |= bottom_3_non_zero >> 8;
  bottom_3_non_zero |= bottom_3_non_zero >> 4;
  bottom_3_non_zero |= bottom_3_non_zero >> 2;
  bottom_3_non_zero |= bottom_3_non_zero >> 1;
  bottom_3_non_zero =
      static_cast<uint32>(static_cast<int32>(bottom_3_non_zero) >> 31);

  // Everything depends on the value of out[3].
  //    If it's > 0xffff000 and top_4_all_ones != 0 then the whole value is >= p
  //    If it's = 0xffff000 and top_4_all_ones != 0 and bottom_3_non_zero != 0,
  //      then the whole value is >= p
  //    If it's < 0xffff000, then the whole value is < p
  uint32 n = out[3] - 0xffff000;
  uint32 out_3_equal = n;
  out_3_equal |= out_3_equal >> 16;
  out_3_equal |= out_3_equal >> 8;
  out_3_equal |= out_3_equal >> 4;
  out_3_equal |= out_3_equal >> 2;
  out_3_equal |= out_3_equal >> 1;
  out_3_equal =
      ~static_cast<uint32>(static_cast<int32>(out_3_equal << 31) >> 31);

  // If out[3] > 0xffff000 then n's MSB will be zero.
  uint32 out_3_gt = ~static_cast<uint32>(static_cast<int32>(n << 31) >> 31);

  uint32 mask = top_4_all_ones & ((out_3_equal & bottom_3_non_zero) | out_3_gt);
  out[0] -= 1 & mask;
  out[3] -= 0xffff000 & mask;
  out[4] -= 0xfffffff & mask;
  out[5] -= 0xfffffff & mask;
  out[6] -= 0xfffffff & mask;
  out[7] -= 0xfffffff & mask;
}


// Group element functions.
//
// These functions deal with group elements. The group is an elliptic curve
// group with a = -3 defined in FIPS 186-3, section D.2.2.

using crypto::p224::Point;

// kB is parameter of the elliptic curve.
const FieldElement kB = {
  55967668, 11768882, 265861671, 185302395,
  39211076, 180311059, 84673715, 188764328,
};

void CopyConditional(Point* out, const Point& a, uint32 mask);
void DoubleJacobian(Point* out, const Point& a);

// AddJacobian computes *out = a+b where a != b.
void AddJacobian(Point *out,
                 const Point& a,
                 const Point& b) {
  // See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl
  FieldElement z1z1, z2z2, u1, u2, s1, s2, h, i, j, r, v;

  uint32 z1_is_zero = IsZero(a.z);
  uint32 z2_is_zero = IsZero(b.z);

  // Z1Z1 = Z1²
  Square(&z1z1, a.z);

  // Z2Z2 = Z2²
  Square(&z2z2, b.z);

  // U1 = X1*Z2Z2
  Mul(&u1, a.x, z2z2);

  // U2 = X2*Z1Z1
  Mul(&u2, b.x, z1z1);

  // S1 = Y1*Z2*Z2Z2
  Mul(&s1, b.z, z2z2);
  Mul(&s1, a.y, s1);

  // S2 = Y2*Z1*Z1Z1
  Mul(&s2, a.z, z1z1);
  Mul(&s2, b.y, s2);

  // H = U2-U1
  Subtract(&h, u2, u1);
  Reduce(&h);
  uint32 x_equal = IsZero(h);

  // I = (2*H)²
  for (int j = 0; j < 8; j++) {
    i[j] = h[j] << 1;
  }
  Reduce(&i);
  Square(&i, i);

  // J = H*I
  Mul(&j, h, i);
  // r = 2*(S2-S1)
  Subtract(&r, s2, s1);
  Reduce(&r);
  uint32 y_equal = IsZero(r);

  if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) {
    // The two input points are the same therefore we must use the dedicated
    // doubling function as the slope of the line is undefined.
    DoubleJacobian(out, a);
    return;
  }

  for (int i = 0; i < 8; i++) {
    r[i] <<= 1;
  }
  Reduce(&r);

  // V = U1*I
  Mul(&v, u1, i);

  // Z3 = ((Z1+Z2)²-Z1Z1-Z2Z2)*H
  Add(&z1z1, z1z1, z2z2);
  Add(&z2z2, a.z, b.z);
  Reduce(&z2z2);
  Square(&z2z2, z2z2);
  Subtract(&out->z, z2z2, z1z1);
  Reduce(&out->z);
  Mul(&out->z, out->z, h);

