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Merge pull request #25 from jhoenicke/comments

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Added comments to the tricky algorithms.
pull/25/head
Pavol Rusnak 9 years ago
parent 38cfebdbfe 7e98c02afd
commit dc31cc50d2
1 changed files with 137 additions and 4 deletions
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141
bignum.c
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@ -177,8 +177,11 @@ void bn_muli(bignum256 *a, uint32_t b)
a->val[8] += t;
}
// x = k * x
// both inputs and result may be bigger than prime but not bigger than 2 * prime
// Compute x := k * x (mod prime)
// both inputs must be smaller than 2 * prime.
// result is reduced to 0 <= x < 2 * prime
// This only works for primes between 2^256-2^196 and 2^256.
// this particular implementation accepts inputs up to 2^263 or 128*prime.
void bn_multiply(const bignum256 *k, bignum256 *x, const bignum256 *prime)
{
int i, j;
@ -204,11 +207,21 @@ void bn_multiply(const bignum256 *k, bignum256 *x, const bignum256 *prime)
temp >>= 30;
}
res[17] = temp;
// res = k * x is a normalized number (every limb < 2^30)
// 0 <= res < 2^526.
// compute modulo p division is only estimated so this may give result greater than prime but not bigger than 2 * prime
for (i = 16; i >= 8; i--) {
// let k = i-8.
// invariants:
// res[0..(i+1)] = k * x (mod prime)
// 0 <= res < 2^(30k + 256) * (2^30 + 1)
// estimate (res / prime)
coef = (res[i] >> 16) + (res[i + 1] << 14);
// substract (coef * prime) from res
// coef = res / 2^(30k + 256) rounded down
// 0 <= coef <= 2^30
// subtract (coef * 2^(30k) * prime) from res
// note that we unrolled the first iteration
temp = 0x1000000000000000ull + res[i - 8] - prime->val[0] * (uint64_t)coef;
res[i - 8] = temp & 0x3FFFFFFF;
for (j = 1; j < 9; j++) {
@ -216,6 +229,16 @@ void bn_multiply(const bignum256 *k, bignum256 *x, const bignum256 *prime)
temp += 0xFFFFFFFC0000000ull + res[i - 8 + j] - prime->val[j] * (uint64_t)coef;
res[i - 8 + j] = temp & 0x3FFFFFFF;
}
// we don't clear res[i+1] but we never read it again.
// we rely on the fact that prime > 2^256 - 2^196
// res = oldres - coef*2^(30k) * prime;
// and
// coef * 2^(30k + 256) <= oldres < (coef+1) * 2^(30k + 256)
// Hence, 0 <= res < 2^30k (2^256 + coef * (2^256 - prime))
// Since coef * (2^256 - prime) < 2^226, we get
// 0 <= res < 2^(30k + 226) (2^30 + 1)
// Thus the invariant holds again.
}
// store the result
for (i = 0; i < 9; i++) {
@ -223,6 +246,8 @@ void bn_multiply(const bignum256 *k, bignum256 *x, const bignum256 *prime)
}
}
// input x can be any normalized number that fits (0 <= x < 2^270).
// prime must be between (2^256 - 2^196) and 2^256
// result is smaller than 2*prime
void bn_fast_mod(bignum256 *x, const bignum256 *prime)
{
@ -233,6 +258,7 @@ void bn_fast_mod(bignum256 *x, const bignum256 *prime)
coef = x->val[8] >> 16;
if (!coef) return;
// substract (coef * prime) from x
// note that we unrolled the first iteration
temp = 0x1000000000000000ull + x->val[0] - prime->val[0] * (uint64_t)coef;
x->val[0] = temp & 0x3FFFFFFF;
for (j = 1; j < 9; j++) {
@ -246,16 +272,26 @@ void bn_fast_mod(bignum256 *x, const bignum256 *prime)
// http://en.wikipedia.org/wiki/Quadratic_residue#Prime_or_prime_power_modulus
void bn_sqrt(bignum256 *x, const bignum256 *prime)
{
// this method compute x^1/2 = x^(prime+1)/4
uint32_t i, j, limb;
bignum256 res, p;
bn_zero(&res); res.val[0] = 1;
// compute p = (prime+1)/4
memcpy(&p, prime, sizeof(bignum256));
p.val[0] += 1;
bn_rshift(&p);
bn_rshift(&p);
for (i = 0; i < 9; i++) {
// invariants:
