/** * Copyright (c) 2013-2014 Tomas Dzetkulic * Copyright (c) 2013-2014 Pavol Rusnak * Copyright (c) 2015 Jochen Hoenicke * * Permission is hereby granted, free of charge, to any person obtaining * a copy of this software and associated documentation files (the "Software"), * to deal in the Software without restriction, including without limitation * the rights to use, copy, modify, merge, publish, distribute, sublicense, * and/or sell copies of the Software, and to permit persons to whom the * Software is furnished to do so, subject to the following conditions: * * The above copyright notice and this permission notice shall be included * in all copies or substantial portions of the Software. * * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS * OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL * THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES * OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, * ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR * OTHER DEALINGS IN THE SOFTWARE. */ #include #include #include #include #include "address.h" #include "base58.h" #include "bignum.h" #include "ecdsa.h" #include "hmac.h" #include "memzero.h" #include "rand.h" #include "rfc6979.h" #include "secp256k1.h" // Set cp2 = cp1 void point_copy(const curve_point *cp1, curve_point *cp2) { *cp2 = *cp1; } // cp2 = cp1 + cp2 void point_add(const ecdsa_curve *curve, const curve_point *cp1, curve_point *cp2) { bignum256 lambda = {0}, inv = {0}, xr = {0}, yr = {0}; if (point_is_infinity(cp1)) { return; } if (point_is_infinity(cp2)) { point_copy(cp1, cp2); return; } if (point_is_equal(cp1, cp2)) { point_double(curve, cp2); return; } if (point_is_negative_of(cp1, cp2)) { point_set_infinity(cp2); return; } // lambda = (y2 - y1) / (x2 - x1) bn_subtractmod(&(cp2->x), &(cp1->x), &inv, &curve->prime); bn_inverse(&inv, &curve->prime); bn_subtractmod(&(cp2->y), &(cp1->y), &lambda, &curve->prime); bn_multiply(&inv, &lambda, &curve->prime); // xr = lambda^2 - x1 - x2 xr = lambda; bn_multiply(&xr, &xr, &curve->prime); yr = cp1->x; bn_addmod(&yr, &(cp2->x), &curve->prime); bn_subtractmod(&xr, &yr, &xr, &curve->prime); bn_fast_mod(&xr, &curve->prime); bn_mod(&xr, &curve->prime); // yr = lambda (x1 - xr) - y1 bn_subtractmod(&(cp1->x), &xr, &yr, &curve->prime); bn_multiply(&lambda, &yr, &curve->prime); bn_subtractmod(&yr, &(cp1->y), &yr, &curve->prime); bn_fast_mod(&yr, &curve->prime); bn_mod(&yr, &curve->prime); cp2->x = xr; cp2->y = yr; } // cp = cp + cp void point_double(const ecdsa_curve *curve, curve_point *cp) { bignum256 lambda = {0}, xr = {0}, yr = {0}; if (point_is_infinity(cp)) { return; } if (bn_is_zero(&(cp->y))) { point_set_infinity(cp); return; } // lambda = (3 x^2 + a) / (2 y) lambda = cp->y; bn_mult_k(&lambda, 2, &curve->prime); bn_fast_mod(&lambda, &curve->prime); bn_mod(&lambda, &curve->prime); bn_inverse(&lambda, &curve->prime); xr = cp->x; bn_multiply(&xr, &xr, &curve->prime); bn_mult_k(&xr, 3, &curve->prime); bn_subi(&xr, -curve->a, &curve->prime); bn_multiply(&xr, &lambda, &curve->prime); // xr = lambda^2 - 2*x xr = lambda; bn_multiply(&xr, &xr, &curve->prime); yr = cp->x; bn_lshift(&yr); bn_subtractmod(&xr, &yr, &xr, &curve->prime); bn_fast_mod(&xr, &curve->prime); bn_mod(&xr, &curve->prime); // yr = lambda (x - xr) - y bn_subtractmod(&(cp->x), &xr, &yr, &curve->prime); bn_multiply(&lambda, &yr, &curve->prime); bn_subtractmod(&yr, &(cp->y), &yr, &curve->prime); bn_fast_mod(&yr, &curve->prime); bn_mod(&yr, &curve->prime); cp->x = xr; cp->y = yr; } // set point to internal representation of point at infinity void point_set_infinity(curve_point *p) { bn_zero(&(p->x)); bn_zero(&(p->y)); } // return true iff p represent point at infinity // both coords are zero in internal representation int point_is_infinity(const curve_point *p) { return bn_is_zero(&(p->x)) && bn_is_zero(&(p->y)); } // return true iff both points are equal int point_is_equal(const curve_point *p, const curve_point *q) { return bn_is_equal(&(p->x), &(q->x)) && bn_is_equal(&(p->y), &(q->y)); } // returns true iff p == -q // expects p and q be valid points on curve other than point at infinity int point_is_negative_of(const curve_point *p, const curve_point *q) { // if P == (x, y), then -P would be (x, -y) on this curve if (!bn_is_equal(&(p->x), &(q->x))) { return 0; } // we shouldn't hit this for a valid point if (bn_is_zero(&(p->y))) { return 0; } return !bn_is_equal(&(p->y), &(q->y)); } typedef struct jacobian_curve_point { bignum256 x, y, z; } jacobian_curve_point; // generate random K for signing/side-channel noise static void generate_k_random(bignum256 *k, const bignum256 *prime) { do { int i = 0; for (i = 0; i < 8; i++) { k->val[i] = random32() & ((1u << BN_BITS_PER_LIMB) - 1); } k->val[8] = random32() & ((1u << BN_BITS_LAST_LIMB) - 1); // check that k is in range and not zero. } while (bn_is_zero(k) || !bn_is_less(k, prime)); } void curve_to_jacobian(const curve_point *p, jacobian_curve_point *jp, const bignum256 *prime) { // randomize z coordinate generate_k_random(&jp->z, prime); jp->x = jp->z; bn_multiply(&jp->z, &jp->x, prime); // x = z^2 jp->y = jp->x; bn_multiply(&jp->z, &jp->y, prime); // y = z^3 bn_multiply(&p->x, &jp->x, prime); bn_multiply(&p->y, &jp->y, prime); } void jacobian_to_curve(const jacobian_curve_point *jp, curve_point *p, const bignum256 *prime) { p->y = jp->z; bn_inverse(&p->y, prime); // p->y = z^-1 p->x = p->y; bn_multiply(&p->x, &p->x, prime); // p->x = z^-2 bn_multiply(&p->x, &p->y, prime); // p->y = z^-3 bn_multiply(&jp->x, &p->x, prime); // p->x = jp->x * z^-2 bn_multiply(&jp->y, &p->y, prime); // p->y = jp->y * z^-3 bn_mod(&p->x, prime); bn_mod(&p->y, prime); } void point_jacobian_add(const curve_point *p1, jacobian_curve_point *p2, const ecdsa_curve *curve) { bignum256 r = {0}, h = {0}, r2 = {0}; bignum256 hcby = {0}, hsqx = {0}; bignum256 xz = {0}, yz = {0}, az = {0}; int is_doubling = 0; const bignum256 *prime = &curve->prime; int a = curve->a; assert(-3 <= a && a <= 0); /* First we bring p1 to the same denominator: * x1' := x1 * z2^2 * y1' := y1 * z2^3 */ /* * lambda = ((y1' - y2)/z2^3) / ((x1' - x2)/z2^2) * = (y1' - y2) / (x1' - x2) z2 * x3/z3^2 = lambda^2 - (x1' + x2)/z2^2 * y3/z3^3 = 1/2 lambda * (2x3/z3^2 - (x1' + x2)/z2^2) + (y1'+y2)/z2^3 * * For the special case x1=x2, y1=y2 (doubling) we have * lambda = 3/2 ((x2/z2^2)^2 + a) / (y2/z2^3) * = 3/2 (x2^2 + a*z2^4) / y2*z2) * * to get rid of fraction we write lambda as * lambda = r / (h*z2) * with r = is_doubling ? 