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Commit 9c18f925 authored by Adam Langley's avatar Adam Langley Committed by Chromium LUCI CQ
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Drop custom P-224 implementation. 

This dates back to when we still used NSS(!) and thus didn't have access
to group operations for any elliptic curves. We can just use BoringSSL
now.

Change-Id: I8bc4bc09fdd26feb25a76c17bf716808eb1cb2a3
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/2623123
Commit-Queue: Adam Langley <agl@chromium.org>
Auto-Submit: Adam Langley <agl@chromium.org>
Reviewed-by: default avatarDavid Benjamin <davidben@chromium.org>
Cr-Commit-Position: refs/heads/master@{#842843}
parent b73da146
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Showing with 111 additions and 1671 deletions
crypto/BUILD.gn
View file @ 9c18f925
...@@ -26,8 +26,6 @@ component("crypto") { ...@@ -26,8 +26,6 @@ component("crypto") {
"hmac.h", "hmac.h",
"openssl_util.cc", "openssl_util.cc",
"openssl_util.h", "openssl_util.h",
"p224.cc",
"p224.h",
"p224_spake.cc", "p224_spake.cc",
"p224_spake.h", "p224_spake.h",
"random.cc", "random.cc",
...@@ -145,7 +143,6 @@ test("crypto_unittests") { ...@@ -145,7 +143,6 @@ test("crypto_unittests") {
"encryptor_unittest.cc", "encryptor_unittest.cc",
"hmac_unittest.cc", "hmac_unittest.cc",
"p224_spake_unittest.cc", "p224_spake_unittest.cc",
"p224_unittest.cc",
"random_unittest.cc", "random_unittest.cc",
"rsa_private_key_unittest.cc", "rsa_private_key_unittest.cc",
"secure_hash_unittest.cc", "secure_hash_unittest.cc",
......
crypto/p224.cc deleted 100644 → 0
View file @ b73da146
// 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 <stddef.h>
#include <stdint.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_t'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_t IsZero(const FieldElement& a) {
FieldElement minimal;
memcpy(&minimal, &a, sizeof(minimal));
Contract(&minimal);
uint32_t 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_t>(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_t kTwo31p3 = (1u << 31) + (1u << 3);
static const uint32_t kTwo31m3 = (1u << 31) - (1u << 3);
static const uint32_t 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_t kTwo63p35 = (1ull << 63) + (1ull << 35);
static const uint64_t kTwo63m35 = (1ull << 63) - (1ull << 35);
static const uint64_t 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_t kZero63ModP[8] = {
kTwo63p35, kTwo63m35, kTwo63m35, kTwo63m35,
kTwo63m35m19, kTwo63m35, kTwo63m35, kTwo63m35,
};
static const uint32_t 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_t 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_t>(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_t>(in[8] & 0xffff) << 12; // "-2**96" term
(*out)[4] += static_cast<uint32_t>(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_t>(in[0] & kBottom28Bits);
(*out)[1] += static_cast<uint32_t>((in[0] >> 28) & kBottom28Bits);
(*out)[2] += static_cast<uint32_t>(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_t>(a[i]) * static_cast<uint64_t>(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_t r = static_cast<uint64_t>(a[i]) * static_cast<uint64_t>(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_t top = a[7] >> 28;
a[7] &= kBottom28Bits;
// top < 2**4
// Constant-time: mask = (top != 0) ? 0xffffffff : 0
uint32_t mask = top;
mask |= mask >> 2;
mask |= mask >> 1;
mask <<= 31;
mask = static_cast<uint32_t>(static_cast<int32_t>(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_t 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_t mask = static_cast<uint32_t>(static_cast<int32_t>(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_t mask = static_cast<uint32_t>(static_cast<int32_t>(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_t 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_t>(static_cast<int32_t>(top_4_all_ones << 31) >> 31);
// Now we test whether the bottom three limbs are non-zero.
uint32_t 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_t>(static_cast<int32_t>(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_t n = out[3] - 0xffff000;
uint32_t 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_t>(static_cast<int32_t>(out_3_equal << 31) >> 31);
// If out[3] > 0xffff000 then n's MSB will be zero.
uint32_t out_3_gt =
~static_cast<uint32_t>(static_cast<int32_t>(n << 31) >> 31);
uint32_t 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_t 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_t z1_is_zero = IsZero(a.z);
uint32_t 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_t x_equal = IsZero(h);
// I = (2*H)²
for (int k = 0; k < 8; k++) {
i[k] = h[k] << 1;
}
Reduce(&i);
Square(&i, i);
// J = H*I
Mul(&j, h, i);
// r = 2*(S2-S1)
Subtract(&r, s2, s1);
Reduce(&r);
uint32_t 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 k = 0; k < 8; k++) {
r[k] <<= 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 k = 0; k < 8; k++) {
z1z1[k] = v[k] << 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 k = 0; k < 8; k++) {
s1[k] <<= 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_t 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_t* 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_t bit = static_cast<uint32_t>(static_cast<int32_t>(
(((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_t* out, const uint32_t* 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_t* out, const uint32_t* 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(base::StringPiece in) {
if (in.size() != 2*28)
return false;
const uint32_t* inwords = reinterpret_cast<const uint32_t*>(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, xx, yy;
// 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(&xx, x, zinv_sq);
Mul(&zinv_sq, zinv_sq, zinv);
Mul(&yy, y, zinv_sq);
Contract(&xx);
Contract(&yy);
uint32_t outwords[14];
Put224Bits(outwords, xx);
Put224Bits(outwords + 7, yy);
return std::string(reinterpret_cast<const char*>(outwords), sizeof(outwords));
}
void ScalarMult(const Point& in, const uint8_t* 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_t* 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
crypto/p224.h deleted 100644 → 0
View file @ b73da146
// 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.
#ifndef CRYPTO_P224_H_
#define CRYPTO_P224_H_
#include <stddef.h>
#include <stdint.h>
#include <string>
#include "base/strings/string_piece.h"
#include "crypto/crypto_export.h"
namespace crypto {
// P224 implements an elliptic curve group, commonly known as P224 and defined
// in FIPS 186-3, section D.2.2.
namespace p224 {
// An element of the field (ℤ/pℤ) is represented with 8, 28-bit limbs in
// little endian order.
typedef uint32_t FieldElement[8];
struct CRYPTO_EXPORT Point {
// SetFromString the value of the point from the 56 byte, external
// representation. The external point representation is an (x, y) pair of a
// point on the curve. Each field element is represented as a big-endian
// number < p.
bool SetFromString(base::StringPiece in);
// ToString returns an external representation of the Point.
std::string ToString() const;
// An Point is represented in Jacobian form (x/z², y/z³).
