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Merge pull request #26 from jhoenicke/bignum_improvements

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Bignum improvements
pull/25/head
Pavol Rusnak 9 years ago
parent 196cabe012 56f5777b68
commit a757693fe3
7 changed files with 2401 additions and 1662 deletions
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bignum.c
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@ -23,6 +23,7 @@
#include <stdio.h>
#include <string.h>
#include <assert.h>
#include "bignum.h"
#include "secp256k1.h"
@ -42,27 +43,37 @@ inline void write_be(uint8_t *data, uint32_t x)
data[3] = x;
}
// convert a raw bigendian 256 bit number to a normalized bignum
void bn_read_be(const uint8_t *in_number, bignum256 *out_number)
{
int i;
uint64_t temp = 0;
uint32_t temp = 0;
for (i = 0; i < 8; i++) {
temp += (((uint64_t)read_be(in_number + (7 - i) * 4)) << (2 * i));
// invariant: temp = (in_number % 2^(32i)) >> 30i
// get next limb = (in_number % 2^(32(i+1))) >> 32i
uint32_t limb = read_be(in_number + (7 - i) * 4);
// temp = (in_number % 2^(32(i+1))) << 30i
temp |= limb << (2*i);
// store 30 bits into val[i]
out_number->val[i]= temp & 0x3FFFFFFF;
temp >>= 30;
// prepare temp for next round
temp = limb >> (30 - 2*i);
}
out_number->val[8] = temp;
}
// convert a normalized bignum to a raw bigendian 256 bit number.
// in_number must be normalized and < 2^256.
void bn_write_be(const bignum256 *in_number, uint8_t *out_number)
{
int i, shift = 30 + 16 - 32;
uint64_t temp = in_number->val[8];
int i;
uint32_t temp = in_number->val[8] << 16;
for (i = 0; i < 8; i++) {
temp <<= 30;
temp |= in_number->val[7 - i];
write_be(out_number + i * 4, temp >> shift);
shift -= 2;
// invariant: temp = (in_number >> 30*(8-i)) << (16 + 2i)
uint32_t limb = in_number->val[7 - i];
temp |= limb >> (14 - 2*i);
write_be(out_number + i * 4, temp);
temp = limb << (18 + 2*i);
}
}
@ -128,6 +139,26 @@ void bn_rshift(bignum256 *a)
a->val[8] >>= 1;
}
// multiply x by 3/2 modulo prime.
// assumes x < 2*prime,
// guarantees x < 4*prime on exit.
void bn_mult_3_2(bignum256 * x, const bignum256 *prime)
{
int j;
uint32_t xodd = -(x->val[0] & 1);
// compute x = 3*x/2 mod prime
// if x is odd compute (3*x+prime)/2
uint32_t tmp1 = (3*x->val[0] + (prime->val[0] & xodd)) >> 1;
for (j = 0; j < 8; j++) {
uint32_t tmp2 = (3*x->val[j+1] + (prime->val[j+1] & xodd));
tmp1 += (tmp2 & 1) << 29;
x->val[j] = tmp1 & 0x3fffffff;
tmp1 >>= 30;
tmp1 += tmp2 >> 1;
}
x->val[8] = tmp1;
}
// assumes x < 2*prime, result < prime
void bn_mod(bignum256 *x, const bignum256 *prime)
{
@ -233,7 +264,6 @@ void bn_fast_mod(bignum256 *x, const bignum256 *prime)
uint64_t temp;
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;
@ -283,10 +313,6 @@ void bn_sqrt(bignum256 *x, const bignum256 *prime)
#if ! USE_INVERSE_FAST
#if USE_PRECOMPUTED_IV
#warning USE_PRECOMPUTED_IV will not be used
#endif
// in field G_prime, small but slow
void bn_inverse(bignum256 *x, const bignum256 *prime)
{
@ -322,285 +348,272 @@ 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
// in field G_prime, big and complicated but fast
// the input must not be 0 mod prime.
// the result is smaller than 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
int i, j, k, cmp;
struct combo {
uint32_t a[9];
int len1;
} us, vr, *odd, *even;
uint32_t pp[8];
uint32_t temp32;
uint64_t temp;
// The algorithm is based on Schroeppel et. al. "Almost Modular Inverse"
// algorithm. We keep four values u,v,r,s in the combo registers
// us and vr. us stores u in the first len1 limbs (little endian)
// and s in the last 9-len1 limbs (big endian). vr stores v and r.
// This is because both u*s and v*r are guaranteed to fit in 8 limbs, so
// their components are guaranteed to fit in 9. During the algorithm,
// the length of u and v shrinks while r and s grow.
// u,v,r,s correspond to F,G,B,C in Schroeppel's algorithm.
// reduce x modulo prime. This is necessary as it has to fit in 8 limbs.
bn_fast_mod(x, prime);
bn_mod(x, prime);
// convert x and prime it to 8x32 bit limb form
bn_write_be(prime, buf);
// convert x and prime to 8x32 bit limb form
temp32 = prime->val[0];
for (i = 0; i < 8; i++) {
u[i] = read_be(buf + 28 - i * 4);
temp32 |= prime->val[i + 1] << (30-2*i);
us.a[i] = pp[i] = temp32;
temp32 = prime->val[i + 1] >> (2+2*i);
}
bn_write_be(x, buf);
temp32 = x->val[0];
for (i = 0; i < 8; i++) {
v[i] = read_be(buf + 28 - i * 4);
temp32 |= x->val[i + 1] << (30-2*i);
vr.a[i] = temp32;
temp32 = x->val[i + 1] >> (2+2*i);
}
len1 = 8;
s[0] = 1;
r[0] = 0;
len2 = 1;
us.len1 = 8;
vr.len1 = 8;
// set s = 1 and r = 0
us.a[8] = 1;
vr.a[8] = 0;
// set k = 0.
k = 0;
// u = prime, v = x len1 = numlimbs(u,v)
// r = 0 , s = 1 len2 = numlimbs(r,s)
// only one of the numbers u,v can be even at any time. We
// let even point to that number and odd to the other.
