aka The Nonce Reuse Attack

Sony uses a private key, typically stored (in an HSM?) at the company's HQ, to mark their Playstation firmwares as valid and unmodified. The PS3 only needs a public key to verify that the signature came from Sony. Normally, this is considered safe; but Sony did a rookie mistake in the implementation of their signing algorithm - **they used the same random number to sign everything**.

Recall how the (public parameter) $r$ in the signature is generated from a (secret) random number $k$, using the formula $kG = R$, $r$ being the x-coordinate of the point $R$.

Given two signatures that use the same $k$, **prove how you can extract the private key** used for signing. Use the signature formula in the ECDSA section. You'll need pen and paper for this.