3/2 x2^2 + az2^4 : (y1 - y2) * h = is_doubling ? y1+y2 : (x1 - x2) * * With z3 = h*z2 (the denominator of lambda) * we get x3 = lambda^2*z3^2 - (x1' + x2)/z2^2*z3^2 * = r^2 - h^2 * (x1' + x2) * and y3 = 1/2 r * (2x3 - h^2*(x1' + x2)) + h^3*(y1' + y2) */ /* h = x1 - x2 * r = y1 - y2 * x3 = r^2 - h^3 - 2*h^2*x2 * y3 = r*(h^2*x2 - x3) - h^3*y2 * z3 = h*z2 */ xz = p2->z; bn_multiply(&xz, &xz, prime); // xz = z2^2 yz = p2->z; bn_multiply(&xz, &yz, prime); // yz = z2^3 if (a != 0) { az = xz; bn_multiply(&az, &az, prime); // az = z2^4 bn_mult_k(&az, -a, prime); // az = -az2^4 } bn_multiply(&p1->x, &xz, prime); // xz = x1' = x1*z2^2; h = xz; bn_subtractmod(&h, &p2->x, &h, prime); bn_fast_mod(&h, prime); // h = x1' - x2; bn_add(&xz, &p2->x); // xz = x1' + x2 // check for h == 0 % prime. Note that h never normalizes to // zero, since h = x1' + 2*prime - x2 > 0 and a positive // multiple of prime is always normalized to prime by // bn_fast_mod. is_doubling = bn_is_equal(&h, prime); bn_multiply(&p1->y, &yz, prime); // yz = y1' = y1*z2^3; bn_subtractmod(&yz, &p2->y, &r, prime); // r = y1' - y2; bn_add(&yz, &p2->y); // yz = y1' + y2 r2 = p2->x; bn_multiply(&r2, &r2, prime); bn_mult_k(&r2, 3, prime); if (a != 0) { // subtract -a z2^4, i.e, add a z2^4 bn_subtractmod(&r2, &az, &r2, prime); } bn_cmov(&r, is_doubling, &r2, &r); bn_cmov(&h, is_doubling, &yz, &h); // hsqx = h^2 hsqx = h; bn_multiply(&hsqx, &hsqx, prime); // hcby = h^3 hcby = h; bn_multiply(&hsqx, &hcby, prime); // hsqx = h^2 * (x1 + x2) bn_multiply(&xz, &hsqx, prime); // hcby = h^3 * (y1 + y2) bn_multiply(&yz, &hcby, prime); // z3 = h*z2 bn_multiply(&h, &p2->z, prime); // x3 = r^2 - h^2 (x1 + x2) p2->x = r; bn_multiply(&p2->x, &p2->x, prime); bn_subtractmod(&p2->x, &hsqx, &p2->x, prime); bn_fast_mod(&p2->x, prime); // y3 = 1/2 (r*(h^2 (x1 + x2) - 2x3) - h^3 (y1 + y2)) bn_subtractmod(&hsqx, &p2->x, &p2->y, prime); bn_subtractmod(&p2->y, &p2->x, &p2->y, prime); bn_multiply(&r, &p2->y, prime); bn_subtractmod(&p2->y, &hcby, &p2->y, prime); bn_mult_half(&p2->y, prime); bn_fast_mod(&p2->y, prime); } void point_jacobian_double(jacobian_curve_point *p, const ecdsa_curve *curve) { bignum256 az4 = {0}, m = {0}, msq = {0}, ysq = {0}, xysq = {0}; const bignum256 *prime = &curve->prime; assert(-3 <= curve->a && curve->a <= 0); /* usual algorithm: * * lambda = (3((x/z^2)^2 + a) / 2y/z^3) = (3x^2 + az^4)/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 x^2 + az^4) / 2 * Hence, * lambda = m / yz = m / z3 * * 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*x^2 + a z^4) / 2 * x3 = m^2 - 2*xy^2 * y3 = m*(xy^2 - x3) - 8y^4 * z3 = y*z */ m = p->x; bn_multiply(&m, &m, prime); bn_mult_k(&m, 3, prime); az4 = p->z; bn_multiply(&az4, &az4, prime); bn_multiply(&az4, &az4, prime); bn_mult_k(&az4, -curve->a, prime); bn_subtractmod(&m, &az4, &m, prime); bn_mult_half(&m, prime); // msq = m^2 msq = m; bn_multiply(&msq, &msq, prime); // ysq = y^2 ysq = p->y; bn_multiply(&ysq, &ysq, prime); // xysq = xy^2 xysq = p->x; bn_multiply(&ysq, &xysq, prime); // z3 = yz bn_multiply(&p->y, &p->z, prime); // x3 = m^2 - 2*xy^2 p->x = xysq; bn_lshift(&p->x); bn_fast_mod(&p->x, prime); bn_subtractmod(&msq, &p->x, &p->x, prime); bn_fast_mod(&p->x, prime); // y3 = m*(xy^2 - x3) - y^4 bn_subtractmod(&xysq, &p->x, &p->y, prime); bn_multiply(&m, &p->y, prime); bn_multiply(&ysq, &ysq, prime); bn_subtractmod(&p->y, &ysq, &p->y, prime); bn_fast_mod(&p->y, prime); } // res = k * p void point_multiply(const ecdsa_curve *curve, 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, &curve->order)); int i = 0, j = 0; static CONFIDENTIAL bignum256 a; uint32_t *aptr = NULL; uint32_t abits = 0; int ashift = 0; uint32_t is_even = (k->val[0] & 1) - 1; uint32_t bits = {0}, sign = {0}, nsign = {0}; static CONFIDENTIAL jacobian_curve_point jres; curve_point pmult[8] = {0}; const bignum256 *prime = &curve->prime; // is_even = 0xffffffff if k is even, 0 otherwise. // add 2^256. // make number odd: subtract curve->order 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 += (BN_BASE - 1) + k->val[j] - (curve->order.val[j] & is_even); a.val[j] = tmp & (BN_BASE - 1); tmp >>= BN_BITS_PER_LIMB; } is_non_zero |= k->val[j]; a.val[j] = tmp + 0xffffff + k->val[j] - (curve->order.