// Initially the prime u is guaranteed to be odd.
odd = &us;
even = &vr;
// u = prime, v = x
// r = 0 , s = 1
// k = 0
for (;;) {
// invariants:
// r,s,u,v >= 0
// let u = limbs us.a[0..u.len1-1] in little endian,
// let s = limbs us.a[u.len..8] in big endian,
// let v = limbs vr.a[0..u.len1-1] in little endian,
// let r = limbs vr.a[u.len..8] in big endian,
// r,s >= 0 ; u,v >= 1
// 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
// max(u,v) <= 2^k (*) see comment at end of loop
// gcd(u,v) = 1
// len1 = numlimbs(u,v)
// len2 = numlimbs(r,s)
// {odd,even} = {&us, &vr}
// odd->a[0] and odd->a[8] are odd
// even->a[0] or even->a[8] is even
//
// 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;
// first u/v are large and r/s small
// later u/v are small and r/s large
assert(odd->a[0] & 1);
assert(odd->a[8] & 1);
// adjust length of even.
while (even->a[even->len1 - 1] == 0) {
even->len1--;
// if input was 0, return.
// This simple check prevents crashing with stack underflow
// or worse undesired behaviour for illegal input.
if (even->len1 < 0)
return;
}
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++;
// reduce even->a while it is even
while (even->a[0] == 0) {
// shift right first part of even by a limb
// and shift left second part of even by a limb.
for (i = 0; i < 8; i++) {
even->a[i] = even->a[i+1];
}
// add i bits to k.
k += i;
even->a[i] = 0;
even->len1--;
k += 32;
}
// 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));
// count up to 32 zero bits of even->a.
j = 0;
while ((even->a[0] & (1 << j)) == 0) {
j++;
}
if (j > 0) {
// shift first part of even right by j bits.
for (i = 0; i + 1 < even->len1; i++) {
even->a[i] = (even->a[i] >> j) | (even->a[i + 1] << (32 - j));
}
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);
even->a[i] = (even->a[i] >> j);
if (even->a[i] == 0) {
even->len1--;
} else {
i++;
}
r[0] = (r[0] & mask) << i;
// update len2 if necessary
if (r[len2]) {
s[len2] = 0;
len2++;
// shift second part of even left by j bits.
for (; i < 8; i++) {
even->a[i] = (even->a[i] << j) | (even->a[i + 1] >> (32 - j));
}
// add i bits to k.
k += i;
even->a[i] = (even->a[i] << j);
// add j bits to k.
k += j;
}
// 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;
for (i = 1; i < len1; i++) {
temp += 0xFFFFFFFFull + u[i] - v[i];
u[i - 1] += (temp & 1) << 31;
u[i] = (temp >> 1) & 0x7FFFFFFF;
temp >>= 32;
}
temp = temp2 = 0;
// r += s;
// s += s;
for (i = 0; i < len2; i++) {
temp += s[i];
temp += r[i];
temp2 += s[i];
temp2 += s[i];
r[i] = temp;
s[i] = temp2;
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
// now both a[0] are odd.
assert(odd->a[0] & 1);
assert(odd->a[8] & 1);
assert(even->a[0] & 1);
assert((even->a[8] & 1) == 0);
// cmp > 0 if us.a[0..len1-1] > vr.a[0..len1-1],
// cmp = 0 if equal, < 0 if less.
cmp = us.len1 - vr.len1;
if (cmp == 0) {
i = us.len1 - 1;
while (i >= 0 && us.a[i] == vr.a[i]) i--;
// both are equal to 1 and we are done.
if (i == -1)
break;
cmp = us.a[i] > vr.a[i] ? 1 : -1;
}
if (cmp > 0) {
even = &us;
odd = &vr;
} else {
// v >= u:
// v = v - u;
temp = 0x100000000ull + v[0] - u[0];
v[0] = (temp >> 1) & 0x7FFFFFFF;
even = &vr;
odd = &us;
}
// now even > odd.
// even->a[0..len1-1] = (even->a[0..len1-1] - odd->a[0..len1-1]);
temp = 1;
for (i = 0; i < odd->len1; i++) {
temp += 0xFFFFFFFFull + even->a[i] - odd->a[i];
even->a[i] = temp & 0xFFFFFFFF;
temp >>= 32;
for (i = 1; i < len1; i++) {
temp += 0xFFFFFFFFull + v[i] - u[i];
v[i - 1] += (temp & 1) << 31;
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];
temp += r[i];
temp2 += r[i];
temp2 += r[i];
s[i] = temp;
r[i] = temp2;
temp >>= 32;
temp2 >>= 32;
}
if (temp != 0 || temp2 != 0) {
s[len2] = temp;
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++;
for (; i < even->len1; i++) {
temp += 0xFFFFFFFFull + even->a[i];
even->a[i] = temp & 0xFFFFFFFF;
temp >>= 32;
}
// odd->a[len1..8] = (odd->b[len1..8] + even->b[len1..8]);
temp = 0;
for (i = 8; i >= even->len1; i--) {
temp += (uint64_t) odd->a[i] + even->a[i];
odd->a[i] = temp & 0xFFFFFFFF;
temp >>= 32;
}
for (; i >= odd->len1; i--) {
temp += (uint64_t) odd->a[i];
odd->a[i] = temp & 0xFFFFFFFF;
temp >>= 32;
}
// note that
// if u > v:
// u'2^k = (u - v) 2^k = x(-r) - xs = x(-(r+s)) = x(-r') mod prime
// u's' + v'r' = (u-v)s + v(r+s) = us + vr
// if u < v:
// v'2^k = (v - u) 2^k = xs - x(-r) = x(s+r) = xs' mod prime
// u's' + v'r' = u(s+r) + (v-u)r = us + vr
// even->a[0] is difference between two odd numbers, hence even.
