fipsmodule/ec/p256.c

/* Copyright (c) 2020, Google Inc.
 *
 * Permission to use, copy, modify, and/or distribute this software for any
 * purpose with or without fee is hereby granted, provided that the above
 * copyright notice and this permission notice appear in all copies.
 *
 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY
 * SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION
 * OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN
 * CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. */

// An implementation of the NIST P-256 elliptic curve point multiplication.
// 256-bit Montgomery form for 64 and 32-bit. Field operations are generated by
// Fiat, which lives in //third_party/fiat.

#include <ring-core/base.h>

#include "../../limbs/limbs.h"
#include "../../limbs/limbs.inl"

#include "p256_shared.h"

#include "../../internal.h"
#include "./util.h"

#if !defined(OPENSSL_USE_NISTZ256)

#if defined(_MSC_VER) && !defined(__clang__)
// '=': conversion from 'int64_t' to 'int32_t', possible loss of data
#pragma warning(disable: 4242)
// '=': conversion from 'int32_t' to 'uint8_t', possible loss of data
#pragma warning(disable: 4244)
// 'initializing': conversion from 'size_t' to 'fiat_p256_limb_t'
#pragma warning(disable: 4267)
#endif

#if defined(__GNUC__) || defined(__clang__)
#pragma GCC diagnostic ignored "-Wconversion"
#pragma GCC diagnostic ignored "-Wsign-conversion"
#endif

#if defined(__GNUC__) && !defined(__clang__)
#pragma GCC diagnostic ignored "-Winline"
#endif

#if defined(BORINGSSL_HAS_UINT128)
#if defined(__GNUC__)
#pragma GCC diagnostic ignored "-Wpedantic"
#endif
#include "../../../third_party/fiat/p256_64.h"
#elif defined(OPENSSL_64_BIT)
#include "../../../third_party/fiat/p256_64_msvc.h"
#else
#include "../../../third_party/fiat/p256_32.h"
#endif


// utility functions, handwritten

#if defined(OPENSSL_64_BIT)
#define FIAT_P256_NLIMBS 4
typedef uint64_t fiat_p256_limb_t;
typedef uint64_t fiat_p256_felem[FIAT_P256_NLIMBS];
static const fiat_p256_felem fiat_p256_one = {0x1, 0xffffffff00000000,
                                              0xffffffffffffffff, 0xfffffffe};
#else  // 64BIT; else 32BIT
#define FIAT_P256_NLIMBS 8
typedef uint32_t fiat_p256_limb_t;
typedef uint32_t fiat_p256_felem[FIAT_P256_NLIMBS];
static const fiat_p256_felem fiat_p256_one = {
    0x1, 0x0, 0x0, 0xffffffff, 0xffffffff, 0xffffffff, 0xfffffffe, 0x0};
#endif  // 64BIT


static fiat_p256_limb_t fiat_p256_nz(
    const fiat_p256_limb_t in1[FIAT_P256_NLIMBS]) {
  fiat_p256_limb_t ret;
  fiat_p256_nonzero(&ret, in1);
  return ret;
}

static void fiat_p256_copy(fiat_p256_limb_t out[FIAT_P256_NLIMBS],
                           const fiat_p256_limb_t in1[FIAT_P256_NLIMBS]) {
  for (size_t i = 0; i < FIAT_P256_NLIMBS; i++) {
    out[i] = in1[i];
  }
}

static void fiat_p256_cmovznz(fiat_p256_limb_t out[FIAT_P256_NLIMBS],
                              fiat_p256_limb_t t,
                              const fiat_p256_limb_t z[FIAT_P256_NLIMBS],
                              const fiat_p256_limb_t nz[FIAT_P256_NLIMBS]) {
  fiat_p256_selectznz(out, !!t, z, nz);
}

// Group operations
// ----------------
//
// Building on top of the field operations we have the operations on the
// elliptic curve group itself. Points on the curve are represented in Jacobian
// coordinates.
//
// Both operations were transcribed to Coq and proven to correspond to naive
// implementations using Affine coordinates, for all suitable fields.  In the
// Coq proofs, issues of constant-time execution and memory layout (aliasing)
// conventions were not considered. Specification of affine coordinates:
// <https://github.com/mit-plv/fiat-crypto/blob/79f8b5f39ed609339f0233098dee1a3c4e6b3080/src/Spec/WeierstrassCurve.v#L28>
// As a sanity check, a proof that these points form a commutative group:
// <https://github.com/mit-plv/fiat-crypto/blob/79f8b5f39ed609339f0233098dee1a3c4e6b3080/src/Curves/Weierstrass/AffineProofs.v#L33>