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 curve->order) 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 curve->order), 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(curve, &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(curve, &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. aptr = &a.val[8]; abits = *aptr; ashift = 256 - (BN_BITS_PER_LIMB * 8) - 4; bits = abits >> ashift; sign = (bits >> 4) - 1; bits ^= sign; bits &= 15; curve_to_jacobian(&pmult[bits >> 1], &jres, prime); 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) // abits >> (ashift - 4) = lowbits(a >> (i*4)) point_jacobian_double(&jres, curve); point_jacobian_double(&jres, curve); point_jacobian_double(&jres, curve); point_jacobian_double(&jres, curve); // get lowest 5 bits of a >> (i*4). ashift -= 4; if (ashift < 0) { // the condition only depends on the iteration number and // leaks no private information to a side-channel. bits = abits << (-ashift); abits = *(--aptr); ashift += BN_BITS_PER_LIMB; bits |= abits >> ashift; } else { bits = abits >> ashift; } bits &= 31; nsign = (bits >> 4) - 1; bits ^= nsign; bits &= 15; // negate last result to make signs of this round and the // last round equal. bn_cnegate((sign ^ nsign) & 1, &jres.z, prime); // add odd factor point_jacobian_add(&pmult[bits >> 1], &jres, curve); sign = nsign; } bn_cnegate(sign & 1, &jres.z, prime); jacobian_to_curve(&jres, res, prime); memzero(&a, sizeof(a)); memzero(&jres, sizeof(jres)); } #if USE_PRECOMPUTED_CP // res = k * G // k must be a normalized number with 0 <= k < curve->order void scalar_multiply(const ecdsa_curve *curve, const bignum256 *k, curve_point *res) { assert(bn_is_less(k, &curve->order)); int i = {0}, j = {0}; static CONFIDENTIAL bignum256 a; uint32_t is_even = (k->val[0] & 1) - 1; uint32_t lowbits = 0; static CONFIDENTIAL jacobian_curve_point jres; const bignum256 *prime = &curve->prime; // is_even = 0xffffffff if k is even, 0 otherwise. // add 2^256. // make number odd: subtract curve->order 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 += (BN_BASE - 1) + k->val[j] - (curve->order.val[j] & is_even); a.val[j] = tmp & (BN_BASE - 1); tmp >>= BN_BITS_PER_LIMB; } is_non_zero |= k->val[j]; a.val[j] = tmp + 0xffffff + k->val[j] - (curve->order.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 curve->order) 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 curve->order), we can compute // k*G = sum_{i=0..63} a[i] 16^i * G // // We have a big table curve->cp that stores all possible // values of |a[i]| 16^i * G. // curve->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(&curve->cp[0][lowbits >> 1], &jres, prime); 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) << (BN_BITS_PER_LIMB - 4)); } 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. bn_cnegate(~lowbits & 1, &jres.y, prime); // add odd factor point_jacobian_add(&curve->cp[i][lowbits >> 1], &jres, curve); } bn_cnegate(~(a.val[0] >> 4) & 1, &jres.y, prime); jacobian_to_curve(&jres, res, prime); memzero(&a, sizeof(a)); memzero(&jres, sizeof(jres)); } #else void scalar_multiply(const ecdsa_curve *curve, const bignum256 *k, curve_point *res) { point_multiply(curve, k, &curve->G, res); } #endif int