  // X3 = r²-J-2*V
  for (int i = 0; i < 8; i++) {
    z1z1[i] = v[i] << 1;
  }
  Add(&z1z1, j, z1z1);
  Reduce(&z1z1);
  Square(&out->x, r);
  Subtract(&out->x, out->x, z1z1);
  Reduce(&out->x);

  // Y3 = r*(V-X3)-2*S1*J
  for (int i = 0; i < 8; i++) {
    s1[i] <<= 1;
  }
  Mul(&s1, s1, j);
  Subtract(&z1z1, v, out->x);
  Reduce(&z1z1);
  Mul(&z1z1, z1z1, r);
  Subtract(&out->y, z1z1, s1);
  Reduce(&out->y);

  CopyConditional(out, a, z2_is_zero);
  CopyConditional(out, b, z1_is_zero);
}

// DoubleJacobian computes *out = a+a.
void DoubleJacobian(Point* out, const Point& a) {
  // See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
  FieldElement delta, gamma, beta, alpha, t;

  Square(&delta, a.z);
  Square(&gamma, a.y);
  Mul(&beta, a.x, gamma);

  // alpha = 3*(X1-delta)*(X1+delta)
  Add(&t, a.x, delta);
  for (int i = 0; i < 8; i++) {
          t[i] += t[i] << 1;
  }
  Reduce(&t);
  Subtract(&alpha, a.x, delta);
  Reduce(&alpha);
  Mul(&alpha, alpha, t);

  // Z3 = (Y1+Z1)²-gamma-delta
  Add(&out->z, a.y, a.z);
  Reduce(&out->z);
  Square(&out->z, out->z);
  Subtract(&out->z, out->z, gamma);
  Reduce(&out->z);
  Subtract(&out->z, out->z, delta);
  Reduce(&out->z);

  // X3 = alpha²-8*beta
  for (int i = 0; i < 8; i++) {
          delta[i] = beta[i] << 3;
  }
  Reduce(&delta);
  Square(&out->x, alpha);
  Subtract(&out->x, out->x, delta);
  Reduce(&out->x);

  // Y3 = alpha*(4*beta-X3)-8*gamma²
  for (int i = 0; i < 8; i++) {
          beta[i] <<= 2;
  }
  Reduce(&beta);
  Subtract(&beta, beta, out->x);
  Reduce(&beta);
  Square(&gamma, gamma);
  for (int i = 0; i < 8; i++) {
          gamma[i] <<= 3;
  }
  Reduce(&gamma);
  Mul(&out->y, alpha, beta);
  Subtract(&out->y, out->y, gamma);
  Reduce(&out->y);
}

// CopyConditional sets *out=a if mask is 0xffffffff. mask must be either 0 of
// 0xffffffff.
void CopyConditional(Point* out,
                     const Point& a,
                     uint32 mask) {
  for (int i = 0; i < 8; i++) {
    out->x[i] ^= mask & (a.x[i] ^ out->x[i]);
    out->y[i] ^= mask & (a.y[i] ^ out->y[i]);
    out->z[i] ^= mask & (a.z[i] ^ out->z[i]);
  }
}

// ScalarMult calculates *out = a*scalar where scalar is a big-endian number of
// length scalar_len and != 0.
void ScalarMult(Point* out, const Point& a,
                const uint8* scalar, size_t scalar_len) {
  memset(out, 0, sizeof(*out));
  Point tmp;

  for (size_t i = 0; i < scalar_len; i++) {
    for (unsigned int bit_num = 0; bit_num < 8; bit_num++) {
      DoubleJacobian(out, *out);
      uint32 bit = static_cast<uint32>(static_cast<int32>(
          (((scalar[i] >> (7 - bit_num)) & 1) << 31) >> 31));
      AddJacobian(&tmp, a, *out);
      CopyConditional(out, tmp, bit);
    }
  }
}

// Get224Bits reads 7 words from in and scatters their contents in
// little-endian form into 8 words at out, 28 bits per output word.
void Get224Bits(uint32* out, const uint32* in) {
  out[0] = NetToHost32(in[6]) & kBottom28Bits;
  out[1] = ((NetToHost32(in[5]) << 4) |
            (NetToHost32(in[6]) >> 28)) & kBottom28Bits;
  out[2] = ((NetToHost32(in[4]) << 8) |
            (NetToHost32(in[5]) >> 24)) & kBottom28Bits;
  out[3] = ((NetToHost32(in[3]) << 12) |
            (NetToHost32(in[4]) >> 20)) & kBottom28Bits;
  out[4] = ((NetToHost32(in[2]) << 16) |
            (NetToHost32(in[3]) >> 16)) & kBottom28Bits;
  out[5] = ((NetToHost32(in[1]) << 20) |
            (NetToHost32(in[2]) >> 12)) & kBottom28Bits;
  out[6] = ((NetToHost32(in[0]) << 24) |
            (NetToHost32(in[1]) >> 8)) & kBottom28Bits;
  out[7] = (NetToHost32(in[0]) >> 4) & kBottom28Bits;
}