FieldElement x, y, z;
};
// kScalarBytes is the number of bytes needed to represent an element of the
// P224 field.
static const size_t kScalarBytes = 28;
// ScalarMult computes *out = in*scalar where scalar is a 28-byte, big-endian
// number.
void CRYPTO_EXPORT ScalarMult(const Point& in,
const uint8_t* scalar,
Point* out);
// ScalarBaseMult computes *out = g*scalar where g is the base point of the
// curve and scalar is a 28-byte, big-endian number.
void CRYPTO_EXPORT ScalarBaseMult(const uint8_t* scalar, Point* out);
// Add computes *out = a+b.
void CRYPTO_EXPORT Add(const Point& a, const Point& b, Point* out);
// Negate calculates out = -a;
void CRYPTO_EXPORT Negate(const Point& a, Point* out);
} // namespace p224
} // namespace crypto
#endif // CRYPTO_P224_H_
crypto/p224_spake.cc
View file @ 9c18f925
...@@ -12,9 +12,11 @@ ...@@ -12,9 +12,11 @@
#include <algorithm> #include <algorithm>
#include "base/logging.h" #include "base/logging.h"
#include "crypto/p224.h"
#include "crypto/random.h" #include "crypto/random.h"
#include "crypto/secure_util.h" #include "crypto/secure_util.h"
#include "third_party/boringssl/src/include/openssl/bn.h"
#include "third_party/boringssl/src/include/openssl/ec.h"
#include "third_party/boringssl/src/include/openssl/obj.h"
namespace { namespace {
...@@ -82,22 +84,87 @@ namespace { ...@@ -82,22 +84,87 @@ namespace {
// return 0; // return 0;
// } // }
const crypto::p224::Point kM = { const uint8_t kM_X962[1 + 28 + 28] = {
{174237515, 77186811, 235213682, 33849492, 0x04, 0x4d, 0x48, 0xc8, 0xea, 0x8d, 0x23, 0x39, 0x2e, 0x07, 0xe8, 0x51,
33188520, 48266885, 177021753, 81038478}, 0xfa, 0x6a, 0xa8, 0x20, 0x48, 0x09, 0x4e, 0x05, 0x13, 0x72, 0x49, 0x9c,
{104523827, 245682244, 266509668, 236196369, 0x6f, 0xba, 0x62, 0xa7, 0x4b, 0x6c, 0x18, 0x5c, 0xab, 0xd5, 0x2e, 0x2e,
28372046, 145351378, 198520366, 113345994}, 0x8a, 0x9e, 0x2d, 0x21, 0xb0, 0xec, 0x4e, 0xe1, 0x41, 0x21, 0x1f, 0xe2,
{1, 0, 0, 0, 0, 0, 0, 0}, 0x9d, 0x64, 0xea, 0x4d, 0x04, 0x46, 0x3a, 0xe8, 0x33,
}; };
const crypto::p224::Point kN = { const uint8_t kN_X962[1 + 28 + 28] = {
{136176322, 263523628, 251628795, 229292285, 0x04, 0x0b, 0x1c, 0xfc, 0x6a, 0x40, 0x7c, 0xdc, 0xb1, 0x5d, 0xc1, 0x70,
5034302, 185981975, 171998428, 11653062}, 0x4c, 0xd1, 0x3e, 0xda, 0xab, 0x8f, 0xde, 0xff, 0x8c, 0xfb, 0xfb, 0x50,
{197567436, 51226044, 60372156, 175772188, 0xd2, 0xc8, 0x1d, 0xe2, 0xc2, 0x3e, 0x14, 0xf6, 0x29, 0x96, 0x08, 0x09,
42075930, 8083165, 160827401, 65097570}, 0x07, 0xb5, 0x6d, 0xd2, 0x82, 0x07, 0x1a, 0xa7, 0xa1, 0x21, 0xc3, 0x99,
{1, 0, 0, 0, 0, 0, 0, 0}, 0x34, 0xbc, 0x30, 0xda, 0x5b, 0xcb, 0xc6, 0xa3, 0xcc,
}; };
// ToBignum returns |big_endian_bytes| interpreted as a big-endian number.
bssl::UniquePtr<BIGNUM> ToBignum(base::span<const uint8_t> big_endian_bytes) {
bssl::UniquePtr<BIGNUM> bn(BN_new());
CHECK(BN_bin2bn(big_endian_bytes.data(), big_endian_bytes.size(), bn.get()));
return bn;
}
// GetPoint decodes and returns the given X.962-encoded point. It will crash if
// |x962| is not a valid P-224 point.
bssl::UniquePtr<EC_POINT> GetPoint(
const EC_GROUP* p224,
base::span<const uint8_t, 1 + 28 + 28> x962) {
bssl::UniquePtr<EC_POINT> point(EC_POINT_new(p224));
CHECK(EC_POINT_oct2point(p224, point.get(), x962.data(), x962.size(),
/*ctx=*/nullptr));
return point;
}
// GetMask returns (M|N)**pw, where the choice of M or N is controlled by
// |use_m|.
bssl::UniquePtr<EC_POINT> GetMask(const EC_GROUP* p224,
bool use_m,
base::span<const uint8_t> pw) {
bssl::UniquePtr<EC_POINT> MN(GetPoint(p224, use_m ? kM_X962 : kN_X962));
bssl::UniquePtr<EC_POINT> MNpw(EC_POINT_new(p224));
bssl::UniquePtr<BIGNUM> pw_bn(ToBignum(pw));
CHECK(EC_POINT_mul(p224, MNpw.get(), nullptr, MN.get(), pw_bn.get(),
/*ctx=*/nullptr));
return MNpw;
}
// ToMessage serialises |in| as a 56-byte string that contains the big-endian
// representations of x and y, or is all zeros if |in| is infinity.