ecdh_multiply(const ecdsa_curve *curve, const uint8_t *priv_key, const uint8_t *pub_key, uint8_t *session_key) { curve_point point = {0}; if (!ecdsa_read_pubkey(curve, pub_key, &point)) { return 1; } bignum256 k = {0}; bn_read_be(priv_key, &k); point_multiply(curve, &k, &point, &point); memzero(&k, sizeof(k)); session_key[0] = 0x04; bn_write_be(&point.x, session_key + 1); bn_write_be(&point.y, session_key + 33); memzero(&point, sizeof(point)); return 0; } // msg is a data to be signed // msg_len is the message length int ecdsa_sign(const ecdsa_curve *curve, HasherType hasher_sign, const uint8_t *priv_key, const uint8_t *msg, uint32_t msg_len, uint8_t *sig, uint8_t *pby, int (*is_canonical)(uint8_t by, uint8_t sig[64])) { uint8_t hash[32] = {0}; hasher_Raw(hasher_sign, msg, msg_len, hash); int res = ecdsa_sign_digest(curve, priv_key, hash, sig, pby, is_canonical); memzero(hash, sizeof(hash)); return res; } // uses secp256k1 curve // priv_key is a 32 byte big endian stored number // sig is 64 bytes long array for the signature // digest is 32 bytes of digest // is_canonical is an optional function that checks if the signature // conforms to additional coin-specific rules. int ecdsa_sign_digest(const ecdsa_curve *curve, const uint8_t *priv_key, const uint8_t *digest, uint8_t *sig, uint8_t *pby, int (*is_canonical)(uint8_t by, uint8_t sig[64])) { int i = 0; curve_point R = {0}; bignum256 k = {0}, z = {0}, randk = {0}; bignum256 *s = &R.y; uint8_t by; // signature recovery byte #if USE_RFC6979 rfc6979_state rng = {0}; init_rfc6979(priv_key, digest, &rng); #endif bn_read_be(digest, &z); for (i = 0; i < 10000; i++) { #if USE_RFC6979 // generate K deterministically generate_k_rfc6979(&k, &rng); // if k is too big or too small, we don't like it if (bn_is_zero(&k) || !bn_is_less(&k, &curve->order)) { continue; } #else // generate random number k generate_k_random(&k, &curve->order); #endif // compute k*G scalar_multiply(curve, &k, &R); by = R.y.val[0] & 1; // r = (rx mod n) if (!bn_is_less(&R.x, &curve->order)) { bn_subtract(&R.x, &curve->order, &R.x); by |= 2; } // if r is zero, we retry if (bn_is_zero(&R.x)) { continue; } // randomize operations to counter side-channel attacks generate_k_random(&randk, &curve->order); bn_multiply(&randk, &k, &curve->order); // k*rand bn_inverse(&k, &curve->order); // (k*rand)^-1 bn_read_be(priv_key, s); // priv bn_multiply(&R.x, s, &curve->order); // R.x*priv bn_add(s, &z); // R.x*priv + z bn_multiply(&k, s, &curve->order); // (k*rand)^-1 (R.x*priv + z) bn_multiply(&randk, s, &curve->order); // k^-1 (R.x*priv + z) bn_mod(s, &curve->order); // if s is zero, we retry if (bn_is_zero(s)) { continue; } // if S > order/2 => S = -S if (bn_is_less(&curve->order_half, s)) { bn_subtract(&curve->order, s, s); by ^= 1; } // we are done, R.x and s is the result signature bn_write_be(&R.x, sig); bn_write_be(s, sig + 32); // check if the signature is acceptable or retry if (is_canonical && !is_canonical(by, sig)) { continue; } if (pby) { *pby = by; } memzero(&k, sizeof(k)); memzero(&randk, sizeof(randk)); #if USE_RFC6979 memzero(&rng, sizeof(rng)); #endif return 0; } // Too many retries without a valid signature // -> fail with an error memzero(&k, sizeof(k)); memzero(&randk, sizeof(randk)); #if USE_RFC6979 memzero(&rng, sizeof(rng)); #endif return -1; } void ecdsa_get_public_key33(const ecdsa_curve *curve, const uint8_t *priv_key, uint8_t *pub_key) { curve_point R = {0}; bignum256 k = {0}; bn_read_be(priv_key, &k); // compute