mirror of
https://github.com/trezor/trezor-firmware.git
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Added comments to the tricky algorithms.
Added invariants for bn_multiply and bn_inverse. Explain that bn_multiply and bn_fast_mod doesn't work for an arbitrary modulus. The modulus must be close to 2^256.
This commit is contained in:
Jochen Hoenicke
parent
92ab7504b2
commit
7e98c02afd
139
bignum.c
139
bignum.c
@ -177,8 +177,11 @@ void bn_muli(bignum256 *a, uint32_t b)
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a->val[8] += t;
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}
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// x = k * x
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// both inputs and result may be bigger than prime but not bigger than 2 * prime
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// Compute x := k * x (mod prime)
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// both inputs must be smaller than 2 * prime.
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// result is reduced to 0 <= x < 2 * prime
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// This only works for primes between 2^256-2^196 and 2^256.
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// this particular implementation accepts inputs up to 2^263 or 128*prime.
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void bn_multiply(const bignum256 *k, bignum256 *x, const bignum256 *prime)
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{
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int i, j;
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@ -204,11 +207,21 @@ void bn_multiply(const bignum256 *k, bignum256 *x, const bignum256 *prime)
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temp >>= 30;
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}
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res[17] = temp;
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// res = k * x is a normalized number (every limb < 2^30)
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// 0 <= res < 2^526.
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// compute modulo p division is only estimated so this may give result greater than prime but not bigger than 2 * prime
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for (i = 16; i >= 8; i--) {
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// let k = i-8.
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// invariants:
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// res[0..(i+1)] = k * x (mod prime)
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// 0 <= res < 2^(30k + 256) * (2^30 + 1)
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// estimate (res / prime)
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coef = (res[i] >> 16) + (res[i + 1] << 14);
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// substract (coef * prime) from res
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// coef = res / 2^(30k + 256) rounded down
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// 0 <= coef <= 2^30
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// subtract (coef * 2^(30k) * prime) from res
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// note that we unrolled the first iteration
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temp = 0x1000000000000000ull + res[i - 8] - prime->val[0] * (uint64_t)coef;
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res[i - 8] = temp & 0x3FFFFFFF;
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for (j = 1; j < 9; j++) {
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@ -216,6 +229,16 @@ void bn_multiply(const bignum256 *k, bignum256 *x, const bignum256 *prime)
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temp += 0xFFFFFFFC0000000ull + res[i - 8 + j] - prime->val[j] * (uint64_t)coef;
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res[i - 8 + j] = temp & 0x3FFFFFFF;
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}
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// we don't clear res[i+1] but we never read it again.
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// we rely on the fact that prime > 2^256 - 2^196
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// res = oldres - coef*2^(30k) * prime;
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// and
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// coef * 2^(30k + 256) <= oldres < (coef+1) * 2^(30k + 256)
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// Hence, 0 <= res < 2^30k (2^256 + coef * (2^256 - prime))
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// Since coef * (2^256 - prime) < 2^226, we get
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// 0 <= res < 2^(30k + 226) (2^30 + 1)
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// Thus the invariant holds again.
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}
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// store the result
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for (i = 0; i < 9; i++) {
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@ -223,6 +246,8 @@ void bn_multiply(const bignum256 *k, bignum256 *x, const bignum256 *prime)
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}
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}
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// input x can be any normalized number that fits (0 <= x < 2^270).