std::string ToMessage(const EC_GROUP* p224, const EC_POINT* in) {
if (EC_POINT_is_at_infinity(p224, in)) {
return std::string(28 + 28, 0);
}
uint8_t x962[1 + 28 + 28];
CHECK(EC_POINT_point2oct(p224, in, POINT_CONVERSION_UNCOMPRESSED, x962,
sizeof(x962), /*ctx=*/nullptr) == sizeof(x962));
return std::string(reinterpret_cast<const char*>(&x962[1]), sizeof(x962) - 1);
}
// FromMessage converts a message, as generated by |ToMessage|, into a point. It
// returns |nullptr| if the input is invalid or not on the curve.
bssl::UniquePtr<EC_POINT> FromMessage(const EC_GROUP* p224,
base::StringPiece in) {
if (in.size() != 56) {
return nullptr;
}
uint8_t x962[1 + 56];
x962[0] = 4;
memcpy(&x962[1], in.data(), sizeof(x962) - 1);
bssl::UniquePtr<EC_POINT> ret(EC_POINT_new(p224));
if (!EC_POINT_oct2point(p224, ret.get(), x962, sizeof(x962),
/*ctx=*/nullptr)) {
return nullptr;
}
return ret;
}
} // anonymous namespace } // anonymous namespace
namespace crypto { namespace crypto {
...@@ -120,19 +187,25 @@ P224EncryptedKeyExchange::P224EncryptedKeyExchange(PeerType peer_type, ...@@ -120,19 +187,25 @@ P224EncryptedKeyExchange::P224EncryptedKeyExchange(PeerType peer_type,
void P224EncryptedKeyExchange::Init() { void P224EncryptedKeyExchange::Init() {
// X = g**x_ // X = g**x_
p224::Point X; bssl::UniquePtr<EC_GROUP> p224(EC_GROUP_new_by_curve_name(NID_secp224r1));
p224::ScalarBaseMult(x_, &X); bssl::UniquePtr<EC_POINT> X(EC_POINT_new(p224.get()));
bssl::UniquePtr<BIGNUM> x_bn(ToBignum(x_));
// x_bn may be >= the order, but |EC_POINT_mul| handles that. It doesn't do so
// in constant-time, but the these values are locally generated and so this
// occurs with negligible probability. (Same with |pw_|, just below.)
CHECK(EC_POINT_mul(p224.get(), X.get(), x_bn.get(), nullptr, nullptr,
/*ctx=*/nullptr));
// The client masks the Diffie-Hellman value, X, by adding M**pw and the // The client masks the Diffie-Hellman value, X, by adding M**pw and the
// server uses N**pw. // server uses N**pw.
p224::Point MNpw; bssl::UniquePtr<EC_POINT> MNpw(GetMask(p224.get(), !is_server_, pw_));
p224::ScalarMult(is_server_ ? kN : kM, pw_, &MNpw);
// X* = X + (N|M)**pw // X* = X + (N|M)**pw
p224::Point Xstar; bssl::UniquePtr<EC_POINT> Xstar(EC_POINT_new(p224.get()));
p224::Add(X, MNpw, &Xstar); CHECK(EC_POINT_add(p224.get(), Xstar.get(), X.get(), MNpw.get(),
/*ctx=*/nullptr));
next_message_ = Xstar.ToString(); next_message_ = ToMessage(p224.get(), Xstar.get());
} }
const std::string& P224EncryptedKeyExchange::GetNextMessage() { const std::string& P224EncryptedKeyExchange::GetNextMessage() {
...@@ -175,26 +248,31 @@ P224EncryptedKeyExchange::Result P224EncryptedKeyExchange::ProcessMessage( ...@@ -175,26 +248,31 @@ P224EncryptedKeyExchange::Result P224EncryptedKeyExchange::ProcessMessage(
return kResultFailed; return kResultFailed;
} }
bssl::UniquePtr<EC_GROUP> p224(EC_GROUP_new_by_curve_name(NID_secp224r1));
// Y* is the other party's masked, Diffie-Hellman value. // Y* is the other party's masked, Diffie-Hellman value.
p224::Point Ystar; bssl::UniquePtr<EC_POINT> Ystar(FromMessage(p224.get(), message));
if (!Ystar.SetFromString(message)) { if (!Ystar) {
error_ = "failed to parse peer's masked Diffie-Hellman value"; error_ = "failed to parse peer's masked Diffie-Hellman value";
return kResultFailed; return kResultFailed;
} }
// We calculate the mask value: (N|M)**pw // We calculate the mask value: (N|M)**pw
p224::Point MNpw, minus_MNpw, Y, k; bssl::UniquePtr<EC_POINT> MNpw(GetMask(p224.get(), is_server_, pw_));
p224::ScalarMult(is_server_ ? kM : kN, pw_, &MNpw);
p224::Negate(MNpw, &minus_MNpw);
// Y = Y* - (N|M)**pw // Y = Y* - (N|M)**pw
p224::Add(Ystar, minus_MNpw, &Y); CHECK(EC_POINT_invert(p224.get(), MNpw.get(), /*ctx=*/nullptr));
bssl::UniquePtr<EC_POINT> Y(EC_POINT_new(p224.get()));
CHECK(EC_POINT_add(p224.get(), Y.get(), Ystar.get(), MNpw.get(),
/*ctx=*/nullptr));
// K = Y**x_ // K = Y**x_
p224::ScalarMult(Y, x_, &k); bssl::UniquePtr<EC_POINT> K(EC_POINT_new(p224.get()));
bssl::UniquePtr<BIGNUM> x_bn(ToBignum(x_));
CHECK(EC_POINT_mul(p224.get(), K.get(), nullptr, Y.get(), x_bn.get(),
/*ctx=*/nullptr));
// If everything worked out, then K is the same for both parties. // If everything worked out, then K is the same for both parties.
key_ = k.ToString(); key_ = ToMessage(p224.get(), K.get());
std::string client_masked_dh, server_masked_dh; std::string client_masked_dh, server_masked_dh;
if (is_server_) { if (is_server_) {
......
crypto/p224_spake.h
View file @ 9c18f925
...@@ -11,7 +11,6 @@ ...@@ -11,7 +11,6 @@
#include "base/gtest_prod_util.h" #include "base/gtest_prod_util.h"
#include "base/strings/string_piece.h" #include "base/strings/string_piece.h"
#include "crypto/p224.h"
#include "crypto/sha2.h" #include "crypto/sha2.h"
namespace crypto { namespace crypto {
...@@ -110,12 +109,14 @@ class CRYPTO_EXPORT P224EncryptedKeyExchange { ...@@ -110,12 +109,14 @@ class CRYPTO_EXPORT P224EncryptedKeyExchange {
const std::string& k, const std::string& k,
uint8_t* out_digest); uint8_t* out_digest);
// kScalarBytes is the number of bytes in a P-224 scalar.
