cryptonite-0.30: cbits/p256/p256.c
/*
* Copyright 2013 The Android Open Source Project
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// This is an implementation of the P256 elliptic curve group. It's written to
// be portable and still constant-time.
//
// WARNING: Implementing these functions in a constant-time manner is far from
// obvious. Be careful when touching this code.
//
// See http://www.imperialviolet.org/2010/12/04/ecc.html ([1]) for background.
#include <assert.h>
#include <stdint.h>
#include <string.h>
#include <stdio.h>
#include "p256/p256.h"
const cryptonite_p256_int cryptonite_SECP256r1_n = // curve order
{{P256_LITERAL(0xfc632551, 0xf3b9cac2), P256_LITERAL(0xa7179e84, 0xbce6faad),
P256_LITERAL(-1, -1), P256_LITERAL(0, -1)}};
const cryptonite_p256_int cryptonite_SECP256r1_p = // curve field size
{{P256_LITERAL(-1, -1), P256_LITERAL(-1, 0),
P256_LITERAL(0, 0), P256_LITERAL(1, -1) }};
const cryptonite_p256_int cryptonite_SECP256r1_b = // curve b
{{P256_LITERAL(0x27d2604b, 0x3bce3c3e), P256_LITERAL(0xcc53b0f6, 0x651d06b0),
P256_LITERAL(0x769886bc, 0xb3ebbd55), P256_LITERAL(0xaa3a93e7, 0x5ac635d8)}};
void cryptonite_p256_init(cryptonite_p256_int* a) {
memset(a, 0, sizeof(*a));
}
void cryptonite_p256_clear(cryptonite_p256_int* a) { cryptonite_p256_init(a); }
int cryptonite_p256_get_bit(const cryptonite_p256_int* scalar, int bit) {
return (P256_DIGIT(scalar, bit / P256_BITSPERDIGIT)
>> (bit & (P256_BITSPERDIGIT - 1))) & 1;
}
int cryptonite_p256_is_zero(const cryptonite_p256_int* a) {
cryptonite_p256_digit result = 0;
int i = 0;
for (i = 0; i < P256_NDIGITS; ++i) result |= P256_DIGIT(a, i);
return result == 0;
}
// top, c[] += a[] * b
// Returns new top
static cryptonite_p256_digit mulAdd(const cryptonite_p256_int* a,
cryptonite_p256_digit b,
cryptonite_p256_digit top,
cryptonite_p256_digit* c) {
int i;
cryptonite_p256_ddigit carry = 0;
for (i = 0; i < P256_NDIGITS; ++i) {
carry += *c;
carry += (cryptonite_p256_ddigit)P256_DIGIT(a, i) * b;
*c++ = (cryptonite_p256_digit)carry;
carry >>= P256_BITSPERDIGIT;
}
return top + (cryptonite_p256_digit)carry;
}
// top, c[] -= top_a, a[]
static cryptonite_p256_digit subTop(cryptonite_p256_digit top_a,
const cryptonite_p256_digit* a,
cryptonite_p256_digit top_c,
cryptonite_p256_digit* c) {
int i;
cryptonite_p256_sddigit borrow = 0;
for (i = 0; i < P256_NDIGITS; ++i) {
borrow += *c;
borrow -= *a++;
*c++ = (cryptonite_p256_digit)borrow;
borrow >>= P256_BITSPERDIGIT;
}
borrow += top_c;
borrow -= top_a;
top_c = (cryptonite_p256_digit)borrow;
assert((borrow >> P256_BITSPERDIGIT) == 0);
return top_c;
}
// top, c[] -= MOD[] & mask (0 or -1)
// returns new top.
static cryptonite_p256_digit subM(const cryptonite_p256_int* MOD,
cryptonite_p256_digit top,
cryptonite_p256_digit* c,
cryptonite_p256_digit mask) {
int i;
cryptonite_p256_sddigit borrow = 0;
for (i = 0; i < P256_NDIGITS; ++i) {
borrow += *c;
borrow -= P256_DIGIT(MOD, i) & mask;
*c++ = (cryptonite_p256_digit)borrow;
borrow >>= P256_BITSPERDIGIT;
}
return top + (cryptonite_p256_digit)borrow;
}
// top, c[] += MOD[] & mask (0 or -1)
// returns new top.