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// prime must be between (2^256 - 2^196) and 2^256
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// result is smaller than 2*prime
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void bn_fast_mod(bignum256 *x, const bignum256 *prime)
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{
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@ -233,6 +258,7 @@ void bn_fast_mod(bignum256 *x, const bignum256 *prime)
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coef = x->val[8] >> 16;
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if (!coef) return;
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// substract (coef * prime) from x
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// note that we unrolled the first iteration
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temp = 0x1000000000000000ull + x->val[0] - prime->val[0] * (uint64_t)coef;
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x->val[0] = temp & 0x3FFFFFFF;
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for (j = 1; j < 9; j++) {
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@ -246,16 +272,26 @@ void bn_fast_mod(bignum256 *x, const bignum256 *prime)
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// http://en.wikipedia.org/wiki/Quadratic_residue#Prime_or_prime_power_modulus
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void bn_sqrt(bignum256 *x, const bignum256 *prime)
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{
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// this method compute x^1/2 = x^(prime+1)/4
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uint32_t i, j, limb;
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bignum256 res, p;
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bn_zero(&res); res.val[0] = 1;
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// compute p = (prime+1)/4
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memcpy(&p, prime, sizeof(bignum256));
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p.val[0] += 1;
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bn_rshift(&p);
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bn_rshift(&p);
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for (i = 0; i < 9; i++) {
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// invariants:
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// x = old(x)^(2^(i*30))
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// res = old(x)^(p % 2^(i*30))
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// get the i-th limb of prime - 2
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limb = p.val[i];
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for (j = 0; j < 30; j++) {
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// invariants:
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// x = old(x)^(2^(i*30+j))
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// res = old(x)^(p % 2^(i*30+j))
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// limb = (p % 2^(i*30+30)) / 2^(i*30+j)
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if (i == 8 && limb == 0) break;
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if (limb & 1) {
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bn_multiply(x, &res, prime);
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@ -277,14 +313,24 @@ void bn_sqrt(bignum256 *x, const bignum256 *prime)
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// in field G_prime, small but slow
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void bn_inverse(bignum256 *x, const bignum256 *prime)
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{
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// this method compute x^-1 = x^(prime-2)
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uint32_t i, j, limb;
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bignum256 res;
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bn_zero(&res); res.val[0] = 1;
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for (i = 0; i < 9; i++) {
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// invariants:
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// x = old(x)^(2^(i*30))
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// res = old(x)^((prime-2) % 2^(i*30))
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// get the i-th limb of prime - 2
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limb = prime->val[i];
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// this is not enough in general but fine for secp256k1 because prime->val[0] > 1
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if (i == 0) limb -= 2;
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for (j = 0; j < 30; j++) {
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// invariants:
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// x = old(x)^(2^(i*30+j))
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// res = old(x)^((prime-2) % 2^(i*30+j))
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// limb = ((prime-2) % 2^(i*30+30)) / 2^(i*30+j)
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// early abort when only zero bits follow
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if (i == 8 && limb == 0) break;
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if (limb & 1) {
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bn_multiply(x, &res, prime);
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@ -300,14 +346,18 @@ void bn_inverse(bignum256 *x, const bignum256 *prime)
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#else
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// in field G_prime, big but fast
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// this algorithm is based on the Euklidean algorithm
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// the result is smaller than 2*prime
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void bn_inverse(bignum256 *x, const bignum256 *prime)
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{
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int i, j, k, len1, len2, mask;
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uint8_t buf[32];
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uint32_t u[8], v[8], s[9], r[10], temp32;
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uint64_t temp, temp2;
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// reduce x modulo prime
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bn_fast_mod(x, prime);
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bn_mod(x, prime);
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// convert x and prime it to 8x32 bit limb form
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bn_write_be(prime, buf);
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for (i = 0; i < 8; i++) {
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u[i] = read_be(buf + 28 - i * 4);
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@ -321,59 +371,98 @@ void bn_inverse(bignum256 *x, const bignum256 *prime)
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r[0] = 0;
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len2 = 1;
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k = 0;
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// u = prime, v = x len1 = numlimbs(u,v)
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// r = 0 , s = 1 len2 = numlimbs(r,s)
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// k = 0
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for (;;) {
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// invariants:
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// r,s,u,v >= 0
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// x*-r = u*2^k mod prime
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// x*s = v*2^k mod prime
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// u*s + v*r = prime
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// floor(log2(u)) + floor(log2(v)) + k <= 510
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// max(u,v) <= 2^k
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// gcd(u,v) = 1
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// len1 = numlimbs(u,v)
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// len2 = numlimbs(r,s)
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//
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// first u,v are large and s,r small
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// later u,v are small and s,r large
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// if (is_zero(v)) break;
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for (i = 0; i < len1; i++) {
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if (v[i]) break;
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}
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if (i == len1) break;
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// reduce u while it is even
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for (;;) {
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// count up to 30 zero bits of u.