static constexpr size_t kScalarBytes = 28;
// x_ is the secret Diffie-Hellman exponent (see paper referenced in .cc // x_ is the secret Diffie-Hellman exponent (see paper referenced in .cc
// file). // file).
uint8_t x_[p224::kScalarBytes]; uint8_t x_[kScalarBytes];
// pw_ is SHA256(P(password), P(session))[:28] where P() prepends a uint32_t, // pw_ is SHA256(P(password), P(session))[:28] where P() prepends a uint32_t,
// big-endian length prefix (see paper referenced in .cc file). // big-endian length prefix (see paper referenced in .cc file).
uint8_t pw_[p224::kScalarBytes]; uint8_t pw_[kScalarBytes];
// expected_authenticator_ is used to store the hash value expected from the // expected_authenticator_ is used to store the hash value expected from the
// other party. // other party.
uint8_t expected_authenticator_[kSHA256Length]; uint8_t expected_authenticator_[kSHA256Length];
......
crypto/p224_unittest.cc deleted 100644 → 0
View file @ b73da146
// 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.
#include <stddef.h>
#include <stdint.h>
#include <stdio.h>
#include <string.h>
#include "crypto/p224.h"
#include "base/stl_util.h"
#include "testing/gtest/include/gtest/gtest.h"
namespace crypto {
using p224::Point;
// kBasePointExternal is the P224 base point in external representation.
static const uint8_t kBasePointExternal[56] = {
0xb7, 0x0e, 0x0c, 0xbd, 0x6b, 0xb4, 0xbf, 0x7f, 0x32, 0x13, 0x90, 0xb9,
0x4a, 0x03, 0xc1, 0xd3, 0x56, 0xc2, 0x11, 0x22, 0x34, 0x32, 0x80, 0xd6,
0x11, 0x5c, 0x1d, 0x21, 0xbd, 0x37, 0x63, 0x88, 0xb5, 0xf7, 0x23, 0xfb,
0x4c, 0x22, 0xdf, 0xe6, 0xcd, 0x43, 0x75, 0xa0, 0x5a, 0x07, 0x47, 0x64,
0x44, 0xd5, 0x81, 0x99, 0x85, 0x00, 0x7e, 0x34,
};
// TestVector represents a test of scalar multiplication of the base point.
// |scalar| is a big-endian scalar and |affine| is the external representation
// of g*scalar.
struct TestVector {
uint8_t scalar[28];
uint8_t affine[28 * 2];
};
static const int kNumNISTTestVectors = 52;
// kNISTTestVectors are the NIST test vectors for P224.
static const TestVector kNISTTestVectors[kNumNISTTestVectors] = {
{
{0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x01},
{0xb7, 0x0e, 0x0c, 0xbd, 0x6b, 0xb4, 0xbf, 0x7f,
0x32, 0x13, 0x90, 0xb9, 0x4a, 0x03, 0xc1, 0xd3,
0x56, 0xc2, 0x11, 0x22, 0x34, 0x32, 0x80, 0xd6,
0x11, 0x5c, 0x1d, 0x21, 0xbd, 0x37, 0x63, 0x88,
0xb5, 0xf7, 0x23, 0xfb, 0x4c, 0x22, 0xdf, 0xe6,
0xcd, 0x43, 0x75, 0xa0, 0x5a, 0x07, 0x47, 0x64,
0x44, 0xd5, 0x81, 0x99, 0x85, 0x00, 0x7e, 0x34
},
},
{
{0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x02, },
{0x70, 0x6a, 0x46, 0xdc, 0x76, 0xdc, 0xb7, 0x67,
0x98, 0xe6, 0x0e, 0x6d, 0x89, 0x47, 0x47, 0x88,
0xd1, 0x6d, 0xc1, 0x80, 0x32, 0xd2, 0x68, 0xfd,
0x1a, 0x70, 0x4f, 0xa6, 0x1c, 0x2b, 0x76, 0xa7,
0xbc, 0x25, 0xe7, 0x70, 0x2a, 0x70, 0x4f, 0xa9,
0x86, 0x89, 0x28, 0x49, 0xfc, 0xa6, 0x29, 0x48,
0x7a, 0xcf, 0x37, 0x09, 0xd2, 0xe4, 0xe8, 0xbb,
},
},
{
{0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x03, },
{0xdf, 0x1b, 0x1d, 0x66, 0xa5, 0x51, 0xd0, 0xd3,
0x1e, 0xff, 0x82, 0x25, 0x58, 0xb9, 0xd2, 0xcc,
0x75, 0xc2, 0x18, 0x02, 0x79, 0xfe, 0x0d, 0x08,
0xfd, 0x89, 0x6d, 0x04, 0xa3, 0xf7, 0xf0, 0x3c,
0xad, 0xd0, 0xbe, 0x44, 0x4c, 0x0a, 0xa5, 0x68,
0x30, 0x13, 0x0d, 0xdf, 0x77, 0xd3, 0x17, 0x34,
0x4e, 0x1a, 0xf3, 0x59, 0x19, 0x81, 0xa9, 0x25,
},
},
{
{0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0x00, 0x04, },
{0xae, 0x99, 0xfe, 0xeb, 0xb5, 0xd2, 0x69, 0x45,
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0x37, 0xbe, 0x04, 0x60, 0xdd, 0x10, 0x7a, 0xe7,
0x77, 0x25, 0x69, 0x6d, 0x21, 0x14, 0x46, 0xc5,
0x60, 0x9b, 0x45, 0x95, 0x97, 0x6b, 0x16, 0xbd,
},
},
{