static cryptonite_p256_digit addM(const cryptonite_p256_int* MOD,
cryptonite_p256_digit top,
cryptonite_p256_digit* c,
cryptonite_p256_digit mask) {
int i;
cryptonite_p256_ddigit carry = 0;
for (i = 0; i < P256_NDIGITS; ++i) {
carry += *c;
carry += P256_DIGIT(MOD, i) & mask;
*c++ = (cryptonite_p256_digit)carry;
carry >>= P256_BITSPERDIGIT;
}
return top + (cryptonite_p256_digit)carry;
}
// c = a * b mod MOD. c can be a and/or b.
void cryptonite_p256_modmul(const cryptonite_p256_int* MOD,
const cryptonite_p256_int* a,
const cryptonite_p256_digit top_b,
const cryptonite_p256_int* b,
cryptonite_p256_int* c) {
cryptonite_p256_digit tmp[P256_NDIGITS * 2 + 1] = { 0 };
cryptonite_p256_digit top = 0;
int i;
// Multiply/add into tmp.
for (i = 0; i < P256_NDIGITS; ++i) {
if (i) tmp[i + P256_NDIGITS - 1] = top;
top = mulAdd(a, P256_DIGIT(b, i), 0, tmp + i);
}
// Multiply/add top digit
tmp[i + P256_NDIGITS - 1] = top;
top = mulAdd(a, top_b, 0, tmp + i);
// Reduce tmp, digit by digit.
for (; i >= 0; --i) {
cryptonite_p256_digit reducer[P256_NDIGITS] = { 0 };
cryptonite_p256_digit top_reducer;
// top can be any value at this point.
// Guestimate reducer as top * MOD, since msw of MOD is -1.
top_reducer = mulAdd(MOD, top, 0, reducer);
#if P256_BITSPERDIGIT > 32
// Correction when msw of MOD has only high 32 bits set
top_reducer += mulAdd(MOD, top >> 32, 0, reducer);
#endif
// Subtract reducer from top | tmp.
top = subTop(top_reducer, reducer, top, tmp + i);
// top is now either 0 or 1. Make it 0, fixed-timing.
assert(top <= 1);
top = subM(MOD, top, tmp + i, ~(top - 1));
assert(top == 0);
// We have now reduced the top digit off tmp. Fetch new top digit.
top = tmp[i + P256_NDIGITS - 1];
}
// tmp might still be larger than MOD, yet same bit length.
// Make sure it is less, fixed-timing.
addM(MOD, 0, tmp, subM(MOD, 0, tmp, -1));
memcpy(c, tmp, P256_NBYTES);
}
int cryptonite_p256_is_odd(const cryptonite_p256_int* a) { return P256_DIGIT(a, 0) & 1; }
int cryptonite_p256_is_even(const cryptonite_p256_int* a) { return !(P256_DIGIT(a, 0) & 1); }
cryptonite_p256_digit cryptonite_p256_shl(const cryptonite_p256_int* a, int n, cryptonite_p256_int* b) {
int i;
cryptonite_p256_digit top = P256_DIGIT(a, P256_NDIGITS - 1);
n %= P256_BITSPERDIGIT;
for (i = P256_NDIGITS - 1; i > 0; --i) {
cryptonite_p256_digit accu = (P256_DIGIT(a, i) << n);
accu |= (P256_DIGIT(a, i - 1) >> (P256_BITSPERDIGIT - n));
P256_DIGIT(b, i) = accu;
}
P256_DIGIT(b, i) = (P256_DIGIT(a, i) << n);
top = (cryptonite_p256_digit)((((cryptonite_p256_ddigit)top) << n) >> P256_BITSPERDIGIT);
return top;
}
void cryptonite_p256_shr(const cryptonite_p256_int* a, int n, cryptonite_p256_int* b) {
int i;
n %= P256_BITSPERDIGIT;
for (i = 0; i < P256_NDIGITS - 1; ++i) {
cryptonite_p256_digit accu = (P256_DIGIT(a, i) >> n);
accu |= (P256_DIGIT(a, i + 1) << (P256_BITSPERDIGIT - n));
P256_DIGIT(b, i) = accu;
}
P256_DIGIT(b, i) = (P256_DIGIT(a, i) >> n);
}
static void cryptonite_p256_shr1(const cryptonite_p256_int* a, int highbit, cryptonite_p256_int* b) {
int i;
for (i = 0; i < P256_NDIGITS - 1; ++i) {
cryptonite_p256_digit accu = (P256_DIGIT(a, i) >> 1);
accu |= (P256_DIGIT(a, i + 1) << (P256_BITSPERDIGIT - 1));
P256_DIGIT(b, i) = accu;
}
P256_DIGIT(b, i) = (P256_DIGIT(a, i) >> 1) |
(((cryptonite_p256_sdigit) highbit) << (P256_BITSPERDIGIT - 1));
}
// Return -1, 0, 1 for a < b, a == b or a > b respectively.