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for (i = 0; i < 30; i++) {
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if (u[0] & (1 << i)) break;
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}
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// if u was odd break
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if (i == 0) break;
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// shift u right by i bits.
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mask = (1 << i) - 1;
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for (j = 0; j + 1 < len1; j++) {
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u[j] = (u[j] >> i) | ((u[j + 1] & mask) << (32 - i));
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}
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u[j] = (u[j] >> i);
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// shift s left by i bits.
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mask = (1 << (32 - i)) - 1;
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s[len2] = s[len2 - 1] >> (32 - i);
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for (j = len2 - 1; j > 0; j--) {
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s[j] = (s[j - 1] >> (32 - i)) | ((s[j] & mask) << i);
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}
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s[0] = (s[0] & mask) << i;
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// update len2 if necessary
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if (s[len2]) {
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r[len2] = 0;
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len2++;
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}
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// add i bits to k.
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k += i;
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}
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// reduce v while it is even
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for (;;) {
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// count up to 30 zero bits of v.
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for (i = 0; i < 30; i++) {
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if (v[0] & (1 << i)) break;
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}
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// if v was odd break
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if (i == 0) break;
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// shift v right by i bits.
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mask = (1 << i) - 1;
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for (j = 0; j + 1 < len1; j++) {
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v[j] = (v[j] >> i) | ((v[j + 1] & mask) << (32 - i));
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}
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v[j] = (v[j] >> i);
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mask = (1 << (32 - i)) - 1;
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// shift r left by i bits.
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r[len2] = r[len2 - 1] >> (32 - i);
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for (j = len2 - 1; j > 0; j--) {
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r[j] = (r[j - 1] >> (32 - i)) | ((r[j] & mask) << i);
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}
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r[0] = (r[0] & mask) << i;
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// update len2 if necessary
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if (r[len2]) {
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s[len2] = 0;
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len2++;
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}
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// add i bits to k.
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k += i;
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}
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// invariant is reestablished.
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i = len1 - 1;
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while (i > 0 && u[i] == v[i]) i--;
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if (u[i] > v[i]) {
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// u > v:
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// u = (u - v)/2;
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temp = 0x100000000ull + u[0] - v[0];
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u[0] = (temp >> 1) & 0x7FFFFFFF;
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temp >>= 32;
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@ -384,6 +473,8 @@ void bn_inverse(bignum256 *x, const bignum256 *prime)
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temp >>= 32;
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}
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temp = temp2 = 0;
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// r += s;
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// s += s;
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for (i = 0; i < len2; i++) {
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temp += s[i];
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temp += r[i];
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@ -394,12 +485,19 @@ void bn_inverse(bignum256 *x, const bignum256 *prime)
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temp >>= 32;
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temp2 >>= 32;
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}
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// expand if necessary.
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if (temp != 0 || temp2 != 0) {
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r[len2] = temp;
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s[len2] = temp2;
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len2++;
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}
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// note that
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// u'2^(k+1) = (u - v) 2^k = x -(r + s) = x -r' mod prime
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// v'2^(k+1) = 2*v 2^k = x (s + s) = x s' mod prime
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// u's' + v'r' = (u-v)/2(2s) + v(r+s) = us + vr
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} else {
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// v >= u:
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// v = v - u;
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temp = 0x100000000ull + v[0] - u[0];
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v[0] = (temp >> 1) & 0x7FFFFFFF;
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temp >>= 32;
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@ -409,6 +507,8 @@ void bn_inverse(bignum256 *x, const bignum256 *prime)
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v[i] = (temp >> 1) & 0x7FFFFFFF;
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temp >>= 32;
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}
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// s = s + r
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// r = r + r
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temp = temp2 = 0;
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for (i = 0; i < len2; i++) {
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temp += s[i];
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@ -425,11 +525,28 @@ void bn_inverse(bignum256 *x, const bignum256 *prime)
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r[len2] = temp2;
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len2++;
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}
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// note that
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// u'2^(k+1) = 2*u 2^k = x -(r + r) = x -r' mod prime
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// v'2^(k+1) = (v - u) 2^k = x (s + r) = x s' mod prime
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// u's' + v'r' = u(r+s) + (v-u)/2(2r) = us + vr
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}
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// adjust len1 if possible.