{0x7f, 0xff, 0xff, 0xc0, 0x00, 0xff, 0xfe, 0x3f,
0xff, 0xfc, 0x10, 0x00, 0x00, 0x20, 0x00, 0x3f,
0xff, 0xff, 0x00, 0x00, 0x00, 0xfc, 0x00, 0x00,
0x3f, 0xff, 0xff, 0xff, },
{0xdc, 0x9f, 0xa7, 0x79, 0x78, 0xa0, 0x05, 0x51,
0x09, 0x80, 0xe9, 0x29, 0xa1, 0x48, 0x5f, 0x63,
0x71, 0x6d, 0xf6, 0x95, 0xd7, 0xa0, 0xc1, 0x8b,
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0xf2, 0xdd, 0xff, 0xc2, 0xa8, 0xc0, 0x15, 0xb1,
0x34, 0x92, 0x82, 0x75, 0xce, 0x09, 0xe5, 0x66,
0x1b, 0x7a, 0xb1, 0x4c, 0xe0, 0xd1, 0xd4, 0x03,
},
},
{
{0x70, 0x01, 0xf0, 0x00, 0x1c, 0x00, 0x01, 0xc0,
0x00, 0x00, 0x1f, 0xff, 0xff, 0xfc, 0x00, 0x00,
0x1f, 0xff, 0xff, 0xf8, 0x00, 0x0f, 0xc0, 0x00,
0x00, 0x01, 0xfc, 0x00, },
{0x49, 0x9d, 0x8b, 0x28, 0x29, 0xcf, 0xb8, 0x79,
0xc9, 0x01, 0xf7, 0xd8, 0x5d, 0x35, 0x70, 0x45,
0xed, 0xab, 0x55, 0x02, 0x88, 0x24, 0xd0, 0xf0,
0x5b, 0xa2, 0x79, 0xba, 0xbf, 0x92, 0x95, 0x37,
0xb0, 0x6e, 0x40, 0x15, 0x91, 0x96, 0x39, 0xd9,
0x4f, 0x57, 0x83, 0x8f, 0xa3, 0x3f, 0xc3, 0xd9,
0x52, 0x59, 0x8d, 0xcd, 0xbb, 0x44, 0xd6, 0x38,
},
},
{
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0x00, 0xff, 0xf0, 0x30, 0x00, 0x1f, 0x00, 0x00,
0xff, 0xff, 0xf0, 0x00, 0x00, 0x38, 0x00, 0x00,
0x00, 0x00, 0x00, 0x02, },
{0x82, 0x46, 0xc9, 0x99, 0x13, 0x71, 0x86, 0x63,
0x2c, 0x5f, 0x9e, 0xdd, 0xf3, 0xb1, 0xb0, 0xe1,
0x76, 0x4c, 0x5e, 0x8b, 0xd0, 0xe0, 0xd8, 0xa5,
0x54, 0xb9, 0xcb, 0x77, 0xe8, 0x0e, 0xd8, 0x66,
0x0b, 0xc1, 0xcb, 0x17, 0xac, 0x7d, 0x84, 0x5b,
0xe4, 0x0a, 0x7a, 0x02, 0x2d, 0x33, 0x06, 0xf1,
0x16, 0xae, 0x9f, 0x81, 0xfe, 0xa6, 0x59, 0x47,
},
},
{
{0x7f, 0xff, 0x80, 0x00, 0x00, 0x00, 0x00, 0x00,
0x07, 0xff, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
0x00, 0x00, 0xff, 0xfe, 0x08, 0x00, 0x00, 0x1f,
0xf0, 0x00, 0x1f, 0xff, },
{0x66, 0x70, 0xc2, 0x0a, 0xfc, 0xce, 0xae, 0xa6,
0x72, 0xc9, 0x7f, 0x75, 0xe2, 0xe9, 0xdd, 0x5c,
0x84, 0x60, 0xe5, 0x4b, 0xb3, 0x85, 0x38, 0xeb,
0xb4, 0xbd, 0x30, 0xeb, 0xf2, 0x80, 0xd8, 0x00,
0x8d, 0x07, 0xa4, 0xca, 0xf5, 0x42, 0x71, 0xf9,
0x93, 0x52, 0x7d, 0x46, 0xff, 0x3f, 0xf4, 0x6f,
0xd1, 0x19, 0x0a, 0x3f, 0x1f, 0xaa, 0x4f, 0x74,
},
},
{
{0x00, 0x00, 0x7f, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xc0, 0x00, 0x07, 0xff, 0xff, 0xe0, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xf8, 0x00, 0xff,
0xff, 0xff, 0xff, 0xff, },
{0x00, 0x0e, 0xca, 0x93, 0x42, 0x47, 0x42, 0x5c,
0xfd, 0x94, 0x9b, 0x79, 0x5c, 0xb5, 0xce, 0x1e,
0xff, 0x40, 0x15, 0x50, 0x38, 0x6e, 0x28, 0xd1,
0xa4, 0xc5, 0xa8, 0xeb, 0xd4, 0xc0, 0x10, 0x40,
0xdb, 0xa1, 0x96, 0x28, 0x93, 0x1b, 0xc8, 0x85,
0x53, 0x70, 0x31, 0x7c, 0x72, 0x2c, 0xbd, 0x9c,
0xa6, 0x15, 0x69, 0x85, 0xf1, 0xc2, 0xe9, 0xce,
},
},
{
{0x7f, 0xff, 0xfc, 0x03, 0xff, 0x80, 0x7f, 0xff,
0xe0, 0x00, 0x1f, 0xff, 0xff, 0x80, 0x0f, 0xff,
0x80, 0x00, 0x01, 0xff, 0xff, 0x00, 0x01, 0xff,
0xff, 0xfe, 0x00, 0x1f, },
{0xef, 0x35, 0x3b, 0xf5, 0xc7, 0x3c, 0xd5, 0x51,
0xb9, 0x6d, 0x59, 0x6f, 0xbc, 0x9a, 0x67, 0xf1,
0x6d, 0x61, 0xdd, 0x9f, 0xe5, 0x6a, 0xf1, 0x9d,
0xe1, 0xfb, 0xa9, 0xcd, 0x21, 0x77, 0x1b, 0x9c,
0xdc, 0xe3, 0xe8, 0x43, 0x0c, 0x09, 0xb3, 0x83,
0x8b, 0xe7, 0x0b, 0x48, 0xc2, 0x1e, 0x15, 0xbc,
0x09, 0xee, 0x1f, 0x2d, 0x79, 0x45, 0xb9, 0x1f,
},
},
{
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0xff, 0xff, 0x01, 0xff, 0xfe, 0x03, 0xff, 0xfe,
0x40, 0x00, 0x38, 0x00, 0x07, 0xe0, 0x00, 0x3f,
0xfe, 0x00, 0x00, 0x00, },
{0x40, 0x36, 0x05, 0x2a, 0x30, 0x91, 0xeb, 0x48,
0x10, 0x46, 0xad, 0x32, 0x89, 0xc9, 0x5d, 0x3a,
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0x38, 0xa2, 0xb9, 0x6f, 0x81, 0x25, 0x86, 0xa9,
0xd5, 0x9d, 0x41, 0x36, 0x03, 0x5d, 0x9c, 0x85,
0x3a, 0x5b, 0xf2, 0xe1, 0xc8, 0x6a, 0x49, 0x93,
},
},
{
{0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0x16, 0xa2,