// odd->a[8] is sum of even and odd number, hence odd.
assert(odd->a[0] & 1);
assert(odd->a[8] & 1);
assert((even->a[0] & 1) == 0);
// The invariants are (almost) reestablished.
// The invariant max(u,v) <= 2^k can be invalidated at this point,
// because odd->a[len1..8] was changed. We only have
//
// odd->a[len1..8] <= 2^{k+1}
//
// Since even->a[0] is even, k will be incremented at the beginning
// of the next loop while odd->a[len1..8] remains unchanged.
// So after that, odd->a[len1..8] <= 2^k will hold again.
}
// 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.
// In the last iteration we had u = v and gcd(u,v) = 1.
// Hence, u=1, v=1, s+r = prime, k <= 510, 2^k > max(s,r) >= prime/2
// This implies 0 <= s < prime and 255 <= k <= 510.
//
// 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++) {
uint32_t q = r[i] >> (30 - 2 * i);
r[i] = ((r[i] << (2 * i)) & 0x3FFFFFFFu) + j;
j=q;
// The invariants also give us x*s = 2^k mod prime,
// hence s = 2^k * x^-1 mod prime.
// We need to compute s/2^k mod prime.
// First we compute inverse = -prime^-1 mod 2^32, which we need later.
// We use the Explicit Quadratic Modular inverse algorithm.
// http://arxiv.org/pdf/1209.6626.pdf
// a^-1 = (2-a) * PROD_i (1 + (a - 1)^(2^i)) mod 2^32
// the product will converge quickly, because (a-1)^(2^i) will be
// zero mod 2^32 after at most five iterations.
// We want to compute -prime^-1 so we start with (pp[0]-2).
assert(pp[0] & 1);
uint32_t amone = pp[0]-1;
uint32_t inverse = pp[0] - 2;
while (amone) {
amone *= amone;
inverse *= (amone + 1);
}
r[i] = j;
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]) {
temp32 = 1;
for (i = 0; i < 9; i++) {
temp32 += 0x3FFFFFFF + r[i] - prime->val[i];
r[i] = temp32 & 0x3FFFFFFF;
temp32 >>= 30;
while (k >= 32) {
// compute s / 2^32 modulo prime.
// Idea: compute factor, such that
// s + factor*prime mod 2^32 == 0
// i.e. factor = s * -1/prime mod 2^32.
// Then compute s + factor*prime and shift right by 32 bits.
uint32_t factor = (inverse * us.a[8]) & 0xffffffff;
temp = us.a[8] + (uint64_t) pp[0] * factor;
assert((temp & 0xffffffff) == 0);
temp >>= 32;
for (i = 0; i < 7; i++) {
temp += us.a[8-(i+1)] + (uint64_t) pp[i+1] * factor;
us.a[8-i] = temp & 0xffffffff;
temp >>= 32;
}
us.a[8-i] = temp & 0xffffffff;
k -= 32;
}
// 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];
if (k > 0) {
// compute s / 2^k modulo prime.
// Same idea: compute factor, such that
// s + factor*prime mod 2^k == 0
// i.e. factor = s * -1/prime mod 2^k.
// Then compute s + factor*prime and shift right by k bits.
uint32_t mask = (1 << k) - 1;
uint32_t factor = (inverse * us.a[8]) & mask;
temp = (us.a[8] + (uint64_t) pp[0] * factor) >> k;
assert(((us.a[8] + pp[0] * factor) & mask) == 0);
for (i = 0; i < 7; i++) {
temp += (us.a[8-(i+1)] + (uint64_t) pp[i+1] * factor) << (32 - k);
us.a[8-i] = temp & 0xffffffff;
temp >>= 32;
}
// 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;
us.a[8-i] = temp & 0xffffffff;
}
#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;
for (i = 1; i < 9; i++) {
temp32 += r[i] + prime->val[i];
r[i - 1] += (temp32 & 1) << 29;
r[i] = (temp32 >> 1) & 0x1FFFFFFF;
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];
}
// convert s to bignum style
temp32 = 0;
for (i = 0; i < 8; i++) {
x->val[i] = ((us.a[8-i] << (2 * i)) & 0x3FFFFFFFu) | temp32;
temp32 = us.a[8-i] >> (30 - 2 * i);
}
x->val[i] = temp32;
}
#endif
@ -632,14 +645,15 @@ void bn_addmodi(bignum256 *a, uint32_t b, const bignum256 *prime) {
bn_mod(a, prime);
}
// res = a - b
// b < 2*prime; result not normalized
void bn_subtractmod(const bignum256 *a, const bignum256 *b, bignum256 *res)
// res = a - b mod prime. More exactly res = a + (2*prime - b).
