With ECDSA, given $(r,s)$ and $m$, is there a way for a verifier to calculate any (boolean) properties of $k$, without knowing $k$ or the private key $D_A$?
(I understand that $k$ should be random, or follow RFC6979, but I'm curious.)
In particular, could the verifier compute, given a signature $(r,s)$ and a message $m$, that:
- $k$ is odd
- $k$ has some mathematical relation to one of the curve parameters
- $k$ has some mathematical relation to the public key $Q_A$
- $k$ has some mathematical relation to the (truncated) message hash $z$
- $k$ was deterministically generated with RFC6979 https://en.wikipedia.org/wiki/Elliptic_Curve_Digital_Signature_Algorithm#cite_note-4
for the value of $k$ which was used to generate the signature.
(Variable names taken from https://en.wikipedia.org/wiki/Elliptic_Curve_Digital_Signature_Algorithm)