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In ECDSA (Elliptic Curve Digital Signature Algorithm) we use a random number $k$ in the signature process.
Signature in ECDSA is:
$$ (x_1,y_1) = [k]G $$ $$r = x_1 \bmod n $$ $$s = k^{-1} (z+r\,d_{a}) \bmod n$$
If we know the private key $d_a$ and signature $(r,s)$ would it be possible to calculate random $k$? If not how hard would it be to calculate that?
This is immediate: $$ k = s^{-1} (z + r \, d_a) \bmod n $$ where $z$ is the hash value of the message being signed.