k*G scalar_multiply(curve, &k, &R); pub_key[0] = 0x02 | (R.y.val[0] & 0x01); bn_write_be(&R.x, pub_key + 1); memzero(&R, sizeof(R)); memzero(&k, sizeof(k)); } void ecdsa_get_public_key65(const ecdsa_curve *curve, const uint8_t *priv_key, uint8_t *pub_key) { curve_point R = {0}; bignum256 k = {0}; bn_read_be(priv_key, &k); // compute k*G scalar_multiply(curve, &k, &R); pub_key[0] = 0x04; bn_write_be(&R.x, pub_key + 1); bn_write_be(&R.y, pub_key + 33); memzero(&R, sizeof(R)); memzero(&k, sizeof(k)); } int ecdsa_uncompress_pubkey(const ecdsa_curve *curve, const uint8_t *pub_key, uint8_t *uncompressed) { curve_point pub = {0}; if (!ecdsa_read_pubkey(curve, pub_key, &pub)) { return 0; } uncompressed[0] = 4; bn_write_be(&pub.x, uncompressed + 1); bn_write_be(&pub.y, uncompressed + 33); return 1; } void ecdsa_get_pubkeyhash(const uint8_t *pub_key, HasherType hasher_pubkey, uint8_t *pubkeyhash) { uint8_t h[HASHER_DIGEST_LENGTH] = {0}; if (pub_key[0] == 0x04) { // uncompressed format hasher_Raw(hasher_pubkey, pub_key, 65, h); } else if (pub_key[0] == 0x00) { // point at infinity hasher_Raw(hasher_pubkey, pub_key, 1, h); } else { // expecting compressed format hasher_Raw(hasher_pubkey, pub_key, 33, h); } memcpy(pubkeyhash, h, 20); memzero(h, sizeof(h)); } void ecdsa_get_address_raw(const uint8_t *pub_key, uint32_t version, HasherType hasher_pubkey, uint8_t *addr_raw) { size_t prefix_len = address_prefix_bytes_len(version); address_write_prefix_bytes(version, addr_raw); ecdsa_get_pubkeyhash(pub_key, hasher_pubkey, addr_raw + prefix_len); } void ecdsa_get_address(const uint8_t *pub_key, uint32_t version, HasherType hasher_pubkey, HasherType hasher_base58, char *addr, int addrsize) { uint8_t raw[MAX_ADDR_RAW_SIZE] = {0}; size_t prefix_len = address_prefix_bytes_len(version); ecdsa_get_address_raw(pub_key, version, hasher_pubkey, raw); base58_encode_check(raw, 20 + prefix_len, hasher_base58, addr, addrsize); // not as important to clear this one, but we might as well memzero(raw, sizeof(raw)); } void ecdsa_get_address_segwit_p2sh_raw(const uint8_t *pub_key, uint32_t version, HasherType hasher_pubkey, uint8_t *addr_raw) { uint8_t buf[32 + 2] = {0}; buf[0] = 0; // version byte buf[1] = 20; // push 20 bytes ecdsa_get_pubkeyhash(pub_key, hasher_pubkey, buf + 2); size_t prefix_len = address_prefix_bytes_len(version); address_write_prefix_bytes(version, addr_raw); hasher_Raw(hasher_pubkey, buf, 22, addr_raw + prefix_len); } void ecdsa_get_address_segwit_p2sh(const uint8_t *pub_key, uint32_t version, HasherType hasher_pubkey, HasherType hasher_base58, char *addr, int addrsize) { uint8_t raw[MAX_ADDR_RAW_SIZE] = {0}; size_t prefix_len = address_prefix_bytes_len(version); ecdsa_get_address_segwit_p2sh_raw(pub_key, version, hasher_pubkey, raw); base58_encode_check(raw, prefix_len + 20, hasher_base58, addr, addrsize); memzero(raw, sizeof(raw)); } void ecdsa_get_wif(const uint8_t *priv_key, uint32_t version, HasherType hasher_base58, char *wif, int wifsize) { uint8_t wif_raw[MAX_WIF_RAW_SIZE] = {0}; size_t prefix_len = address_prefix_bytes_len(version); address_write_prefix_bytes(version, wif_raw); memcpy(wif_raw + prefix_len, priv_key, 32); wif_raw[prefix_len + 32] = 0x01; base58_encode_check(wif_raw, prefix_len + 32 + 1, hasher_base58, wif, wifsize); // private keys running around our stack can cause trouble memzero(wif_raw, sizeof(wif_raw)); } int ecdsa_address_decode(const char *addr, uint32_t version, HasherType hasher_base58, uint8_t *out) { if (!addr) return 0; int prefix_len = address_prefix_bytes_len(version); return base58_decode_check(addr, hasher_base58, out, 20 + prefix_len) == 20 + prefix_len && address_check_prefix(out, version); } void compress_coords(const curve_point *cp, uint8_t *compressed) { compressed[0] = bn_is_odd(&cp->y) ? 