// fiat_p256_point_double calculates 2*(x_in, y_in, z_in)
//
// The method is taken from:
//   http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
//
// Coq transcription and correctness proof:
// <https://github.com/mit-plv/fiat-crypto/blob/79f8b5f39ed609339f0233098dee1a3c4e6b3080/src/Curves/Weierstrass/Jacobian.v#L93>
// <https://github.com/mit-plv/fiat-crypto/blob/79f8b5f39ed609339f0233098dee1a3c4e6b3080/src/Curves/Weierstrass/Jacobian.v#L201>
//
// Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed.
// while x_out == y_in is not (maybe this works, but it's not tested).
static void fiat_p256_point_double(fiat_p256_felem x_out, fiat_p256_felem y_out,
                                   fiat_p256_felem z_out,
                                   const fiat_p256_felem x_in,
                                   const fiat_p256_felem y_in,
                                   const fiat_p256_felem z_in) {
  fiat_p256_felem delta, gamma, beta, ftmp, ftmp2, tmptmp, alpha, fourbeta;
  // delta = z^2
  fiat_p256_square(delta, z_in);
  // gamma = y^2
  fiat_p256_square(gamma, y_in);
  // beta = x*gamma
  fiat_p256_mul(beta, x_in, gamma);

  // alpha = 3*(x-delta)*(x+delta)
  fiat_p256_sub(ftmp, x_in, delta);
  fiat_p256_add(ftmp2, x_in, delta);

  fiat_p256_add(tmptmp, ftmp2, ftmp2);
  fiat_p256_add(ftmp2, ftmp2, tmptmp);
  fiat_p256_mul(alpha, ftmp, ftmp2);

  // x' = alpha^2 - 8*beta
  fiat_p256_square(x_out, alpha);
  fiat_p256_add(fourbeta, beta, beta);
  fiat_p256_add(fourbeta, fourbeta, fourbeta);
  fiat_p256_add(tmptmp, fourbeta, fourbeta);
  fiat_p256_sub(x_out, x_out, tmptmp);

  // z' = (y + z)^2 - gamma - delta
  fiat_p256_add(delta, gamma, delta);
  fiat_p256_add(ftmp, y_in, z_in);
  fiat_p256_square(z_out, ftmp);
  fiat_p256_sub(z_out, z_out, delta);

  // y' = alpha*(4*beta - x') - 8*gamma^2
  fiat_p256_sub(y_out, fourbeta, x_out);
  fiat_p256_add(gamma, gamma, gamma);
  fiat_p256_square(gamma, gamma);
  fiat_p256_mul(y_out, alpha, y_out);
  fiat_p256_add(gamma, gamma, gamma);
  fiat_p256_sub(y_out, y_out, gamma);
}

// fiat_p256_point_add calculates (x1, y1, z1) + (x2, y2, z2)
//
// The method is taken from:
//   http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl,
// adapted for mixed addition (z2 = 1, or z2 = 0 for the point at infinity).
//
// Coq transcription and correctness proof:
// <https://github.com/mit-plv/fiat-crypto/blob/79f8b5f39ed609339f0233098dee1a3c4e6b3080/src/Curves/Weierstrass/Jacobian.v#L135>
// <https://github.com/mit-plv/fiat-crypto/blob/79f8b5f39ed609339f0233098dee1a3c4e6b3080/src/Curves/Weierstrass/Jacobian.v#L205>
//
// This function includes a branch for checking whether the two input points
// are equal, (while not equal to the point at infinity). This case never
// happens during single point multiplication, so there is no timing leak for
// ECDH or ECDSA signing.
static void fiat_p256_point_add(fiat_p256_felem x3, fiat_p256_felem y3,
                                fiat_p256_felem z3, const fiat_p256_felem x1,
                                const fiat_p256_felem y1,
                                const fiat_p256_felem z1, const int mixed,
                                const fiat_p256_felem x2,
                                const fiat_p256_felem y2,
                                const fiat_p256_felem z2) {
  fiat_p256_felem x_out, y_out, z_out;
  fiat_p256_limb_t z1nz = fiat_p256_nz(z1);
  fiat_p256_limb_t z2nz = fiat_p256_nz(z2);

  // z1z1 = z1z1 = z1**2
  fiat_p256_felem z1z1;
  fiat_p256_square(z1z1, z1);

  fiat_p256_felem u1, s1, two_z1z2;
  if (!mixed) {
    // z2z2 = z2**2
    fiat_p256_felem z2z2;
    fiat_p256_square(z2z2, z2);