int cryptonite_p256_cmp(const cryptonite_p256_int* a, const cryptonite_p256_int* b) {
int i;
cryptonite_p256_sddigit borrow = 0;
cryptonite_p256_digit notzero = 0;
for (i = 0; i < P256_NDIGITS; ++i) {
borrow += (cryptonite_p256_sddigit)P256_DIGIT(a, i) - P256_DIGIT(b, i);
// Track whether any result digit is ever not zero.
// Relies on !!(non-zero) evaluating to 1, e.g., !!(-1) evaluating to 1.
notzero |= !!((cryptonite_p256_digit)borrow);
borrow >>= P256_BITSPERDIGIT;
}
return (int)borrow | notzero;
}
// c = a - b. Returns borrow: 0 or -1.
int cryptonite_p256_sub(const cryptonite_p256_int* a, const cryptonite_p256_int* b, cryptonite_p256_int* c) {
int i;
cryptonite_p256_sddigit borrow = 0;
for (i = 0; i < P256_NDIGITS; ++i) {
borrow += (cryptonite_p256_sddigit)P256_DIGIT(a, i) - P256_DIGIT(b, i);
if (c) P256_DIGIT(c, i) = (cryptonite_p256_digit)borrow;
borrow >>= P256_BITSPERDIGIT;
}
return (int)borrow;
}
// c = a + b. Returns carry: 0 or 1.
int cryptonite_p256_add(const cryptonite_p256_int* a, const cryptonite_p256_int* b, cryptonite_p256_int* c) {
int i;
cryptonite_p256_ddigit carry = 0;
for (i = 0; i < P256_NDIGITS; ++i) {
carry += (cryptonite_p256_ddigit)P256_DIGIT(a, i) + P256_DIGIT(b, i);
if (c) P256_DIGIT(c, i) = (cryptonite_p256_digit)carry;
carry >>= P256_BITSPERDIGIT;
}
return (int)carry;
}
// b = a + d. Returns carry, 0 or 1.
int cryptonite_p256_add_d(const cryptonite_p256_int* a, cryptonite_p256_digit d, cryptonite_p256_int* b) {
int i;
cryptonite_p256_ddigit carry = d;
for (i = 0; i < P256_NDIGITS; ++i) {
carry += (cryptonite_p256_ddigit)P256_DIGIT(a, i);
if (b) P256_DIGIT(b, i) = (cryptonite_p256_digit)carry;
carry >>= P256_BITSPERDIGIT;
}
return (int)carry;
}
// b = 1/a mod MOD, binary euclid.
void cryptonite_p256_modinv_vartime(const cryptonite_p256_int* MOD,
const cryptonite_p256_int* a,
cryptonite_p256_int* b) {
cryptonite_p256_int R = P256_ZERO;
cryptonite_p256_int S = P256_ONE;
cryptonite_p256_int U = *MOD;
cryptonite_p256_int V = *a;
for (;;) {
if (cryptonite_p256_is_even(&U)) {
cryptonite_p256_shr1(&U, 0, &U);
if (cryptonite_p256_is_even(&R)) {
cryptonite_p256_shr1(&R, 0, &R);
} else {
// R = (R+MOD)/2
cryptonite_p256_shr1(&R, cryptonite_p256_add(&R, MOD, &R), &R);
}
} else if (cryptonite_p256_is_even(&V)) {
cryptonite_p256_shr1(&V, 0, &V);
if (cryptonite_p256_is_even(&S)) {
cryptonite_p256_shr1(&S, 0, &S);
} else {
// S = (S+MOD)/2
cryptonite_p256_shr1(&S, cryptonite_p256_add(&S, MOD, &S) , &S);
}
} else { // U,V both odd.