// precondition: 0 <= b < 2*prime, 0 <= a < prime
// res < 3*prime
void bn_subtractmod(const bignum256 *a, const bignum256 *b, bignum256 *res, const bignum256 *prime)
{
int i;
uint32_t temp = 0;
for (i = 0; i < 9; i++) {
temp += a->val[i] + 2u * prime256k1.val[i] - b->val[i];
temp += a->val[i] + 2u * prime->val[i] - b->val[i];
res->val[i] = temp & 0x3FFFFFFF;
temp >>= 30;
}
@ -665,9 +679,17 @@ void bn_divmod58(bignum256 *a, uint32_t *r)
rem = a->val[8] % 58;
a->val[8] /= 58;
for (i = 7; i >= 0; i--) {
// invariants:
// rem = old(a) >> 30(i+1) % 58
// a[i+1..8] = old(a[i+1..8])/58
// a[0..i] = old(a[0..i])
// 2^30 == 18512790*58 + 4
tmp = rem * 4 + a->val[i];
// set a[i] = (rem * 2^30 + a[i])/58
// = rem * 18512790 + (rem * 4 + a[i])/58
a->val[i] = rem * 18512790 + (tmp / 58);
// set rem = (rem * 2^30 + a[i]) mod 58
// = (rem * 4 + a[i]) mod 58
rem = tmp % 58;
}
*r = rem;

4
bignum.h
Unescape View File

@ -57,6 +57,8 @@ void bn_lshift(bignum256 *a);
void bn_rshift(bignum256 *a);
void bn_mult_3_2(bignum256 *x, const bignum256 *prime);
void bn_mod(bignum256 *x, const bignum256 *prime);
void bn_multiply(const bignum256 *k, bignum256 *x, const bignum256 *prime);
@ -73,7 +75,7 @@ void bn_addmod(bignum256 *a, const bignum256 *b, const bignum256 *prime);
void bn_addmodi(bignum256 *a, uint32_t b, const bignum256 *prime);
void bn_subtractmod(const bignum256 *a, const bignum256 *b, bignum256 *res);
void bn_subtractmod(const bignum256 *a, const bignum256 *b, bignum256 *res, const bignum256 *prime);
void bn_subtract(const bignum256 *a, const bignum256 *b, bignum256 *res);

553
ecdsa.c
Unescape View File

@ -24,6 +24,7 @@
#include <stdint.h>
#include <stdlib.h>
#include <string.h>
#include <assert.h>
#include "bignum.h"
#include "rand.h"
@ -36,8 +37,7 @@
// Set cp2 = cp1
void point_copy(const curve_point *cp1, curve_point *cp2)
{
memcpy(&(cp2->x), &(cp1->x), sizeof(bignum256));
memcpy(&(cp2->y), &(cp1->y), sizeof(bignum256));
*cp2 = *cp1;
}
// cp2 = cp1 + cp2
@ -63,29 +63,32 @@ void point_add(const curve_point *cp1, curve_point *cp2)
return;
}
bn_subtractmod(&(cp2->x), &(cp1->x), &inv);
bn_subtractmod(&(cp2->x), &(cp1->x), &inv, &prime256k1);
bn_inverse(&inv, &prime256k1);
bn_subtractmod(&(cp2->y), &(cp1->y), &lambda);
bn_subtractmod(&(cp2->y), &(cp1->y), &lambda, &prime256k1);
bn_multiply(&inv, &lambda, &prime256k1);
memcpy(&xr, &lambda, sizeof(bignum256));
// xr = lambda^2 - x1 - x2
xr = lambda;
bn_multiply(&xr, &xr, &prime256k1);
temp = 0;
temp = 1;
for (i = 0; i < 9; i++) {
temp += xr.val[i] + 3u * prime256k1.val[i] - cp1->x.val[i] - cp2->x.val[i];
temp += 0x3FFFFFFF + xr.val[i] + 2u * prime256k1.val[i] - cp1->x.val[i] - cp2->x.val[i];
xr.val[i] = temp & 0x3FFFFFFF;
temp >>= 30;
}
bn_fast_mod(&xr, &prime256k1);
bn_subtractmod(&(cp1->x), &xr, &yr);
// no need to fast_mod here
// bn_fast_mod(&yr);
bn_mod(&xr, &prime256k1);
// yr = lambda (x1 - xr) - y1
bn_subtractmod(&(cp1->x), &xr, &yr, &prime256k1);
bn_multiply(&lambda, &yr, &prime256k1);
bn_subtractmod(&yr, &(cp1->y), &yr);
bn_subtractmod(&yr, &(cp1->y), &yr, &prime256k1);
bn_fast_mod(&yr, &prime256k1);
memcpy(&(cp2->x), &xr, sizeof(bignum256));
memcpy(&(cp2->y), &yr, sizeof(bignum256));
bn_mod(&(cp2->x), &prime256k1);
bn_mod(&(cp2->y), &prime256k1);
bn_mod(&yr, &prime256k1);
cp2->x = xr;
cp2->y = yr;
}
// cp = cp + cp
@ -93,7 +96,7 @@ void point_double(curve_point *cp)
{
int i;
uint32_t temp;
bignum256 lambda, inverse_y, xr, yr;
bignum256 lambda, xr, yr;
if (point_is_infinity(cp)) {
return;
@ -103,56 +106,34 @@ void point_double(curve_point *cp)
return;
}
memcpy(&inverse_y, &(cp->y), sizeof(bignum256));
bn_inverse(&inverse_y, &prime256k1);
memcpy(&lambda, &three_over_two256k1, sizeof(bignum256));
bn_multiply(&inverse_y, &lambda, &prime256k1);
bn_multiply(&(cp->x), &lambda, &prime256k1);
bn_multiply(&(cp->x), &lambda, &prime256k1);
memcpy(&xr, &lambda, sizeof(bignum256));