    // u1 = x1*z2z2
    fiat_p256_mul(u1, x1, z2z2);

    // two_z1z2 = (z1 + z2)**2 - (z1z1 + z2z2) = 2z1z2
    fiat_p256_add(two_z1z2, z1, z2);
    fiat_p256_square(two_z1z2, two_z1z2);
    fiat_p256_sub(two_z1z2, two_z1z2, z1z1);
    fiat_p256_sub(two_z1z2, two_z1z2, z2z2);

    // s1 = y1 * z2**3
    fiat_p256_mul(s1, z2, z2z2);
    fiat_p256_mul(s1, s1, y1);
  } else {
    // We'll assume z2 = 1 (special case z2 = 0 is handled later).

    // u1 = x1*z2z2
    fiat_p256_copy(u1, x1);
    // two_z1z2 = 2z1z2
    fiat_p256_add(two_z1z2, z1, z1);
    // s1 = y1 * z2**3
    fiat_p256_copy(s1, y1);
  }

  // u2 = x2*z1z1
  fiat_p256_felem u2;
  fiat_p256_mul(u2, x2, z1z1);

  // h = u2 - u1
  fiat_p256_felem h;
  fiat_p256_sub(h, u2, u1);

  fiat_p256_limb_t xneq = fiat_p256_nz(h);

  // z_out = two_z1z2 * h
  fiat_p256_mul(z_out, h, two_z1z2);

  // z1z1z1 = z1 * z1z1
  fiat_p256_felem z1z1z1;
  fiat_p256_mul(z1z1z1, z1, z1z1);

  // s2 = y2 * z1**3
  fiat_p256_felem s2;
  fiat_p256_mul(s2, y2, z1z1z1);

  // r = (s2 - s1)*2
  fiat_p256_felem r;
  fiat_p256_sub(r, s2, s1);
  fiat_p256_add(r, r, r);

  fiat_p256_limb_t yneq = fiat_p256_nz(r);

  fiat_p256_limb_t is_nontrivial_double = constant_time_is_zero_w(xneq | yneq) &
                                          ~constant_time_is_zero_w(z1nz) &
                                          ~constant_time_is_zero_w(z2nz);
  if (constant_time_declassify_w(is_nontrivial_double)) {
    fiat_p256_point_double(x3, y3, z3, x1, y1, z1);
    return;
  }

  // I = (2h)**2
  fiat_p256_felem i;
  fiat_p256_add(i, h, h);
  fiat_p256_square(i, i);

  // J = h * I
  fiat_p256_felem j;
  fiat_p256_mul(j, h, i);

  // V = U1 * I
  fiat_p256_felem v;
  fiat_p256_mul(v, u1, i);

  // x_out = r**2 - J - 2V
  fiat_p256_square(x_out, r);
  fiat_p256_sub(x_out, x_out, j);
  fiat_p256_sub(x_out, x_out, v);
  fiat_p256_sub(x_out, x_out, v);

  // y_out = r(V-x_out) - 2 * s1 * J
  fiat_p256_sub(y_out, v, x_out);
  fiat_p256_mul(y_out, y_out, r);
  fiat_p256_felem s1j;
  fiat_p256_mul(s1j, s1, j);
  fiat_p256_sub(y_out, y_out, s1j);
  fiat_p256_sub(y_out, y_out, s1j);

  fiat_p256_cmovznz(x_out, z1nz, x2, x_out);
  fiat_p256_cmovznz(x3, z2nz, x1, x_out);
  fiat_p256_cmovznz(y_out, z1nz, y2, y_out);
  fiat_p256_cmovznz(y3, z2nz, y1, y_out);
  fiat_p256_cmovznz(z_out, z1nz, z2, z_out);
  fiat_p256_cmovznz(z3, z2nz, z1, z_out);
}

#include "./p256_table.h"

// fiat_p256_select_point_affine selects the |idx-1|th point from a
// precomputation table and copies it to out. If |idx| is zero, the output is
// the point at infinity.
static void fiat_p256_select_point_affine(
    const fiat_p256_limb_t idx, size_t size,
    const fiat_p256_felem pre_comp[/*size*/][2], fiat_p256_felem out[3]) {
  OPENSSL_memset(out, 0, sizeof(fiat_p256_felem) * 3);
  for (size_t i = 0; i < size; i++) {
    fiat_p256_limb_t mismatch = i ^ (idx - 1);
    fiat_p256_cmovznz(out[0], mismatch, pre_comp[i][0], out[0]);
    fiat_p256_cmovznz(out[1], mismatch, pre_comp[i][1], out[1]);
  }
  fiat_p256_cmovznz(out[2], idx, out[2], fiat_p256_one);
}