if (!cryptonite_p256_sub(&V, &U, NULL)) {
cryptonite_p256_sub(&V, &U, &V);
if (cryptonite_p256_sub(&S, &R, &S)) cryptonite_p256_add(&S, MOD, &S);
if (cryptonite_p256_is_zero(&V)) break; // done.
} else {
cryptonite_p256_sub(&U, &V, &U);
if (cryptonite_p256_sub(&R, &S, &R)) cryptonite_p256_add(&R, MOD, &R);
}
}
}
cryptonite_p256_mod(MOD, &R, b);
}
void cryptonite_p256_mod(const cryptonite_p256_int* MOD,
const cryptonite_p256_int* in,
cryptonite_p256_int* out) {
if (out != in) *out = *in;
addM(MOD, 0, P256_DIGITS(out), subM(MOD, 0, P256_DIGITS(out), -1));
}
// Verify y^2 == x^3 - 3x + b mod p
// and 0 < x < p and 0 < y < p
int cryptonite_p256_is_valid_point(const cryptonite_p256_int* x, const cryptonite_p256_int* y) {
cryptonite_p256_int y2, x3;
if (cryptonite_p256_cmp(&cryptonite_SECP256r1_p, x) <= 0 ||
cryptonite_p256_cmp(&cryptonite_SECP256r1_p, y) <= 0 ||
cryptonite_p256_is_zero(x) ||
cryptonite_p256_is_zero(y)) return 0;
cryptonite_p256_modmul(&cryptonite_SECP256r1_p, y, 0, y, &y2); // y^2
cryptonite_p256_modmul(&cryptonite_SECP256r1_p, x, 0, x, &x3); // x^2
cryptonite_p256_modmul(&cryptonite_SECP256r1_p, x, 0, &x3, &x3); // x^3
if (cryptonite_p256_sub(&x3, x, &x3)) cryptonite_p256_add(&x3, &cryptonite_SECP256r1_p, &x3); // x^3 - x
if (cryptonite_p256_sub(&x3, x, &x3)) cryptonite_p256_add(&x3, &cryptonite_SECP256r1_p, &x3); // x^3 - 2x
if (cryptonite_p256_sub(&x3, x, &x3)) cryptonite_p256_add(&x3, &cryptonite_SECP256r1_p, &x3); // x^3 - 3x
if (cryptonite_p256_add(&x3, &cryptonite_SECP256r1_b, &x3)) // x^3 - 3x + b
cryptonite_p256_sub(&x3, &cryptonite_SECP256r1_p, &x3);
return cryptonite_p256_cmp(&y2, &x3) == 0;
}
void cryptonite_p256_from_bin(const uint8_t src[P256_NBYTES], cryptonite_p256_int* dst) {
int i, n;
const uint8_t* p = &src[0];
for (i = P256_NDIGITS - 1; i >= 0; --i) {
cryptonite_p256_digit dig = 0;
n = P256_BITSPERDIGIT;
while (n > 0) {
n -= 8;
dig |= ((cryptonite_p256_digit) *(p++)) << n;
}
P256_DIGIT(dst, i) = dig;
}
}
void cryptonite_p256_to_bin(const cryptonite_p256_int* src, uint8_t dst[P256_NBYTES])
{
int i, n;
uint8_t* p = &dst[0];
for (i = P256_NDIGITS -1; i >= 0; --i) {
const cryptonite_p256_digit dig = P256_DIGIT(src, i);
n = P256_BITSPERDIGIT;
while (n > 0) {
n -= 8;
*(p++) = dig >> n;
}
}
}
/*
"p256e" functions are not part of the original source
*/
#define MSB_COMPLEMENT(x) (((x) >> (P256_BITSPERDIGIT - 1)) - 1)
// c = a + b mod MOD
void cryptonite_p256e_modadd(const cryptonite_p256_int* MOD, const cryptonite_p256_int* a, const cryptonite_p256_int* b, cryptonite_p256_int* c) {
