$(4)($Elgamal public key cryptosystem$)$

Start with the prime $p=29$ and the primitive root $g=2.$

Alice chooses the secret key $a=5$ and sends Bob the public key $A=3.$

$($a$)$ Bob wants to send the message $m_1=15$ to Alice.

Bob chooses the random element $k=5.$

Using the Elgamal public key cryptosystem, compute the pair of num$-$

bers $(c_1,c_2)$ that Bob sends to Alice.

$($b$)$ Bob encrypts another message $m_2$ and sends the pair of numbers $(4,16)$

to Alice. Using Elgamal public key decryption, compute the plaintext

message $m_2$ from the ciphertext $(4,16).$