The DH algorithm can be used for encryption as well using the Elgamal scheme. Demonstrate this encryption scheme using a numerical example as follows.

Alice wish to encrypt a secret message, M $=400$ to Bob. They have chosen the parammeters and private keys as follows:

Bob: private key b $=221,$ generator G$=3,$ prime modulus p $=659.$

Alice: private key a $=122$

Demonstrate how the scheme works by showing what each party computes and sends to each other, showing clearly the cipher texts, and the decrypted messages,

$($i$)$ using the above numbers for M$,$ a$,$ b $(5$ marks$)$

$($ii$)$ using your own choice of numbers for M$,$ a$,$ b