assert(c); /* avoid repeated checks inside inlined cryptonite_p256_add */
cryptonite_p256_digit top = cryptonite_p256_add(a, b, c);
top = subM(MOD, top, P256_DIGITS(c), -1);
top = subM(MOD, top, P256_DIGITS(c), MSB_COMPLEMENT(top));
addM(MOD, 0, P256_DIGITS(c), top);
}
// c = a - b mod MOD
void cryptonite_p256e_modsub(const cryptonite_p256_int* MOD, const cryptonite_p256_int* a, const cryptonite_p256_int* b, cryptonite_p256_int* c) {
assert(c); /* avoid repeated checks inside inlined cryptonite_p256_sub */
cryptonite_p256_digit top = cryptonite_p256_sub(a, b, c);
top = addM(MOD, top, P256_DIGITS(c), ~MSB_COMPLEMENT(top));
top = subM(MOD, top, P256_DIGITS(c), MSB_COMPLEMENT(top));
addM(MOD, 0, P256_DIGITS(c), top);
}
#define NTH_DOUBLE_THEN_ADD(i, a, nth, b, out) \
cryptonite_p256e_montmul(a, a, out); \
for (i = 1; i < nth; i++) \
cryptonite_p256e_montmul(out, out, out); \
cryptonite_p256e_montmul(out, b, out);
const cryptonite_p256_int cryptonite_SECP256r1_r2 = // r^2 mod n
{{P256_LITERAL(0xBE79EEA2, 0x83244C95), P256_LITERAL(0x49BD6FA6, 0x4699799C),
P256_LITERAL(0x2B6BEC59, 0x2845B239), P256_LITERAL(0xF3D95620, 0x66E12D94)}};
const cryptonite_p256_int cryptonite_SECP256r1_one = {{1}};
// Montgomery multiplication, i.e. c = ab/r mod n with r = 2^256.
// Implementation is adapted from 'sc_montmul' in libdecaf.
static void cryptonite_p256e_montmul(const cryptonite_p256_int* a, const cryptonite_p256_int* b, cryptonite_p256_int* c) {
int i, j, borrow;
cryptonite_p256_digit accum[P256_NDIGITS+1] = {0};
cryptonite_p256_digit hi_carry = 0;
for (i=0; i<P256_NDIGITS; i++) {
cryptonite_p256_digit mand = P256_DIGIT(a, i);
const cryptonite_p256_digit *mier = P256_DIGITS(b);
cryptonite_p256_ddigit chain = 0;
for (j=0; j<P256_NDIGITS; j++) {
chain += ((cryptonite_p256_ddigit)mand)*mier[j] + accum[j];
accum[j] = chain;
chain >>= P256_BITSPERDIGIT;
}
accum[j] = chain;
mand = accum[0] * P256_MONTGOMERY_FACTOR;
chain = 0;
mier = P256_DIGITS(&cryptonite_SECP256r1_n);
for (j=0; j<P256_NDIGITS; j++) {
chain += (cryptonite_p256_ddigit)mand*mier[j] + accum[j];
if (j) accum[j-1] = chain;
chain >>= P256_BITSPERDIGIT;
}
chain += accum[j];
chain += hi_carry;
accum[j-1] = chain;
hi_carry = chain >> P256_BITSPERDIGIT;
}
memcpy(P256_DIGITS(c), accum, sizeof(*c));
borrow = cryptonite_p256_sub(c, &cryptonite_SECP256r1_n, c);
addM(&cryptonite_SECP256r1_n, 0, P256_DIGITS(c), borrow + hi_carry);
}
// b = 1/a mod n, using Fermat's little theorem.