// fiat_p256_select_point selects the |idx|th point from a precomputation table
// and copies it to out.
static void fiat_p256_select_point(const fiat_p256_limb_t idx, size_t size,
                                   const fiat_p256_felem pre_comp[/*size*/][3],
                                   fiat_p256_felem out[3]) {
  OPENSSL_memset(out, 0, sizeof(fiat_p256_felem) * 3);
  for (size_t i = 0; i < size; i++) {
    fiat_p256_limb_t mismatch = i ^ idx;
    fiat_p256_cmovznz(out[0], mismatch, pre_comp[i][0], out[0]);
    fiat_p256_cmovznz(out[1], mismatch, pre_comp[i][1], out[1]);
    fiat_p256_cmovznz(out[2], mismatch, pre_comp[i][2], out[2]);
  }
}

// fiat_p256_get_bit returns the |i|th bit in |in|
static crypto_word_t fiat_p256_get_bit(const Limb in[P256_LIMBS], int i) {
  if (i < 0 || i >= 256) {
    return 0;
  }
#if defined(OPENSSL_64_BIT)
  OPENSSL_STATIC_ASSERT(sizeof(Limb) == 8, "BN_ULONG was not 64-bit");
  return (in[i >> 6] >> (i & 63)) & 1;
#else
  OPENSSL_STATIC_ASSERT(sizeof(Limb) == 4, "BN_ULONG was not 32-bit");
  return (in[i >> 5] >> (i & 31)) & 1;
#endif
}

void p256_point_mul(P256_POINT *r, const Limb scalar[P256_LIMBS],
                        const Limb p_x[P256_LIMBS], const Limb p_y[P256_LIMBS]) {
  debug_assert_nonsecret(r != NULL);
  debug_assert_nonsecret(scalar != NULL);
  debug_assert_nonsecret(p_x != NULL);
  debug_assert_nonsecret(p_y != NULL);

  fiat_p256_felem p_pre_comp[17][3];
  OPENSSL_memset(&p_pre_comp, 0, sizeof(p_pre_comp));
  // Precompute multiples.
  limbs_copy(&p_pre_comp[1][0][0], p_x, P256_LIMBS);
  limbs_copy(&p_pre_comp[1][1][0], p_y, P256_LIMBS);
  limbs_copy(&p_pre_comp[1][2][0], fiat_p256_one, P256_LIMBS);

  for (size_t j = 2; j <= 16; ++j) {
    if (j & 1) {
      fiat_p256_point_add(p_pre_comp[j][0], p_pre_comp[j][1], p_pre_comp[j][2],
                          p_pre_comp[1][0], p_pre_comp[1][1], p_pre_comp[1][2],
                          0, p_pre_comp[j - 1][0], p_pre_comp[j - 1][1],
                          p_pre_comp[j - 1][2]);
    } else {
      fiat_p256_point_double(p_pre_comp[j][0], p_pre_comp[j][1],
                             p_pre_comp[j][2], p_pre_comp[j / 2][0],
                             p_pre_comp[j / 2][1], p_pre_comp[j / 2][2]);
    }
  }

  // Set nq to the point at infinity.
  fiat_p256_felem nq[3] = {{0}, {0}, {0}}, ftmp, tmp[3];

  // Loop over |scalar| msb-to-lsb, incorporating |p_pre_comp| every 5th round.
  int skip = 1;  // Save two point operations in the first round.
  for (size_t i = 255; i < 256; i--) {
    // double
    if (!skip) {
      fiat_p256_point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);
    }

    // do other additions every 5 doublings
    if (i % 5 == 0) {
      crypto_word_t bits = fiat_p256_get_bit(scalar, i + 4) << 5;
      bits |= fiat_p256_get_bit(scalar, i + 3) << 4;
      bits |= fiat_p256_get_bit(scalar, i + 2) << 3;
      bits |= fiat_p256_get_bit(scalar, i + 1) << 2;
      bits |= fiat_p256_get_bit(scalar, i) << 1;
      bits |= fiat_p256_get_bit(scalar, i - 1);
      crypto_word_t sign, digit;
      recode_scalar_bits(&sign, &digit, bits);