// lambda = 3/2 x^2 / y
lambda = cp->y;
bn_inverse(&lambda, &prime256k1);
bn_multiply(&cp->x, &lambda, &prime256k1);
bn_multiply(&cp->x, &lambda, &prime256k1);
bn_mult_3_2(&lambda, &prime256k1);
// xr = lambda^2 - 2*x
xr = lambda;
bn_multiply(&xr, &xr, &prime256k1);
temp = 0;
temp = 1;
for (i = 0; i < 9; i++) {
temp += xr.val[i] + 3u * prime256k1.val[i] - 2u * cp->x.val[i];
temp += 0x3FFFFFFF + xr.val[i] + 2u * (prime256k1.val[i] - cp->x.val[i]);
xr.val[i] = temp & 0x3FFFFFFF;
temp >>= 30;
}
bn_fast_mod(&xr, &prime256k1);
bn_subtractmod(&(cp->x), &xr, &yr);
// no need to fast_mod here
// bn_fast_mod(&yr);
bn_mod(&xr, &prime256k1);
// yr = lambda (x - xr) - y
bn_subtractmod(&(cp->x), &xr, &yr, &prime256k1);
bn_multiply(&lambda, &yr, &prime256k1);
bn_subtractmod(&yr, &(cp->y), &yr);
bn_subtractmod(&yr, &(cp->y), &yr, &prime256k1);
bn_fast_mod(&yr, &prime256k1);
memcpy(&(cp->x), &xr, sizeof(bignum256));
memcpy(&(cp->y), &yr, sizeof(bignum256));
bn_mod(&(cp->x), &prime256k1);
bn_mod(&(cp->y), &prime256k1);
}
bn_mod(&yr, &prime256k1);
// res = k * p
void point_multiply(const bignum256 *k, const curve_point *p, curve_point *res)
{
int i, j;
// result is zero
int is_zero = 1;
curve_point curr;
// initial res
memcpy(&curr, p, sizeof(curve_point));
for (i = 0; i < 9; i++) {
for (j = 0; j < 30; j++) {
if (i == 8 && (k->val[i] >> j) == 0) break;
if (k->val[i] & (1u << j)) {
if (is_zero) {
memcpy(res, &curr, sizeof(curve_point));
is_zero = 0;
} else {
point_add(&curr, res);
}
}
point_double(&curr);
}
}
cp->x = xr;
cp->y = yr;
}
// set point to internal representation of point at infinity
@ -192,48 +173,421 @@ int point_is_negative_of(const curve_point *p, const curve_point *q)
return !bn_is_equal(&(p->y), &(q->y));
}
// res = k * G
void scalar_multiply(const bignum256 *k, curve_point *res)
// Negate a (modulo prime) if cond is 0xffffffff, keep it if cond is 0.
// The timing of this function does not depend on cond.
static void conditional_negate(uint32_t cond, bignum256 *a, const bignum256 *prime)
{
int j;
uint32_t tmp = 1;
for (j = 0; j < 8; j++) {
tmp += 0x3fffffff + prime->val[j] - a->val[j];
a->val[j] = ((tmp & 0x3fffffff) & cond) | (a->val[j] & ~cond);
tmp >>= 30;
}
tmp += 0x3fffffff + prime->val[j] - a->val[j];
a->val[j] = ((tmp & 0x3fffffff) & cond) | (a->val[j] & ~cond);
}
typedef struct jacobian_curve_point {
bignum256 x, y, z;
} jacobian_curve_point;
static void curve_to_jacobian(const curve_point *p, jacobian_curve_point *jp) {
int i;
// result is zero
int is_zero = 1;
curve_point curr;
// initial res
memcpy(&curr, &G256k1, sizeof(curve_point));
for (i = 0; i < 256; i++) {
if (k->val[i / 30] & (1u << (i % 30))) {
if (is_zero) {
#if USE_PRECOMPUTED_CP
if (i < 255 && (k->val[(i + 1) / 30] & (1u << ((i + 1) % 30)))) {
memcpy(res, secp256k1_cp2 + i, sizeof(curve_point));
i++;
} else {
memcpy(res, secp256k1_cp + i, sizeof(curve_point));
}
#else
memcpy(res, &curr, sizeof(curve_point));
#endif
is_zero = 0;
} else {
// randomize z coordinate
for (i = 0; i < 8; i++) {
jp->z.val[i] = random32() & 0x3FFFFFFF;
}
jp->z.val[8] = (random32() & 0x7fff) + 1;
jp->x = jp->z;
bn_multiply(&jp->z, &jp->x, &prime256k1);
// x = z^2
jp->y = jp->x;
bn_multiply(&jp->z, &jp->y, &prime256k1);
// y = z^3
bn_multiply(&p->x, &jp->x, &prime256k1);
bn_multiply(&p->y, &jp->y, &prime256k1);
bn_mod(&jp->x, &prime256k1);
bn_mod(&jp->y, &prime256k1);
}
static void jacobian_to_curve(const jacobian_curve_point *jp, curve_point *p) {
p->y = jp->z;
bn_mod(&p->y, &prime256k1);
bn_inverse(&p->y, &prime256k1);
// p->y = z^-1
p->x = p->y;
bn_multiply(&p->x, &p->x, &prime256k1);
// p->x = z^-2
bn_multiply(&p->x, &p->y, &prime256k1);
// p->y = z^-3
bn_multiply(&jp->x, &p->x, &prime256k1);
// p->x = jp->x * z^-2
bn_multiply(&jp->y, &p->y, &prime256k1);
// p->y = jp->y * z^-3
bn_mod(&p->x, &prime256k1);
bn_mod(&p->y, &prime256k1);
}
static void point_jacobian_add(const curve_point *p1, jacobian_curve_point *p2) {