0xe0, 0xb8, 0xf0, 0x3e, 0x13, 0xdd, 0x29, 0x45,
0x5c, 0x5c, 0x2a, 0x29, },
{0xfc, 0xc7, 0xf2, 0xb4, 0x5d, 0xf1, 0xcd, 0x5a,
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0xfe, 0x78, 0x24, 0x55, 0xf2, 0xa2, 0x8e, 0xef,
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0x74, 0xf2, 0xc6, 0xb2, 0xc8, 0x95, 0xaa, 0x8c,
0x24, 0x94, 0x1f, 0x40, 0xd8, 0xb8, 0x0a, 0xd1,
},
},
{
{0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0x16, 0xa2,
0xe0, 0xb8, 0xf0, 0x3e, 0x13, 0xdd, 0x29, 0x45,
0x5c, 0x5c, 0x2a, 0x2a, },
{0xa1, 0xe8, 0x1c, 0x04, 0xf3, 0x0c, 0xe2, 0x01,
0xc7, 0xc9, 0xac, 0xe7, 0x85, 0xed, 0x44, 0xcc,
0x33, 0xb4, 0x55, 0xa0, 0x22, 0xf2, 0xac, 0xdb,
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0x24, 0xf6, 0x38, 0xf5, 0x33, 0xda, 0xc6, 0xe2,
0xb6, 0xd0, 0x1d, 0xa3, 0xb5, 0xe7, 0xf4, 0x54,
0x29, 0x31, 0x5c, 0xa9, 0x3f, 0xb8, 0xe6, 0x34,
},
},
{
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0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0x16, 0xa2,
0xe0, 0xb8, 0xf0, 0x3e, 0x13, 0xdd, 0x29, 0x45,
0x5c, 0x5c, 0x2a, 0x2b, },
{0xc9, 0xff, 0x61, 0xb0, 0x40, 0x87, 0x4c, 0x05,
0x68, 0x47, 0x92, 0x16, 0x82, 0x4a, 0x15, 0xea,
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0x24, 0x9e, 0x44, 0xfb, 0xa5, 0x93, 0x1f, 0xff,
},
},
{
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0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0x16, 0xa2,
0xe0, 0xb8, 0xf0, 0x3e, 0x13, 0xdd, 0x29, 0x45,
0x5c, 0x5c, 0x2a, 0x2c, },
{0xb8, 0x35, 0x7c, 0x3a, 0x6c, 0xee, 0xf2, 0x88,
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0x99, 0xf9, 0x59, 0x42, 0xdf, 0x10, 0x82, 0xe4,
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0x1a, 0xe8, 0xb2, 0x49, 0x33, 0xaa, 0xb5, 0xdb,
},
},
{
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0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0x16, 0xa2,
0xe0, 0xb8, 0xf0, 0x3e, 0x13, 0xdd, 0x29, 0x45,
0x5c, 0x5c, 0x2a, 0x2d, },
{0x0b, 0x6e, 0xc4, 0xfe, 0x17, 0x77, 0x38, 0x24,
0x04, 0xef, 0x67, 0x99, 0x97, 0xba, 0x8d, 0x1c,
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0x90, 0xc4, 0xc6, 0x6d, 0xcc, 0x66, 0x2b, 0x9b,
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0x10, 0xbd, 0x6d, 0xdd, 0x0d, 0x13, 0x8d, 0xf2,
0xd0, 0x99, 0xa2, 0x82, 0x15, 0x2a, 0x4b, 0x7f,
},
},
{
{0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0x16, 0xa2,
0xe0, 0xb8, 0xf0, 0x3e, 0x13, 0xdd, 0x29, 0x45,
0x5c, 0x5c, 0x2a, 0x2e, },
{0xba, 0xa4, 0xd8, 0x63, 0x55, 0x11, 0xa7, 0xd2,
0x88, 0xae, 0xbe, 0xed, 0xd1, 0x2c, 0xe5, 0x29,
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0x21, 0x91, 0x6b, 0xf9, 0x68, 0x65, 0xa0, 0xb8,
0xa6, 0x07, 0xf0, 0xb0, 0x4b, 0x13, 0xd1, 0xcb,
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0xca, 0x18, 0x49, 0xef, 0xb9, 0xed, 0x86, 0x78,
},
},
{
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0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0x16, 0xa2,
0xe0, 0xb8, 0xf0, 0x3e, 0x13, 0xdd, 0x29, 0x45,
0x5c, 0x5c, 0x2a, 0x2f, },
{0xa5, 0x36, 0x40, 0xc8, 0x3d, 0xc2, 0x08, 0x60,
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0x63, 0x6b, 0x96, 0x43, 0x70, 0xdc, 0x33, 0x52,
},
},
{
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0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0x16, 0xa2,
0xe0, 0xb8, 0xf0, 0x3e, 0x13, 0xdd, 0x29, 0x45,
0x5c, 0x5c, 0x2a, 0x30, },
{0x34, 0xe8, 0xe1, 0x7a, 0x43, 0x0e, 0x43, 0x28,
0x97, 0x93, 0xc3, 0x83, 0xfa, 0xc9, 0x77, 0x42,
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0xc8, 0x2c, 0x82, 0xcb, 0xc9, 0x28, 0x01, 0x46,
0x9c, 0x8d, 0xb0, 0x20, 0x4f, 0x13, 0x48, 0x9a,
},
},
{
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0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0x16, 0xa2,
