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) in the signature is generated from a (secret) random number , using the formula , being the x-coordinate of the point .
Given two signatures that use the same , 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.