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if (u[len1 - 1] == 0 && v[len1 - 1] == 0) len1--;
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// increase k
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k++;
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}
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// In the last iteration just before the comparison and subtraction
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// we had u=1, v=1, s+r = prime, k <= 510, 2^k > max(s,r) >= prime/2
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// hence 0 <= r < prime and 255 <= k <= 510.
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//
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// Afterwards r is doubled, k is incremented by 1.
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// Hence 0 <= r < 2*prime and 256 <= k < 512.
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//
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// The invariants give us x*-r = 2^k mod prime,
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// hence r = -2^k * x^-1 mod prime.
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// We need to compute -r/2^k mod prime.
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// convert r to bignum style
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j = r[0] >> 30;
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r[0] = r[0] & 0x3FFFFFFFu;
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for (i = 1; i < len2; i++) {
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@ -441,6 +558,7 @@ void bn_inverse(bignum256 *x, const bignum256 *prime)
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i++;
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for (; i < 9; i++) r[i] = 0;
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// r = r mod prime, note that r<2*prime.
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i = 8;
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while (i > 0 && r[i] == prime->val[i]) i--;
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if (r[i] >= prime->val[i]) {
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@ -451,26 +569,39 @@ void bn_inverse(bignum256 *x, const bignum256 *prime)
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temp32 >>= 30;
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}
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}
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// negate r: r = prime - r
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temp32 = 1;
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for (i = 0; i < 9; i++) {
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temp32 += 0x3FFFFFFF + prime->val[i] - r[i];
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r[i] = temp32 & 0x3FFFFFFF;
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temp32 >>= 30;
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}
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// now: r = 2^k * x^-1 mod prime
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// compute r/2^k, 256 <= k < 511
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int done = 0;
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#if USE_PRECOMPUTED_IV
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if (prime == &prime256k1) {
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for (j = 0; j < 9; j++) {
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x->val[j] = r[j];
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}
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// secp256k1_iv[k-256] = 2^-k mod prime
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bn_multiply(secp256k1_iv + k - 256, x, prime);
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// bn_fast_mod is unnecessary as bn_multiply already
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// guarantees x < 2*prime
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bn_fast_mod(x, prime);
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// We don't guarantee x < prime!
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// the slow variant and the slow case below guarantee
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// this.
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done = 1;
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}
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#endif
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if (!done) {
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// compute r = r/2^k mod prime
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for (j = 0; j < k; j++) {
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// invariant: r = 2^(k-j) * x^-1 mod prime
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// in each iteration divide r by 2 modulo prime.
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if (r[0] & 1) {
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// r is odd; compute r = (prime + r)/2
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temp32 = r[0] + prime->val[0];
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r[0] = (temp32 >> 1) & 0x1FFFFFFF;
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temp32 >>= 30;
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@ -481,12 +612,14 @@ void bn_inverse(bignum256 *x, const bignum256 *prime)
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temp32 >>= 30;
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}
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} else {
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// r = r / 2
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for (i = 0; i < 8; i++) {
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r[i] = (r[i] >> 1) | ((r[i + 1] & 1) << 29);
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}
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r[8] = r[8] >> 1;
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}
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}
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// r = x^-1 mod prime, since j = k
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for (j = 0; j < 9; j++) {
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x->val[j] = r[j];
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}
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