The following is a first attempt at an elliptic curve signature scheme. We have a global elliptic curve, prime \(p\), and "generator" \(G\). Alice picks a private signing key \(X_{A}\) and forms the public verifying key \(Y_{A}=X_{A} G\). To sign a message \(M\) :

- Alice picks a value \(k\).

- Alice sends Bob \(M, k\) and the signature \(S=M-k X_{A} G\).

- Bob verifies that \(M=S+k Y_{A}\).

a. Show that this scheme works. That is, show that the verification process produces an equality if the signature is valid.

b. Show that the scheme is unacceptable by describing a simple technique for forging a user's signature on an arbitrary message.