Elliptic-curve Merkle-Diffie-Hellman. Alice and Bob want to share a key and they decide to use the elliptic-curve Merkle-Diffie-Hellman key agreement. They have decided on the subgroup of the elliptic curve E : y ^2 = x^ 3 + 4x + 4 over the field F31 generated by G = (1, 28).

(a) Determine Alice's public key, given that Alice's private key is a = 2.

(b) Give the compressed point representation of Alice's public key.

(c) Suppose Bob's public key B = (5, 1) is given in compressed point representation. Convert Bob's public key to affine coordinates.

(d) Using Bob's public key B, compute the shared secret of Alice and Bob.