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// This is an implementation of the P256 elliptic curve group. It's written to
// be portable 32-bit, although it's 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 "constrainedcrypto/p256.h"
const p256_int SECP256r1_n = // curve order
{{0xfc632551, 0xf3b9cac2, 0xa7179e84, 0xbce6faad, -1, -1, 0, -1}};
const p256_int SECP256r1_p = // curve field size
{{-1, -1, -1, 0, 0, 0, 1, -1 }};
const p256_int SECP256r1_b = // curve b
{{0x27d2604b, 0x3bce3c3e, 0xcc53b0f6, 0x651d06b0,
0x769886bc, 0xb3ebbd55, 0xaa3a93e7, 0x5ac635d8}};
void p256_init(p256_int* a) {
memset(a, 0, sizeof(*a));
}
void p256_clear(p256_int* a) { p256_init(a); }
int p256_get_bit(const p256_int* scalar, int bit) {
return (P256_DIGIT(scalar, bit / P256_BITSPERDIGIT)
>> (bit & (P256_BITSPERDIGIT - 1))) & 1;
}
int p256_is_zero(const p256_int* a) {
int i, result = 0;
for (i = 0; i < P256_NDIGITS; ++i) result |= P256_DIGIT(a, i);
return !result;
}
// top, c[] += a[] * b
// Returns new top
static p256_digit mulAdd(const p256_int* a,
p256_digit b,
p256_digit top,
p256_digit* c) {
int i;
p256_ddigit carry = 0;
for (i = 0; i < P256_NDIGITS; ++i) {
carry += *c;
carry += (p256_ddigit)P256_DIGIT(a, i) * b;
*c++ = (p256_digit)carry;
carry >>= P256_BITSPERDIGIT;
}
return top + (p256_digit)carry;
}
// top, c[] -= top_a, a[]
static p256_digit subTop(p256_digit top_a,
const p256_digit* a,
p256_digit top_c,
p256_digit* c) {
int i;
p256_sddigit borrow = 0;
for (i = 0; i < P256_NDIGITS; ++i) {
borrow += *c;
borrow -= *a++;
*c++ = (p256_digit)borrow;
borrow >>= P256_BITSPERDIGIT;
}
borrow += top_c;
borrow -= top_a;
top_c = (p256_digit)borrow;
assert((borrow >> P256_BITSPERDIGIT) == 0);
return top_c;
}
// top, c[] -= MOD[] & mask (0 or -1)
// returns new top.
static p256_digit subM(const p256_int* MOD,
p256_digit top,
p256_digit* c,
p256_digit mask) {
int i;
p256_sddigit borrow = 0;
for (i = 0; i < P256_NDIGITS; ++i) {
borrow += *c;
borrow -= P256_DIGIT(MOD, i) & mask;
*c++ = (p256_digit)borrow;
borrow >>= P256_BITSPERDIGIT;
}
return top + (p256_digit)borrow;
}
// top, c[] += MOD[] & mask (0 or -1)
// returns new top.
static p256_digit addM(const p256_int* MOD,
p256_digit top,
p256_digit* c,
p256_digit mask) {
int i;
p256_ddigit carry = 0;
for (i = 0; i < P256_NDIGITS; ++i) {
carry += *c;
carry += P256_DIGIT(MOD, i) & mask;
*c++ = (p256_digit)carry;
carry >>= P256_BITSPERDIGIT;
}
return top + (p256_digit)carry;
}
// c = a * b mod MOD. c can be a and/or b.
void p256_modmul(const p256_int* MOD,
const p256_int* a,
const p256_digit top_b,
const p256_int* b,
p256_int* c) {
p256_digit tmp[P256_NDIGITS * 2 + 1] = { 0 };
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) {
p256_digit reducer[P256_NDIGITS] = { 0 };
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);
// 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 p256_is_odd(const p256_int* a) { return P256_DIGIT(a, 0) & 1; }
int p256_is_even(const p256_int* a) { return !(P256_DIGIT(a, 0) & 1); }
p256_digit p256_shl(const p256_int* a, int n, p256_int* b) {
int i;
p256_digit top = P256_DIGIT(a, P256_NDIGITS - 1);
n %= P256_BITSPERDIGIT;
for (i = P256_NDIGITS - 1; i > 0; --i) {
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 = (p256_digit)((((p256_ddigit)top) << n) >> P256_BITSPERDIGIT);
return top;
}
void p256_shr(const p256_int* a, int n, p256_int* b) {
int i;
n %= P256_BITSPERDIGIT;
for (i = 0; i < P256_NDIGITS - 1; ++i) {
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 p256_shr1(const p256_int* a, int highbit, p256_int* b) {
int i;
for (i = 0; i < P256_NDIGITS - 1; ++i) {
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) |
(highbit << (P256_BITSPERDIGIT - 1));
}
// Return -1, 0, 1 for a < b, a == b or a > b respectively.
