[curves] Schnorr NIZK over Curve 25519
Ben Smith
hyperelliptic at gmail.com
Wed Mar 21 11:35:55 PDT 2018
Hi all,
2018-03-20 21:55 GMT+01:00 Stojan Dimitrovski <sdimitrovski at gmail.com>:
> At the end of the protocol, Bob performs the following checks. If
> any check fails, the verification is unsuccessful.
>
> 1. To verify A is a valid point on the curve and A x [h] is not the
> point at infinity;
>
> 2. To verify V = G x [r] + A x [c].
>
> The first check ensures that A is a valid public key, hence the
> discrete logarithm of A with respect to the base G actually exists.
That's not quite correct. If A is a legitimate multiple of G and T is
a point of order 2, say, then A+T also passes this test, but it has no
discrete log w.r.t. G (and is therefore not a valid public key). What
Test 1 is really telling you is that A is a point on the curve and
that the order of A is not a divisor of h. In this case, where the
curve order is h*prime, this lets you deduce that the order of A is
divisible by the prime---but that's all (there might be bits of h left
over). Multiplying everything by 8 pushes everything right into the
interesting subgroup, and removes that sort of ambiguity.
ben
--
You know we all became mathematicians for the same reason: we were lazy.
--Max Rosenlicht
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