0x03 : 0x02; bn_write_be(&cp->x, compressed + 1); } void uncompress_coords(const ecdsa_curve *curve, uint8_t odd, const bignum256 *x, bignum256 *y) { // y^2 = x^3 + a*x + b memcpy(y, x, sizeof(bignum256)); // y is x bn_multiply(x, y, &curve->prime); // y is x^2 bn_subi(y, -curve->a, &curve->prime); // y is x^2 + a bn_multiply(x, y, &curve->prime); // y is x^3 + ax bn_add(y, &curve->b); // y is x^3 + ax + b bn_sqrt(y, &curve->prime); // y = sqrt(y) if ((odd & 0x01) != (y->val[0] & 1)) { bn_subtract(&curve->prime, y, y); // y = -y } } int ecdsa_read_pubkey(const ecdsa_curve *curve, const uint8_t *pub_key, curve_point *pub) { if (!curve) { curve = &secp256k1; } if (pub_key[0] == 0x04) { bn_read_be(pub_key + 1, &(pub->x)); bn_read_be(pub_key + 33, &(pub->y)); return ecdsa_validate_pubkey(curve, pub); } if (pub_key[0] == 0x02 || pub_key[0] == 0x03) { // compute missing y coords bn_read_be(pub_key + 1, &(pub->x)); uncompress_coords(curve, pub_key[0], &(pub->x), &(pub->y)); return ecdsa_validate_pubkey(curve, pub); } // error return 0; } // Verifies that: // - pub is not the point at infinity. // - pub->x and pub->y are in range [0,p-1]. // - pub is on the curve. // We assume that all curves using this code have cofactor 1, so there is no // need to verify that pub is a scalar multiple of G. int ecdsa_validate_pubkey(const ecdsa_curve *curve, const curve_point *pub) { bignum256 y_2 = {0}, x3_ax_b = {0}; if (point_is_infinity(pub)) { return 0; } if (!bn_is_less(&(pub->x), &curve->prime) || !bn_is_less(&(pub->y), &curve->prime)) { return 0; } memcpy(&y_2, &(pub->y), sizeof(bignum256)); memcpy(&x3_ax_b, &(pub->x), sizeof(bignum256)); // y^2 bn_multiply(&(pub->y), &y_2, &curve->prime); bn_mod(&y_2, &curve->prime); // x^3 + ax + b bn_multiply(&(pub->x), &x3_ax_b, &curve->prime); // x^2 bn_subi(&x3_ax_b, -curve->a, &curve->prime); // x^2 + a bn_multiply(&(pub->x), &x3_ax_b, &curve->prime); // x^3 + ax bn_addmod(&x3_ax_b, &curve->b, &curve->prime); // x^3 + ax + b bn_mod(&x3_ax_b, &curve->prime); if (!bn_is_equal(&x3_ax_b, &y_2)) { return 0; } return 1; } // uses secp256k1 curve // pub_key - 65 bytes uncompressed key // signature - 64 bytes signature // msg is a data that was signed // msg_len is the message length int ecdsa_verify(const ecdsa_curve *curve, HasherType hasher_sign, const uint8_t *pub_key, const uint8_t *sig, const uint8_t *msg, uint32_t msg_len) { uint8_t hash[32] = {0}; hasher_Raw(hasher_sign, msg, msg_len, hash); int res = ecdsa_verify_digest(curve, pub_key, sig, hash); memzero(hash, sizeof(hash)); return res; } // Compute public key from signature and recovery id. // returns 0 if the key is successfully recovered int ecdsa_recover_pub_from_sig(const ecdsa_curve *curve, uint8_t *pub_key, const uint8_t *sig, const uint8_t *digest, int recid) { bignum256 r = {0}, s = {0}, e = {0}; curve_point cp = {0}, cp2 = {0}; // read r and s bn_read_be(sig, &r); bn_read_be(sig + 32, &s); if (!bn_is_less(&r, &curve->order) || bn_is_zero(&r)) { return 1; } if (!bn_is_less(&s, &curve->order) || bn_is_zero(&s)) { return 1; } // cp = R = k * G (k is secret nonce when signing) memcpy(&cp.x, &r, sizeof(bignum256)); if (recid & 