// x = old(x)^(2^(i*30))
// res = old(x)^(p % 2^(i*30))
// get the i-th limb of prime - 2
limb = p.val[i];
for (j = 0; j < 30; j++) {
// invariants:
// x = old(x)^(2^(i*30+j))
// res = old(x)^(p % 2^(i*30+j))
// limb = (p % 2^(i*30+30)) / 2^(i*30+j)
if (i == 8 && limb == 0) break;
if (limb & 1) {
bn_multiply(x, &res, prime);
@ -277,14 +313,24 @@ void bn_sqrt(bignum256 *x, const bignum256 *prime)
// in field G_prime, small but slow
void bn_inverse(bignum256 *x, const bignum256 *prime)
{
// this method compute x^-1 = x^(prime-2)
uint32_t i, j, limb;
bignum256 res;
bn_zero(&res); res.val[0] = 1;
for (i = 0; i < 9; i++) {
// invariants:
// x = old(x)^(2^(i*30))
// res = old(x)^((prime-2) % 2^(i*30))
// get the i-th limb of prime - 2
limb = prime->val[i];
// this is not enough in general but fine for secp256k1 because prime->val[0] > 1
if (i == 0) limb -= 2;
for (j = 0; j < 30; j++) {
// invariants:
// x = old(x)^(2^(i*30+j))
// res = old(x)^((prime-2) % 2^(i*30+j))
// limb = ((prime-2) % 2^(i*30+30)) / 2^(i*30+j)
// early abort when only zero bits follow
if (i == 8 && limb == 0) break;
if (limb & 1) {
bn_multiply(x, &res, prime);
@ -300,14 +346,18 @@ void bn_inverse(bignum256 *x, const bignum256 *prime)
#else
// in field G_prime, big but fast
// this algorithm is based on the Euklidean algorithm
// the result is smaller than 2*prime
void bn_inverse(bignum256 *x, const bignum256 *prime)
{
int i, j, k, len1, len2, mask;
uint8_t buf[32];
uint32_t u[8], v[8], s[9], r[10], temp32;
uint64_t temp, temp2;
// reduce x modulo prime
bn_fast_mod(x, prime);
bn_mod(x, prime);
// convert x and prime it to 8x32 bit limb form
bn_write_be(prime, buf);
for (i = 0; i < 8; i++) {
u[i] = read_be(buf + 28 - i * 4);
@ -321,59 +371,98 @@ void bn_inverse(bignum256 *x, const bignum256 *prime)
r[0] = 0;
len2 = 1;
k = 0;
// u = prime, v = x len1 = numlimbs(u,v)
// r = 0 , s = 1 len2 = numlimbs(r,s)
// k = 0
for (;;) {
// invariants:
// r,s,u,v >= 0
// x*-r = u*2^k mod prime
// x*s = v*2^k mod prime
// u*s + v*r = prime
// floor(log2(u)) + floor(log2(v)) + k <= 510
// max(u,v) <= 2^k
// gcd(u,v) = 1
// len1 = numlimbs(u,v)
// len2 = numlimbs(r,s)
//
// first u,v are large and s,r small
// later u,v are small and s,r large
// if (is_zero(v)) break;
for (i = 0; i < len1; i++) {
if (v[i]) break;
}
if (i == len1) break;
// reduce u while it is even
for (;;) {
// count up to 30 zero bits of u.
for (i = 0; i < 30; i++) {
if (u[0] & (1 << i)) break;
}
// if u was odd break
if (i == 0) break;
// shift u right by i bits.
mask = (1 << i) - 1;
for (j = 0; j + 1 < len1; j++) {
u[j] = (u[j] >> i) | ((u[j + 1] & mask) << (32 - i));
}
u[j] = (u[j] >> i);
// shift s left by i bits.
mask = (1 << (32 - i)) - 1;
s[len2] = s[len2 - 1] >> (32 - i);
for (j = len2 - 1; j > 0; j--) {
s[j] = (s[j - 1] >> (32 - i)) | ((s[j] & mask) << i);
}
s[0] = (s[0] & mask) << i;
// update len2 if necessary
if (s[len2]) {
r[len2] = 0;
len2++;
}
// add i bits to k.
k += i;
}
// reduce v while it is even
for (;;) {
// count up to 30 zero bits of v.
for (i = 0; i < 30; i++) {
if (v[0] & (1 << i)) break;
}
// if v was odd break
if (i == 0) break;
// shift v right by i bits.
mask = (1 << i) - 1;
for (j = 0; j + 1 < len1; j++) {
v[j] = (v[j] >> i) | ((v[j + 1] & mask) << (32 - i));
}
v[j] = (v[j] >> i);
mask = (1 << (32 - i)) - 1;
// shift r left by i bits.
r[len2] = r[len2 - 1] >> (32 - i);
for (j = len2 - 1; j > 0; j--) {
r[j] = (r[j - 1] >> (32 - i)) | ((r[j] & mask) << i);
}
r[0] = (r[0] & mask) << i;
// update len2 if necessary
if (r[len2]) {
s[len2] = 0;
len2++;
}
// add i bits to k.
k += i;
}
// invariant is reestablished.
