With ECDSA is there a way for the verifier to calculate any properties of $k$? 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)
There are no relation we are currently aware of.
The reason is as follows. The map
$$k \mapsto (k G).x$$ is assumed to be a good pseudo random number generator.
(The NSA infiltration of the Dual EC drgb has nothing to do with that fact).
This basically says that k and r can be seen as independant random variables.