# Calculating random number in ECDSA based on private key and signature

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.