i = len1 - 1;
while (i > 0 && u[i] == v[i]) i--;
if (u[i] > v[i]) {
// u > v:
// u = (u - v)/2;
temp = 0x100000000ull + u[0] - v[0];
u[0] = (temp >> 1) & 0x7FFFFFFF;
temp >>= 32;
@ -384,6 +473,8 @@ void bn_inverse(bignum256 *x, const bignum256 *prime)
temp >>= 32;
}
temp = temp2 = 0;
// r += s;
// s += s;
for (i = 0; i < len2; i++) {
temp += s[i];
temp += r[i];
@ -394,12 +485,19 @@ void bn_inverse(bignum256 *x, const bignum256 *prime)
temp >>= 32;
temp2 >>= 32;
}
// expand if necessary.
if (temp != 0 || temp2 != 0) {
r[len2] = temp;
s[len2] = temp2;
len2++;
}
// note that
// u'2^(k+1) = (u - v) 2^k = x -(r + s) = x -r' mod prime
// v'2^(k+1) = 2*v 2^k = x (s + s) = x s' mod prime
// u's' + v'r' = (u-v)/2(2s) + v(r+s) = us + vr
} else {
// v >= u:
// v = v - u;
temp = 0x100000000ull + v[0] - u[0];
v[0] = (temp >> 1) & 0x7FFFFFFF;
temp >>= 32;
@ -409,6 +507,8 @@ void bn_inverse(bignum256 *x, const bignum256 *prime)
v[i] = (temp >> 1) & 0x7FFFFFFF;
temp >>= 32;
}
// s = s + r
// r = r + r
temp = temp2 = 0;
for (i = 0; i < len2; i++) {
temp += s[i];
@ -425,11 +525,28 @@ void bn_inverse(bignum256 *x, const bignum256 *prime)
r[len2] = temp2;
len2++;
}
// note that
// u'2^(k+1) = 2*u 2^k = x -(r + r) = x -r' mod prime
// v'2^(k+1) = (v - u) 2^k = x (s + r) = x s' mod prime
// u's' + v'r' = u(r+s) + (v-u)/2(2r) = us + vr
}
// adjust len1 if possible.
if (u[len1 - 1] == 0 && v[len1 - 1] == 0) len1--;
// increase k
k++;
}
// In the last iteration just before the comparison and subtraction
// we had u=1, v=1, s+r = prime, k <= 510, 2^k > max(s,r) >= prime/2
// hence 0 <= r < prime and 255 <= k <= 510.
//
// Afterwards r is doubled, k is incremented by 1.
// Hence 0 <= r < 2*prime and 256 <= k < 512.
//
// The invariants give us x*-r = 2^k mod prime,
// hence r = -2^k * x^-1 mod prime.
// We need to compute -r/2^k mod prime.
// convert r to bignum style
j = r[0] >> 30;
r[0] = r[0] & 0x3FFFFFFFu;
for (i = 1; i < len2; i++) {
@ -441,6 +558,7 @@ void bn_inverse(bignum256 *x, const bignum256 *prime)
i++;
for (; i < 9; i++) r[i] = 0;
// r = r mod prime, note that r<2*prime.
i = 8;
while (i > 0 && r[i] == prime->val[i]) i--;
if (r[i] >= prime->val[i]) {
@ -451,26 +569,39 @@ void bn_inverse(bignum256 *x, const bignum256 *prime)
temp32 >>= 30;
}
}
// negate r: r = prime - r
temp32 = 1;
for (i = 0; i < 9; i++) {
temp32 += 0x3FFFFFFF + prime->val[i] - r[i];
r[i] = temp32 & 0x3FFFFFFF;
temp32 >>= 30;
}
// now: r = 2^k * x^-1 mod prime
// compute r/2^k, 256 <= k < 511
int done = 0;
#if USE_PRECOMPUTED_IV
if (prime == &prime256k1) {
for (j = 0; j < 9; j++) {
x->val[j] = r[j];
}
// secp256k1_iv[k-256] = 2^-k mod prime
bn_multiply(secp256k1_iv + k - 256, x, prime);
// bn_fast_mod is unnecessary as bn_multiply already
// guarantees x < 2*prime
bn_fast_mod(x, prime);
// We don't guarantee x < prime!
// the slow variant and the slow case below guarantee
// this.
done = 1;
}
#endif
if (!done) {
// compute r = r/2^k mod prime
for (j = 0; j < k; j++) {
// invariant: r = 2^(k-j) * x^-1 mod prime
// in each iteration divide r by 2 modulo prime.
if (r[0] & 1) {
// r is odd; compute r = (prime + r)/2
temp32 = r[0] + prime->val[0];
r[0] = (temp32 >> 1) & 0x1FFFFFFF;
temp32 >>= 30;
@ -481,12 +612,14 @@ void bn_inverse(bignum256 *x, const bignum256 *prime)
temp32 >>= 30;
}
} else {
// r = r / 2
for (i = 0; i < 8; i++) {
r[i] = (r[i] >> 1) | ((r[i + 1] & 1) << 29);
}
r[8] = r[8] >> 1;
}
}
// r = x^-1 mod prime, since j = k
for (j = 0; j < 9; j++) {
x->val[j] = r[j];
}

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