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Add explanation about how inversion can be avoided

This commit is contained in:
Pieter Wuille 2014-12-01 13:29:47 +01:00
parent ce7eb6fb3d
commit 13278f642c
1 changed files with 19 additions and 1 deletions
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20
src/ecdsa_impl.h
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@ -126,15 +126,33 @@ static int secp256k1_ecdsa_sig_verify(const secp256k1_ecdsa_sig_t *sig, const se
secp256k1_scalar_get_b32(c, &sig->r);
secp256k1_fe_t xr;
secp256k1_fe_set_b32(&xr, c);
// We now have the recomputed R point in pr, and its claimed x coordinate (modulo n)
// in xr. Naively, we would extract the x coordinate from pr (requiring a inversion modulo p),
// compute the remainder modulo n, and compare it to xr. However:
//
// xr == X(pr) mod n
// <=> exists h. (xr + h * n < p && xr + h * n == X(pr))
// [Since 2 * n > p, h can only be 0 or 1]
// <=> (xr == X(pr)) || (xr + n < p && xr + n == X(pr))
// [In Jacobian coordinates, X(pr) is pr.x / pr.z^2 mod p]
// <=> (xr == pr.x / pr.z^2 mod p) || (xr + n < p && xr + n == pr.x / pr.z^2 mod p)
// [Multiplying both sides of the equations by pr.z^2 mod p]
// <=> (xr * pr.z^2 mod p == pr.x) || (xr + n < p && (xr + n) * pr.z^2 mod p == pr.x)
//
// Thus, we can avoid the inversion, but we have to check both cases separately.
// secp256k1_gej_eq_x implements the (xr * pr.z^2 mod p == pr.x) test.
if (secp256k1_gej_eq_x_var(&xr, &pr)) {
// xr.x == xr * xr.z^2 mod p, so the signature is valid.
return 1;
}
if (secp256k1_fe_cmp_var(&xr, &secp256k1_ecdsa_consts->p_minus_order) >= 0) {
// We can't add the order to r. This will be the case for almost every r.
// xr + p >= n, so we can skip testing the second case.
return 0;
}
secp256k1_fe_add(&xr, &secp256k1_ecdsa_consts->order_as_fe);
if (secp256k1_gej_eq_x_var(&xr, &pr)) {
// (xr + n) * pr.z^2 mod p == pr.x, so the signature is valid.
return 1;
}
return 0;