bignum256 r, h;
bignum256 rsq, hcb, hcby2, hsqx2;
int j;
uint64_t tmp1;
/* usual algorithm:
*
* lambda = (y1 - y2/z2^3) / (x1 - x2/z2^2)
* x3/z3^2 = lambda^2 - x1 - x2/z2^2
* y3/z3^3 = lambda * (x2/z2^2 - x3/z3^2) - y2/z2^3
*
* to get rid of fraction we set
* r = (y1 * z2^3 - y2) (the numerator of lambda * z2^3)
* h = (x1 * z2^2 - x2) (the denominator of lambda * z2^2)
* Hence,
* lambda = r / (h*z2)
*
* With z3 = h*z2 (the denominator of lambda)
* we get x3 = lambda^2*z3^2 - x1*z3^2 - x2/z2^2*z3^2
* = r^2 - x1*h^2*z2^2 - x2*h^2
* = r^2 - h^2*(x1*z2^2 + x2)
* = r^2 - h^2*(h + 2*x2)
* = r^2 - h^3 - 2*h^2*x2
* and y3 = (lambda * (x2/z2^2 - x3/z3^2) - y2/z2^3) * z3^3
* = r * (h^2*x2 - x3) - h^3*y2
*/
/* h = x1*z2^2 - x2
* r = y1*z2^3 - y2
* x3 = r^2 - h^3 - 2*h^2*x2
* y3 = r*(h^2*x2 - x3) - h^3*y2
* z3 = h*z2
*/
// h = x1 * z2^2 - x2;
// r = y1 * z2^3 - y2;
h = p2->z;
bn_multiply(&h, &h, &prime256k1); // h = z2^2
r = p2->z;
bn_multiply(&h, &r, &prime256k1); // r = z2^3
bn_multiply(&p1->x, &h, &prime256k1);
bn_subtractmod(&h, &p2->x, &h, &prime256k1);
// h = x1 * z2^2 - x2;
bn_multiply(&p1->y, &r, &prime256k1);
bn_subtractmod(&r, &p2->y, &r, &prime256k1);
// r = y1 * z2^3 - y2;
// hsqx2 = h^2
hsqx2 = h;
bn_multiply(&hsqx2, &hsqx2, &prime256k1);
// hcb = h^3
hcb = h;
bn_multiply(&hsqx2, &hcb, &prime256k1);
// hsqx2 = h^2 * x2
bn_multiply(&p2->x, &hsqx2, &prime256k1);
// hcby2 = h^3 * y2
hcby2 = hcb;
bn_multiply(&p2->y, &hcby2, &prime256k1);
// rsq = r^2
rsq = r;
bn_multiply(&rsq, &rsq, &prime256k1);
// z3 = h*z2
bn_multiply(&h, &p2->z, &prime256k1);
bn_mod(&p2->z, &prime256k1);
// x3 = r^2 - h^3 - 2h^2x2
tmp1 = 0;
for (j = 0; j < 9; j++) {
tmp1 += (uint64_t) rsq.val[j] + 4*prime256k1.val[j] - hcb.val[j] - 2*hsqx2.val[j];
assert(tmp1 < 5 * 0x40000000ull);
p2->x.val[j] = tmp1 & 0x3fffffff;
tmp1 >>= 30;
}
bn_fast_mod(&p2->x, &prime256k1);
bn_mod(&p2->x, &prime256k1);
// y3 = r*(h^2x2 - x3) - y2*h^3
bn_subtractmod(&hsqx2, &p2->x, &p2->y, &prime256k1);
bn_multiply(&r, &p2->y, &prime256k1);
bn_subtractmod(&p2->y, &hcby2, &p2->y, &prime256k1);
bn_fast_mod(&p2->y, &prime256k1);
bn_mod(&p2->y, &prime256k1);
}
static void point_jacobian_double(jacobian_curve_point *p) {
bignum256 m, msq, ysq, xysq;
int j;
uint32_t tmp1;
/* usual algorithm:
*
* lambda = (3(x/z^2)^2 / 2y/z^3) = 3x^2/2yz
* x3/z3^2 = lambda^2 - 2x/z^2
* y3/z3^3 = lambda * (x/z^2 - x3/z3^2) - y/z^3
*
* to get rid of fraction we set
* m = 3/2 x^2
* Hence,
* lambda = m / yz
*
* With z3 = yz (the denominator of lambda)
* we get x3 = lambda^2*z3^2 - 2*x/z^2*z3^2
* = m^2 - 2*xy^2
* and y3 = (lambda * (x/z^2 - x3/z3^2) - y/z^3) * z3^3
* = m * (xy^2 - x3) - y^4
*/
/* m = 3/2*x*x
* x3 = m^2 - 2*xy^2
* y3 = m*(xy^2 - x3) - 8y^4
* z3 = y*z
*/
m = p->x;
bn_multiply(&m, &m, &prime256k1);
bn_mult_3_2(&m, &prime256k1);
// msq = m^2
msq = m;
bn_multiply(&msq, &msq, &prime256k1);
// ysq = y^2
ysq = p->y;
bn_multiply(&ysq, &ysq, &prime256k1);
// xysq = xy^2
xysq = p->x;
bn_multiply(&ysq, &xysq, &prime256k1);
// z3 = yz
bn_multiply(&p->y, &p->z, &prime256k1);
bn_mod(&p->z, &prime256k1);
// x3 = m^2 - 2*xy^2
tmp1 = 0;
for (j = 0; j < 9; j++) {
tmp1 += msq.val[j] + 3*prime256k1.val[j] - 2*xysq.val[j];
p->x.val[j] = tmp1 & 0x3fffffff;
tmp1 >>= 30;
}
bn_fast_mod(&p->x, &prime256k1);
bn_mod(&p->x, &prime256k1);
// y3 = m*(xy^2 - x3) - y^4
bn_subtractmod(&xysq, &p->x, &p->y, &prime256k1);
bn_multiply(&m, &p->y, &prime256k1);
bn_multiply(&ysq, &ysq, &prime256k1);
bn_subtractmod(&p->y, &ysq, &p->y, &prime256k1);
bn_fast_mod(&p->y, &prime256k1);
bn_mod(&p->y, &prime256k1);
}
// res = k * p
void point_multiply(const bignum256 *k, const curve_point *p, curve_point *res)
{
// this algorithm is loosely based on
// Katsuyuki Okeya and Tsuyoshi Takagi, The Width-w NAF Method Provides
// Small Memory and Fast Elliptic Scalar Multiplications Secure against
// Side Channel Attacks.