void cryptonite_p256e_scalar_invert(const cryptonite_p256_int* a, cryptonite_p256_int* b) {
cryptonite_p256_int _1, _10, _11, _101, _111, _1010, _1111;
cryptonite_p256_int _10101, _101010, _101111, x6, x8, x16, x32;
int i;
// Montgomerize
cryptonite_p256e_montmul(a, &cryptonite_SECP256r1_r2, &_1);
// P-256 (secp256r1) Scalar Inversion
// <https://briansmith.org/ecc-inversion-addition-chains-01>
cryptonite_p256e_montmul(&_1 , &_1 , &_10);
cryptonite_p256e_montmul(&_10 , &_1 , &_11);
cryptonite_p256e_montmul(&_10 , &_11 , &_101);
cryptonite_p256e_montmul(&_10 , &_101 , &_111);
cryptonite_p256e_montmul(&_101 , &_101 , &_1010);
cryptonite_p256e_montmul(&_101 , &_1010 , &_1111);
NTH_DOUBLE_THEN_ADD(i, &_1010, 1 , &_1 , &_10101);
cryptonite_p256e_montmul(&_10101 , &_10101 , &_101010);
cryptonite_p256e_montmul(&_101 , &_101010, &_101111);
cryptonite_p256e_montmul(&_10101 , &_101010, &x6);
NTH_DOUBLE_THEN_ADD(i, &x6 , 2 , &_11 , &x8);
NTH_DOUBLE_THEN_ADD(i, &x8 , 8 , &x8 , &x16);
NTH_DOUBLE_THEN_ADD(i, &x16 , 16 , &x16 , &x32);
NTH_DOUBLE_THEN_ADD(i, &x32 , 32+32, &x32 , b);
NTH_DOUBLE_THEN_ADD(i, b , 32, &x32 , b);
NTH_DOUBLE_THEN_ADD(i, b , 6, &_101111, b);
NTH_DOUBLE_THEN_ADD(i, b , 2 + 3, &_111 , b);
NTH_DOUBLE_THEN_ADD(i, b , 2 + 2, &_11 , b);
NTH_DOUBLE_THEN_ADD(i, b , 1 + 4, &_1111 , b);
NTH_DOUBLE_THEN_ADD(i, b , 5, &_10101 , b);
NTH_DOUBLE_THEN_ADD(i, b , 1 + 3, &_101 , b);
NTH_DOUBLE_THEN_ADD(i, b , 3, &_101 , b);
NTH_DOUBLE_THEN_ADD(i, b , 3, &_101 , b);
NTH_DOUBLE_THEN_ADD(i, b , 2 + 3, &_111 , b);
NTH_DOUBLE_THEN_ADD(i, b , 3 + 6, &_101111, b);
NTH_DOUBLE_THEN_ADD(i, b , 2 + 4, &_1111 , b);
NTH_DOUBLE_THEN_ADD(i, b , 1 + 1, &_1 , b);
NTH_DOUBLE_THEN_ADD(i, b , 4 + 1, &_1 , b);
NTH_DOUBLE_THEN_ADD(i, b , 2 + 4, &_1111 , b);
NTH_DOUBLE_THEN_ADD(i, b , 2 + 3, &_111 , b);
NTH_DOUBLE_THEN_ADD(i, b , 1 + 3, &_111 , b);
NTH_DOUBLE_THEN_ADD(i, b , 2 + 3, &_111 , b);
NTH_DOUBLE_THEN_ADD(i, b , 2 + 3, &_101 , b);
NTH_DOUBLE_THEN_ADD(i, b , 1 + 2, &_11 , b);
NTH_DOUBLE_THEN_ADD(i, b , 4 + 6, &_101111, b);
NTH_DOUBLE_THEN_ADD(i, b , 2, &_11 , b);
NTH_DOUBLE_THEN_ADD(i, b , 3 + 2, &_11 , b);
NTH_DOUBLE_THEN_ADD(i, b , 3 + 2, &_11 , b);
NTH_DOUBLE_THEN_ADD(i, b , 2 + 1, &_1 , b);
NTH_DOUBLE_THEN_ADD(i, b , 2 + 5, &_10101 , b);
NTH_DOUBLE_THEN_ADD(i, b , 2 + 4, &_1111 , b);
// Demontgomerize
cryptonite_p256e_montmul(b, &cryptonite_SECP256r1_one, b);
}