0xe0, 0xb8, 0xf0, 0x3e, 0x13, 0xdd, 0x29, 0x45,
0x5c, 0x5c, 0x2a, 0x31, },
{0x6e, 0x31, 0xee, 0x1d, 0xc1, 0x37, 0xf8, 0x1b,
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0x72, 0x5f, 0x94, 0xc3, 0x41, 0x0d, 0xdb, 0x56,
0xc5, 0x4e, 0x16, 0x1f, 0xc9, 0xef, 0x62, 0xee,
},
},
{
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0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0x16, 0xa2,
0xe0, 0xb8, 0xf0, 0x3e, 0x13, 0xdd, 0x29, 0x45,
0x5c, 0x5c, 0x2a, 0x32, },
{0xef, 0x53, 0xb6, 0x29, 0x4a, 0xca, 0x43, 0x1f,
0x0f, 0x3c, 0x22, 0xdc, 0x82, 0xeb, 0x90, 0x50,
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0x44, 0x8e, 0x50, 0x7c, 0xdf, 0x4a, 0xef, 0xff,
0xbf, 0x6d, 0x16, 0x99, 0xc9, 0x30, 0x48, 0x1c,
0xd1, 0x02, 0x12, 0x7c, 0x9a, 0x3d, 0x99, 0x20,
0x48, 0xab, 0x05, 0x92, 0x9b, 0x6e, 0x59, 0x27,
},
},
{
{0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0x16, 0xa2,
0xe0, 0xb8, 0xf0, 0x3e, 0x13, 0xdd, 0x29, 0x45,
0x5c, 0x5c, 0x2a, 0x33, },
{0xae, 0xa9, 0xe1, 0x7a, 0x30, 0x65, 0x17, 0xeb,
0x89, 0x15, 0x2a, 0xa7, 0x09, 0x6d, 0x2c, 0x38,
0x1e, 0xc8, 0x13, 0xc5, 0x1a, 0xa8, 0x80, 0xe7,
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0x4c, 0xc8, 0x1f, 0x5a, 0xde, 0x49, 0x34, 0x5e,
0x54, 0x1b, 0x4d, 0x4b, 0x5c, 0x1a, 0xdb, 0x3e,
0xb5, 0xc0, 0x1c, 0x14, 0xee, 0x94, 0x9a, 0xa2,
},
},
{
{0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0x16, 0xa2,
0xe0, 0xb8, 0xf0, 0x3e, 0x13, 0xdd, 0x29, 0x45,
0x5c, 0x5c, 0x2a, 0x34, },
{0x2f, 0xdc, 0xcc, 0xfe, 0xe7, 0x20, 0xa7, 0x7e,
0xf6, 0xcb, 0x3b, 0xfb, 0xb4, 0x47, 0xf9, 0x38,
0x31, 0x17, 0xe3, 0xda, 0xa4, 0xa0, 0x7e, 0x36,
0xed, 0x15, 0xf7, 0x8d, 0xc8, 0xe8, 0xcd, 0x1b,
0x0b, 0xe4, 0x0b, 0x08, 0x77, 0xcf, 0xca, 0x19,
0x58, 0x60, 0x31, 0x22, 0xf1, 0xe6, 0x91, 0x4f,
0x84, 0xb7, 0xe8, 0xe9, 0x68, 0xae, 0x8b, 0x9e,
},
},
{
{0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0x16, 0xa2,
0xe0, 0xb8, 0xf0, 0x3e, 0x13, 0xdd, 0x29, 0x45,
0x5c, 0x5c, 0x2a, 0x35, },
{0x85, 0x8e, 0x6f, 0x9c, 0xc6, 0xc1, 0x2c, 0x31,
0xf5, 0xdf, 0x12, 0x4a, 0xa7, 0x77, 0x67, 0xb0,
0x5c, 0x8b, 0xc0, 0x21, 0xbd, 0x68, 0x3d, 0x2b,
0x55, 0x57, 0x15, 0x50, 0xfb, 0x92, 0x32, 0xc1,
0x5a, 0x3b, 0xc7, 0x67, 0x3a, 0x3a, 0x03, 0xb0,
0x25, 0x38, 0x24, 0xc5, 0x3d, 0x0f, 0xd1, 0x41,
0x1b, 0x1c, 0xab, 0xe2, 0xe1, 0x87, 0xfb, 0x87,
},
},
{
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0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0x16, 0xa2,
0xe0, 0xb8, 0xf0, 0x3e, 0x13, 0xdd, 0x29, 0x45,
0x5c, 0x5c, 0x2a, 0x36, },
{0xdb, 0x2f, 0x6b, 0xe6, 0x30, 0xe2, 0x46, 0xa5,
0xcf, 0x7d, 0x99, 0xb8, 0x51, 0x94, 0xb1, 0x23,
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0x4e, 0x9c, 0x10, 0x72, 0xac, 0x48, 0xea, 0x2e,
0x74, 0x4b, 0x1b, 0x7f, 0x72, 0xfd, 0x46, 0x9e,
},
},
{
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0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0x16, 0xa2,
0xe0, 0xb8, 0xf0, 0x3e, 0x13, 0xdd, 0x29, 0x45,
0x5c, 0x5c, 0x2a, 0x37, },
{0x1f, 0x24, 0x83, 0xf8, 0x25, 0x72, 0x25, 0x1f,
0xca, 0x97, 0x5f, 0xea, 0x40, 0xdb, 0x82, 0x1d,
0xf8, 0xad, 0x82, 0xa3, 0xc0, 0x02, 0xee, 0x6c,
0x57, 0x11, 0x24, 0x08, 0x76, 0x05, 0x0f, 0x33,
0x48, 0xaf, 0x26, 0x64, 0xaa, 0xc3, 0xa8, 0xb0,
0x52, 0x81, 0x30, 0x4e, 0xbc, 0x7a, 0x79, 0x14,
0xc6, 0xad, 0x50, 0xa4, 0xb4, 0xea, 0xc3, 0x83,
},
},
{
{0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0x16, 0xa2,
0xe0, 0xb8, 0xf0, 0x3e, 0x13, 0xdd, 0x29, 0x45,
0x5c, 0x5c, 0x2a, 0x38, },
{0x31, 0xc4, 0x9a, 0xe7, 0x5b, 0xce, 0x78, 0x07,
0xcd, 0xff, 0x22, 0x05, 0x5d, 0x94, 0xee, 0x90,
0x21, 0xfe, 0xdb, 0xb5, 0xab, 0x51, 0xc5, 0x75,
0x26, 0xf0, 0x11, 0xaa, 0xd8, 0x17, 0x40, 0x0e,