int p256_cmp(const p256_int* a, const p256_int* b) {
int i;
p256_sddigit borrow = 0;
p256_digit notzero = 0;
for (i = 0; i < P256_NDIGITS; ++i) {
borrow += (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 |= !!((p256_digit)borrow);
borrow >>= P256_BITSPERDIGIT;
}
return (int)borrow | notzero;
}
// c = a - b. Returns borrow: 0 or -1.
int p256_sub(const p256_int* a, const p256_int* b, p256_int* c) {
int i;
p256_sddigit borrow = 0;
for (i = 0; i < P256_NDIGITS; ++i) {
borrow += (p256_sddigit)P256_DIGIT(a, i) - P256_DIGIT(b, i);
if (c) P256_DIGIT(c, i) = (p256_digit)borrow;
borrow >>= P256_BITSPERDIGIT;
}
return (int)borrow;
}
// c = a + b. Returns carry: 0 or 1.
int p256_add(const p256_int* a, const p256_int* b, p256_int* c) {
int i;
p256_ddigit carry = 0;
for (i = 0; i < P256_NDIGITS; ++i) {
carry += (p256_ddigit)P256_DIGIT(a, i) + P256_DIGIT(b, i);
if (c) P256_DIGIT(c, i) = (p256_digit)carry;
carry >>= P256_BITSPERDIGIT;
}
return (int)carry;
}
// b = a + d. Returns carry, 0 or 1.
int p256_add_d(const p256_int* a, p256_digit d, p256_int* b) {
int i;
p256_ddigit carry = d;
for (i = 0; i < P256_NDIGITS; ++i) {
carry += (p256_ddigit)P256_DIGIT(a, i);
if (b) P256_DIGIT(b, i) = (p256_digit)carry;
carry >>= P256_BITSPERDIGIT;
}
return (int)carry;
}
// b = 1/a mod MOD, binary euclid.
void p256_modinv_vartime(const p256_int* MOD,
const p256_int* a,
p256_int* b) {
p256_int R = P256_ZERO;
p256_int S = P256_ONE;
p256_int U = *MOD;
p256_int V = *a;
for (;;) {
if (p256_is_even(&U)) {
p256_shr1(&U, 0, &U);
if (p256_is_even(&R)) {
p256_shr1(&R, 0, &R);
} else {
// R = (R+MOD)/2
p256_shr1(&R, p256_add(&R, MOD, &R), &R);
}
} else if (p256_is_even(&V)) {
p256_shr1(&V, 0, &V);
if (p256_is_even(&S)) {
p256_shr1(&S, 0, &S);
} else {
// S = (S+MOD)/2
p256_shr1(&S, p256_add(&S, MOD, &S) , &S);
}
} else { // U,V both odd.
if (!p256_sub(&V, &U, NULL)) {
p256_sub(&V, &U, &V);
if (p256_sub(&S, &R, &S)) p256_add(&S, MOD, &S);
if (p256_is_zero(&V)) break; // done.
} else {
p256_sub(&U, &V, &U);
if (p256_sub(&R, &S, &R)) p256_add(&R, MOD, &R);
}
}
}
p256_mod(MOD, &R, b);
}
void p256_mod(const p256_int* MOD,
const p256_int* in,
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 p256_is_valid_point(const p256_int* x, const p256_int* y) {
p256_int y2, x3;
if (p256_cmp(&SECP256r1_p, x) <= 0 ||
p256_cmp(&SECP256r1_p, y) <= 0 ||
p256_is_zero(x) ||
p256_is_zero(y)) return 0;
p256_modmul(&SECP256r1_p, y, 0, y, &y2); // y^2
p256_modmul(&SECP256r1_p, x, 0, x, &x3); // x^2
p256_modmul(&SECP256r1_p, x, 0, &x3, &x3); // x^3
if (p256_sub(&x3, x, &x3)) p256_add(&x3, &SECP256r1_p, &x3); // x^3 - x
if (p256_sub(&x3, x, &x3)) p256_add(&x3, &SECP256r1_p, &x3); // x^3 - 2x
if (p256_sub(&x3, x, &x3)) p256_add(&x3, &SECP256r1_p, &x3); // x^3 - 3x
if (p256_add(&x3, &SECP256r1_b, &x3)) // x^3 - 3x + b
p256_sub(&x3, &SECP256r1_p, &x3);
return p256_cmp(&y2, &x3) == 0;
}
void p256_from_bin(const uint8_t src[P256_NBYTES], p256_int* dst) {
int i;
const uint8_t* p = &src[0];
for (i = P256_NDIGITS - 1; i >= 0; --i) {
P256_DIGIT(dst, i) =
(p[0] << 24) |
(p[1] << 16) |
(p[2] << 8) |
p[3];
p += 4;
}
}