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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:

$$ (x_1,y_1) = k * G $$ $$r = x_1 \mod n $$ $$s = k^{-1} (z+rd_{a}) \mod n$$

If we know the private key $d_a$ and signature (r,s) would it be possible to calculate random k? If no how hard would it be to calculate that?

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This is immediate: $$ k = s^{-1} (z + r \, d_a) \bmod n $$ where $z$ is the hash value of the message being signed.

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