      // select the point to add or subtract, in constant time.
      fiat_p256_select_point((fiat_p256_limb_t)digit, 17,
        RING_CORE_POINTLESS_ARRAY_CONST_CAST((const fiat_p256_felem(*)[3]))p_pre_comp,
        tmp);
      fiat_p256_opp(ftmp, tmp[1]);  // (X, -Y, Z) is the negative point.
      fiat_p256_cmovznz(tmp[1], (fiat_p256_limb_t)sign, tmp[1], ftmp);

      if (!skip) {
        fiat_p256_point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2],
                            0 /* mixed */, tmp[0], tmp[1], tmp[2]);
      } else {
        fiat_p256_copy(nq[0], tmp[0]);
        fiat_p256_copy(nq[1], tmp[1]);
        fiat_p256_copy(nq[2], tmp[2]);
        skip = 0;
      }
    }
  }

  limbs_copy(r->X, nq[0], P256_LIMBS);
  limbs_copy(r->Y, nq[1], P256_LIMBS);
  limbs_copy(r->Z, nq[2], P256_LIMBS);
}

void p256_point_mul_base(P256_POINT *r, const Limb scalar[P256_LIMBS]) {
  // Set nq to the point at infinity.
  fiat_p256_felem nq[3] = {{0}, {0}, {0}}, tmp[3];

  int skip = 1;  // Save two point operations in the first round.
  for (size_t i = 31; i < 32; i--) {
    if (!skip) {
      fiat_p256_point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);
    }

    // First, look 32 bits upwards.
    crypto_word_t bits = fiat_p256_get_bit(scalar, i + 224) << 3;
    bits |= fiat_p256_get_bit(scalar, i + 160) << 2;
    bits |= fiat_p256_get_bit(scalar, i + 96) << 1;
    bits |= fiat_p256_get_bit(scalar, i + 32);
    // Select the point to add, in constant time.
    fiat_p256_select_point_affine((fiat_p256_limb_t)bits, 15,
                                  fiat_p256_g_pre_comp[1], tmp);

    if (!skip) {
      fiat_p256_point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2],
                          1 /* mixed */, tmp[0], tmp[1], tmp[2]);
    } else {
      fiat_p256_copy(nq[0], tmp[0]);
      fiat_p256_copy(nq[1], tmp[1]);
      fiat_p256_copy(nq[2], tmp[2]);
      skip = 0;
    }

    // Second, look at the current position.
    bits = fiat_p256_get_bit(scalar, i + 192) << 3;
    bits |= fiat_p256_get_bit(scalar, i + 128) << 2;
    bits |= fiat_p256_get_bit(scalar, i + 64) << 1;
    bits |= fiat_p256_get_bit(scalar, i);
    // Select the point to add, in constant time.
    fiat_p256_select_point_affine((fiat_p256_limb_t)bits, 15,
                                  fiat_p256_g_pre_comp[0], tmp);
    fiat_p256_point_add(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2], 1 /* mixed */,
                        tmp[0], tmp[1], tmp[2]);
  }

  limbs_copy(r->X, nq[0], P256_LIMBS);
  limbs_copy(r->Y, nq[1], P256_LIMBS);
  limbs_copy(r->Z, nq[2], P256_LIMBS);
}

void p256_mul_mont(Limb r[P256_LIMBS], const Limb a[P256_LIMBS],
                       const Limb b[P256_LIMBS]) {
  fiat_p256_mul(r, a, b);
}

void p256_sqr_mont(Limb r[P256_LIMBS], const Limb a[P256_LIMBS]) {
  fiat_p256_square(r, a);
}

void p256_point_add(P256_POINT *r, const P256_POINT *a, const P256_POINT *b) {
  fiat_p256_point_add(r->X, r->Y, r->Z,
                      a->X, a->Y, a->Z,
                      0,
                      b->X, b->Y, b->Z);
}

void p256_point_double(P256_POINT *r, const P256_POINT *a) {
  fiat_p256_point_double(r->X, r->Y, r->Z,
                         a->X, a->Y, a->Z);
}

// For testing only.
void p256_point_add_affine(P256_POINT *r, const P256_POINT *a,
                               const BN_ULONG b[P256_LIMBS * 2]) {
  const Limb *b_x = &b[0];
  const Limb *b_y = &b[P256_LIMBS];
  fiat_p256_felem b_z = {0};
  crypto_word_t b_is_inf = constant_time_select_w(
      LIMBS_are_zero(b_x, P256_LIMBS), LIMBS_are_zero(b_y, P256_LIMBS), 0);
  fiat_p256_cmovznz(b_z, constant_time_is_zero_w(b_is_inf), b_z, fiat_p256_one);
  fiat_p256_point_add(r->X, r->Y, r->Z,
                      a->X, a->Y, a->Z,
                      1,
                      b_x, b_y, b_z);
}

#endif