ECDSA: Revealing the private key, from four signed messages, two keys and shared nonces (SECP256k1)

It’s well known that the ECDSA signature (as used in Bitcoin) needs to be carefully created, or the private key can be discovered. The nightmare would thus be to sign a few messages and for Eve to then discover the private key from these. In this case, we will look at the case where Alice has two key pairs and signs four…