assert (bn_is_less(k, &order256k1));
int i, j;
int pos, shift;
bignum256 a;
uint32_t is_even = (k->val[0] & 1) - 1;
uint32_t bits, sign, nsign;
jacobian_curve_point jres;
curve_point pmult[8];
// is_even = 0xffffffff if k is even, 0 otherwise.
// add 2^256.
// make number odd: subtract order256k1 if even
uint32_t tmp = 1;
uint32_t is_non_zero = 0;
for (j = 0; j < 8; j++) {
is_non_zero |= k->val[j];
tmp += 0x3fffffff + k->val[j] - (order256k1.val[j] & is_even);
a.val[j] = tmp & 0x3fffffff;
tmp >>= 30;
}
is_non_zero |= k->val[j];
a.val[j] = tmp + 0xffff + k->val[j] - (order256k1.val[j] & is_even);
assert((a.val[0] & 1) != 0);
// special case 0*p: just return zero. We don't care about constant time.
if (!is_non_zero) {
point_set_infinity(res);
return;
}
// Now a = k + 2^256 (mod order256k1) and a is odd.
//
// The idea is to bring the new a into the form.
// sum_{i=0..64} a[i] 16^i, where |a[i]| < 16 and a[i] is odd.
// a[0] is odd, since a is odd. If a[i] would be even, we can
// add 1 to it and subtract 16 from a[i-1]. Afterwards,
// a[64] = 1, which is the 2^256 that we added before.
//
// Since k = a - 2^256 (mod order256k1), we can compute
// k*p = sum_{i=0..63} a[i] 16^i * p
//
// We compute |a[i]| * p in advance for all possible
// values of |a[i]| * p. pmult[i] = (2*i+1) * p
// We compute p, 3*p, ..., 15*p and store it in the table pmult.
// store p^2 temporarily in pmult[7]
pmult[7] = *p;
point_double(&pmult[7]);
// compute 3*p, etc by repeatedly adding p^2.
pmult[0] = *p;
for (i = 1; i < 8; i++) {
pmult[i] = pmult[7];
point_add(&pmult[i-1], &pmult[i]);
}
// now compute res = sum_{i=0..63} a[i] * 16^i * p step by step,
// starting with i = 63.
// initialize jres = |a[63]| * p.
// Note that a[i] = a>>(4*i) & 0xf if (a&0x10) != 0
// and - (16 - (a>>(4*i) & 0xf)) otherwise. We can compute this as
// ((a ^ (((a >> 4) & 1) - 1)) & 0xf) >> 1
// since a is odd.
bits = a.val[8] >> 12;
sign = (bits >> 4) - 1;
bits ^= sign;
bits &= 15;
curve_to_jacobian(&pmult[bits>>1], &jres);
for (i = 62; i >= 0; i--) {
// sign = sign(a[i+1]) (0xffffffff for negative, 0 for positive)
// invariant jres = (-1)^sign sum_{j=i+1..63} (a[j] * 16^{j-i-1} * p)
point_jacobian_double(&jres);
point_jacobian_double(&jres);
point_jacobian_double(&jres);
point_jacobian_double(&jres);
// get lowest 5 bits of a >> (i*4).
pos = i*4/30; shift = i*4 % 30;
bits = (a.val[pos+1]<<(30-shift) | a.val[pos] >> shift) & 31;
nsign = (bits >> 4) - 1;
bits ^= nsign;
bits &= 15;
// negate last result to make signs of this round and the
// last round equal.
conditional_negate(sign ^ nsign, &jres.z, &prime256k1);
// add odd factor
point_jacobian_add(&pmult[bits >> 1], &jres);
sign = nsign;
}
conditional_negate(sign, &jres.z, &prime256k1);
jacobian_to_curve(&jres, res);
}
#if USE_PRECOMPUTED_CP
if (i < 255 && (k->val[(i + 1) / 30] & (1u << ((i + 1) % 30)))) {
point_add(secp256k1_cp2 + i, res);
i++;
} else {
point_add(secp256k1_cp + i, res);
}
#else
point_add(&curr, res);
#endif
}
}
#if ! USE_PRECOMPUTED_CP
point_double(&curr);
#endif
// res = k * G
// k must be a normalized number with 0 <= k < order256k1
void scalar_multiply(const bignum256 *k, curve_point *res)
{
assert (bn_is_less(k, &order256k1));
int i, j;
bignum256 a;
uint32_t is_even = (k->val[0] & 1) - 1;
uint32_t lowbits;
jacobian_curve_point jres;
// is_even = 0xffffffff if k is even, 0 otherwise.
// add 2^256.