2) { bn_add(&cp.x, &curve->order); if (!bn_is_less(&cp.x, &curve->prime)) { return 1; } } // compute y from x uncompress_coords(curve, recid & 1, &cp.x, &cp.y); if (!ecdsa_validate_pubkey(curve, &cp)) { return 1; } // e = -digest bn_read_be(digest, &e); bn_mod(&e, &curve->order); bn_subtract(&curve->order, &e, &e); // r = r^-1 bn_inverse(&r, &curve->order); // e = -digest * r^-1 bn_multiply(&r, &e, &curve->order); bn_mod(&e, &curve->order); // s = s * r^-1 bn_multiply(&r, &s, &curve->order); bn_mod(&s, &curve->order); // cp = s * r^-1 * k * G point_multiply(curve, &s, &cp, &cp); // cp2 = -digest * r^-1 * G scalar_multiply(curve, &e, &cp2); // cp = (s * r^-1 * k - digest * r^-1) * G = Pub point_add(curve, &cp2, &cp); pub_key[0] = 0x04; bn_write_be(&cp.x, pub_key + 1); bn_write_be(&cp.y, pub_key + 33); return 0; } // returns 0 if verification succeeded int ecdsa_verify_digest(const ecdsa_curve *curve, const uint8_t *pub_key, const uint8_t *sig, const uint8_t *digest) { curve_point pub = {0}, res = {0}; bignum256 r = {0}, s = {0}, z = {0}; int result = 0; if (!ecdsa_read_pubkey(curve, pub_key, &pub)) { result = 1; } if (result == 0) { bn_read_be(sig, &r); bn_read_be(sig + 32, &s); bn_read_be(digest, &z); if (bn_is_zero(&r) || bn_is_zero(&s) || (!bn_is_less(&r, &curve->order)) || (!bn_is_less(&s, &curve->order))) { result = 2; } if (bn_is_zero(&z)) { // The digest was all-zero. The probability of this happening by chance is // infinitesimal, but it could be induced by a fault injection. In this // case the signature (r,s) can be forged by taking r := (t * Q).x mod n // and s := r * t^-1 mod n for any t in [1, n-1]. We fail verification, // because there is no guarantee that the signature was created by the // owner of the private key. result = 3; } } if (result == 0) { bn_inverse(&s, &curve->order); // s = s^-1 bn_multiply(&s, &z, &curve->order); // z = z * s [u1 = z * s^-1 mod n] bn_mod(&z, &curve->order); } if (result == 0) { bn_multiply(&r, &s, &curve->order); // s = r * s [u2 = r * s^-1 mod n] bn_mod(&s, &curve->order); scalar_multiply(curve, &z, &res); // res = z * G [= u1 * G] point_multiply(curve, &s, &pub, &pub); // pub = s * pub [= u2 * Q] point_add(curve, &pub, &res); // res = pub + res [R = u1 * G + u2 * Q] if (point_is_infinity(&res)) { // R == Infinity result = 4; } } if (result == 0) { bn_mod(&(res.x), &curve->order); if (!bn_is_equal(&res.x, &r)) { // R.x != r // signature does not match result = 5; } } memzero(&pub, sizeof(pub)); memzero(&res, sizeof(res)); memzero(&r, sizeof(r)); memzero(&s, sizeof(s)); memzero(&z, sizeof(z)); // all OK return result; } int ecdsa_sig_to_der(const uint8_t *sig, uint8_t *der) { int i = 0; uint8_t *p = der, *len = NULL, *len1 = NULL, *len2 = NULL; *p = 0x30; p++; // sequence *p = 0x00; len = p; p++; // len(sequence) *p = 0x02; p++; // integer *p = 0x00; len1 = p; p++; // len(integer) // process R i = 0; while (sig[i] == 0 && i < 32) { i++; } // skip leading zeroes if (sig[i] >= 0x80) { // put zero in output if MSB set *p = 0x00; p++; *len1 = *len1 + 1; } while (i < 32) { // copy bytes to output *p = sig[i]; p++; *len1 = *len1 + 1; i++; } *p = 0x02; p++; // integer *p = 0x00; len2 = p; p++; // len(integer) // process S i = 32; while (sig[i] == 0 && i < 64) { i++; } // skip leading zeroes if (sig[i] >= 0x80) { // put zero in output if MSB set *p = 0x00; p++; *len2 = *len2 + 1; } while (i < 64) { // copy bytes to output *p = sig[i]; p++; *len2 = *len2 + 1; i++; } *len = *len1 + *len2 + 4; return *len + 2; }