0x8b, 0xa9, 0xca, 0x13, 0xa4, 0x5f, 0x36, 0x0e,
0x3d, 0x12, 0x1e, 0xaa, 0xeb, 0x39, 0xaf, 0x82,
0xd6, 0x00, 0x1c, 0x81, 0x86, 0xf5, 0xf8, 0x66,
},
},
{
{0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0x16, 0xa2,
0xe0, 0xb8, 0xf0, 0x3e, 0x13, 0xdd, 0x29, 0x45,
0x5c, 0x5c, 0x2a, 0x39, },
{0xae, 0x99, 0xfe, 0xeb, 0xb5, 0xd2, 0x69, 0x45,
0xb5, 0x48, 0x92, 0x09, 0x2a, 0x8a, 0xee, 0x02,
0x91, 0x29, 0x30, 0xfa, 0x41, 0xcd, 0x11, 0x4e,
0x40, 0x44, 0x73, 0x01, 0xfb, 0x7d, 0xa7, 0xf5,
0xf1, 0x3a, 0x43, 0xb8, 0x17, 0x74, 0x37, 0x3c,
0x87, 0x9c, 0xd3, 0x2d, 0x69, 0x34, 0xc0, 0x5f,
0xa7, 0x58, 0xee, 0xb1, 0x4f, 0xcf, 0xab, 0x38,
},
},
{
{0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0x16, 0xa2,
0xe0, 0xb8, 0xf0, 0x3e, 0x13, 0xdd, 0x29, 0x45,
0x5c, 0x5c, 0x2a, 0x3a, },
{0xdf, 0x1b, 0x1d, 0x66, 0xa5, 0x51, 0xd0, 0xd3,
0x1e, 0xff, 0x82, 0x25, 0x58, 0xb9, 0xd2, 0xcc,
0x75, 0xc2, 0x18, 0x02, 0x79, 0xfe, 0x0d, 0x08,
0xfd, 0x89, 0x6d, 0x04, 0x5c, 0x08, 0x0f, 0xc3,
0x52, 0x2f, 0x41, 0xbb, 0xb3, 0xf5, 0x5a, 0x97,
0xcf, 0xec, 0xf2, 0x1f, 0x88, 0x2c, 0xe8, 0xcb,
0xb1, 0xe5, 0x0c, 0xa6, 0xe6, 0x7e, 0x56, 0xdc,
},
},
{
{0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0x16, 0xa2,
0xe0, 0xb8, 0xf0, 0x3e, 0x13, 0xdd, 0x29, 0x45,
0x5c, 0x5c, 0x2a, 0x3b, },
{0x70, 0x6a, 0x46, 0xdc, 0x76, 0xdc, 0xb7, 0x67,
0x98, 0xe6, 0x0e, 0x6d, 0x89, 0x47, 0x47, 0x88,
0xd1, 0x6d, 0xc1, 0x80, 0x32, 0xd2, 0x68, 0xfd,
0x1a, 0x70, 0x4f, 0xa6, 0xe3, 0xd4, 0x89, 0x58,
0x43, 0xda, 0x18, 0x8f, 0xd5, 0x8f, 0xb0, 0x56,
0x79, 0x76, 0xd7, 0xb5, 0x03, 0x59, 0xd6, 0xb7,
0x85, 0x30, 0xc8, 0xf6, 0x2d, 0x1b, 0x17, 0x46,
},
},
{
{0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0x16, 0xa2,
0xe0, 0xb8, 0xf0, 0x3e, 0x13, 0xdd, 0x29, 0x45,
0x5c, 0x5c, 0x2a, 0x3c, },
{0xb7, 0x0e, 0x0c, 0xbd, 0x6b, 0xb4, 0xbf, 0x7f,
0x32, 0x13, 0x90, 0xb9, 0x4a, 0x03, 0xc1, 0xd3,
0x56, 0xc2, 0x11, 0x22, 0x34, 0x32, 0x80, 0xd6,
0x11, 0x5c, 0x1d, 0x21, 0x42, 0xc8, 0x9c, 0x77,
0x4a, 0x08, 0xdc, 0x04, 0xb3, 0xdd, 0x20, 0x19,
0x32, 0xbc, 0x8a, 0x5e, 0xa5, 0xf8, 0xb8, 0x9b,
0xbb, 0x2a, 0x7e, 0x66, 0x7a, 0xff, 0x81, 0xcd,
},
},
};
TEST(P224, ExternalToInternalAndBack) {
Point point;
EXPECT_TRUE(point.SetFromString(base::StringPiece(
reinterpret_cast<const char *>(kBasePointExternal),
sizeof(kBasePointExternal))));
const std::string external = point.ToString();
ASSERT_EQ(external.size(), 56u);
EXPECT_EQ(0, memcmp(external.data(), kBasePointExternal,
sizeof(kBasePointExternal)));
}
TEST(P224, ScalarBaseMult) {
Point point;
for (size_t i = 0; i < base::size(kNISTTestVectors); i++) {
p224::ScalarBaseMult(kNISTTestVectors[i].scalar, &point);
const std::string external = point.ToString();
ASSERT_EQ(external.size(), 56u);
EXPECT_EQ(0, memcmp(external.data(), kNISTTestVectors[i].affine,
external.size()));
}
}
TEST(P224, Addition) {
Point a, b, minus_b, sum, a_again;
ASSERT_TRUE(a.SetFromString(base::StringPiece(
reinterpret_cast<const char *>(kNISTTestVectors[10].affine), 56)));
ASSERT_TRUE(b.SetFromString(base::StringPiece(
reinterpret_cast<const char *>(kNISTTestVectors[11].affine), 56)));
p224::Negate(b, &minus_b);
p224::Add(a, b, &sum);
EXPECT_NE(0, memcmp(&sum, &a, sizeof(sum)));
p224::Add(minus_b, sum, &a_again);
EXPECT_EQ(a_again.ToString(), a.ToString());
}
TEST(P224, Infinity) {
char zeros[56];
memset(zeros, 0, sizeof(zeros));
// Test that x^0 = ∞.
Point a;
p224::ScalarBaseMult(reinterpret_cast<const uint8_t*>(zeros), &a);
EXPECT_EQ(0, memcmp(zeros, a.ToString().data(), sizeof(zeros)));
// We shouldn't allow ∞ to be imported.
EXPECT_FALSE(a.SetFromString(std::string(zeros, sizeof(zeros))));
}
} // namespace crypto
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