// make number odd: subtract order256k1 if even
uint32_t tmp = 1;
uint32_t is_non_zero = 0;
for (j = 0; j < 8; j++) {
is_non_zero |= k->val[j];
tmp += 0x3fffffff + k->val[j] - (order256k1.val[j] & is_even);
a.val[j] = tmp & 0x3fffffff;
tmp >>= 30;
}
is_non_zero |= k->val[j];
a.val[j] = tmp + 0xffff + k->val[j] - (order256k1.val[j] & is_even);
assert((a.val[0] & 1) != 0);
// special case 0*G: just return zero. We don't care about constant time.
if (!is_non_zero) {
point_set_infinity(res);
return;
}
// Now a = k + 2^256 (mod order256k1) and a is odd.
//
// The idea is to bring the new a into the form.
// sum_{i=0..64} a[i] 16^i, where |a[i]| < 16 and a[i] is odd.
// a[0] is odd, since a is odd. If a[i] would be even, we can
// add 1 to it and subtract 16 from a[i-1]. Afterwards,
// a[64] = 1, which is the 2^256 that we added before.
//
// Since k = a - 2^256 (mod order256k1), we can compute
// k*G = sum_{i=0..63} a[i] 16^i * G
//
// We have a big table secp256k1_cp that stores all possible
// values of |a[i]| 16^i * G.
// secp256k1_cp[i][j] = (2*j+1) * 16^i * G
// now compute res = sum_{i=0..63} a[i] * 16^i * G step by step.
// initial res = |a[0]| * G. Note that a[0] = a & 0xf if (a&0x10) != 0
// and - (16 - (a & 0xf)) otherwise. We can compute this as
// ((a ^ (((a >> 4) & 1) - 1)) & 0xf) >> 1
// since a is odd.
lowbits = a.val[0] & ((1 << 5) - 1);
lowbits ^= (lowbits >> 4) - 1;
lowbits &= 15;
curve_to_jacobian(&secp256k1_cp[0][lowbits >> 1], &jres);
for (i = 1; i < 64; i ++) {
// invariant res = sign(a[i-1]) sum_{j=0..i-1} (a[j] * 16^j * G)
// shift a by 4 places.
for (j = 0; j < 8; j++) {
a.val[j] = (a.val[j] >> 4) | ((a.val[j + 1] & 0xf) << 26);
}
a.val[j] >>= 4;
// a = old(a)>>(4*i)
// a is even iff sign(a[i-1]) = -1
lowbits = a.val[0] & ((1 << 5) - 1);
lowbits ^= (lowbits >> 4) - 1;
lowbits &= 15;
// negate last result to make signs of this round and the
// last round equal.
conditional_negate((lowbits & 1) - 1, &jres.y, &prime256k1);
// add odd factor
point_jacobian_add(&secp256k1_cp[i][lowbits >> 1], &jres);
}
conditional_negate(((a.val[0] >> 4) & 1) - 1, &jres.y, &prime256k1);
jacobian_to_curve(&jres, res);
}
#else
void scalar_multiply(const bignum256 *k, curve_point *res)
{
point_multiply(k, &G256k1, res);
}
#endif
// generate random K for signing
int generate_k_random(bignum256 *k) {
int i, j;
@ -531,7 +885,6 @@ int ecdsa_verify_double(const uint8_t *pub_key, const uint8_t *sig, const uint8_
// returns 0 if verification succeeded
int ecdsa_verify_digest(const uint8_t *pub_key, const uint8_t *sig, const uint8_t *digest)
{
int i, j;
curve_point pub, res;
bignum256 r, s, z;
@ -562,16 +915,8 @@ int ecdsa_verify_digest(const uint8_t *pub_key, const uint8_t *sig, const uint8_
}
// both pub and res can be infinity, can have y = 0 OR can be equal -> false negative
for (i = 0; i < 9; i++) {
for (j = 0; j < 30; j++) {
if (i == 8 && (s.val[i] >> j) == 0) break;
if (s.val[i] & (1u << j)) {
point_add(&pub, &res);
}
point_double(&pub);
}
}
point_multiply(&s, &pub, &pub);
point_add(&pub, &res);
bn_mod(&(res.x), &order256k1);
// signature does not match

5
options.h
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@ -23,11 +23,6 @@
#ifndef __OPTIONS_H__
#define __OPTIONS_H__
// use precomputed Inverse Values of powers of two
#ifndef USE_PRECOMPUTED_IV
#define USE_PRECOMPUTED_IV 1
#endif
// use precomputed Curve Points (some scalar multiples of curve base point G)
#ifndef USE_PRECOMPUTED_CP
#define USE_PRECOMPUTED_CP 1

2952
secp256k1.c
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File diff suppressed because it is too large Load Diff

7
secp256k1.h
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@ -48,13 +48,8 @@ extern const bignum256 order256k1_half;
// 3/2 in G_p
extern const bignum256 three_over_two256k1;
#if USE_PRECOMPUTED_IV
extern const bignum256 secp256k1_iv[256];
#endif
#if USE_PRECOMPUTED_CP
extern const curve_point secp256k1_cp[256];
extern const curve_point secp256k1_cp2[255];
extern const curve_point secp256k1_cp[64][8];
#endif
#endif

2
tests.c
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@ -34,6 +34,7 @@
#include "bip39.h"
#include "ecdsa.h"
#include "pbkdf2.h"
#include "rand.h"
#include "sha2.h"
#include "options.h"
@ -1271,6 +1272,7 @@ Suite *test_suite(void)
int main(void)
{
int number_failed;
init_rand(); // needed for scalar_multiply()
Suite *s = test_suite();
SRunner *sr = srunner_create(s);
srunner_run_all(sr, CK_VERBOSE);

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