// Put224Bits performs the inverse operation to Get224Bits: taking 28 bits from
// each of 8 input words and writing them in big-endian order to 7 words at
// out.
void Put224Bits(uint32* out, const uint32* in) {
  out[6] = HostToNet32((in[0] >> 0) | (in[1] << 28));
  out[5] = HostToNet32((in[1] >> 4) | (in[2] << 24));
  out[4] = HostToNet32((in[2] >> 8) | (in[3] << 20));
  out[3] = HostToNet32((in[3] >> 12) | (in[4] << 16));
  out[2] = HostToNet32((in[4] >> 16) | (in[5] << 12));
  out[1] = HostToNet32((in[5] >> 20) | (in[6] << 8));
  out[0] = HostToNet32((in[6] >> 24) | (in[7] << 4));
}

} // anonymous namespace

namespace crypto {

namespace p224 {

bool Point::SetFromString(const base::StringPiece& in) {
  if (in.size() != 2*28)
    return false;
  const uint32* inwords = reinterpret_cast<const uint32*>(in.data());
  Get224Bits(x, inwords);
  Get224Bits(y, inwords + 7);
  memset(&z, 0, sizeof(z));
  z[0] = 1;

  // Check that the point is on the curve, i.e. that y² = x³ - 3x + b.
  FieldElement lhs;
  Square(&lhs, y);
  Contract(&lhs);

  FieldElement rhs;
  Square(&rhs, x);
  Mul(&rhs, x, rhs);

  FieldElement three_x;
  for (int i = 0; i < 8; i++) {
    three_x[i] = x[i] * 3;
  }
  Reduce(&three_x);
  Subtract(&rhs, rhs, three_x);
  Reduce(&rhs);

  ::Add(&rhs, rhs, kB);
  Contract(&rhs);
  return memcmp(&lhs, &rhs, sizeof(lhs)) == 0;
}

std::string Point::ToString() const {
  FieldElement zinv, zinv_sq, x, y;

  // If this is the point at infinity we return a string of all zeros.
  if (IsZero(this->z)) {
    static const char zeros[56] = {0};
    return std::string(zeros, sizeof(zeros));
  }

  Invert(&zinv, this->z);
  Square(&zinv_sq, zinv);
  Mul(&x, this->x, zinv_sq);
  Mul(&zinv_sq, zinv_sq, zinv);
  Mul(&y, this->y, zinv_sq);

  Contract(&x);
  Contract(&y);

  uint32 outwords[14];
  Put224Bits(outwords, x);
  Put224Bits(outwords + 7, y);
  return std::string(reinterpret_cast<const char*>(outwords), sizeof(outwords));
}

void ScalarMult(const Point& in, const uint8* scalar, Point* out) {
  ::ScalarMult(out, in, scalar, 28);
}

// kBasePoint is the base point (generator) of the elliptic curve group.
static const Point kBasePoint = {
  {22813985, 52956513, 34677300, 203240812,
   12143107, 133374265, 225162431, 191946955},
  {83918388, 223877528, 122119236, 123340192,
   266784067, 263504429, 146143011, 198407736},
  {1, 0, 0, 0, 0, 0, 0, 0},
};

void ScalarBaseMult(const uint8* scalar, Point* out) {
  ::ScalarMult(out, kBasePoint, scalar, 28);
}

void Add(const Point& a, const Point& b, Point* out) {
  AddJacobian(out, a, b);
}

void Negate(const Point& in, Point* out) {
  // Guide to elliptic curve cryptography, page 89 suggests that (X : X+Y : Z)
  // is the negative in Jacobian coordinates, but it doesn't actually appear to
  // be true in testing so this performs the negation in affine coordinates.
  FieldElement zinv, zinv_sq, y;
  Invert(&zinv, in.z);
  Square(&zinv_sq, zinv);
  Mul(&out->x, in.x, zinv_sq);
  Mul(&zinv_sq, zinv_sq, zinv);
  Mul(&y, in.y, zinv_sq);

  Subtract(&out->y, kP, y);
  Reduce(&out->y);

  memset(&out->z, 0, sizeof(out->z));
  out->z[0] = 1;
}

}  // namespace p224

}  // namespace crypto