### Instruction: Consider the ElGamal public key encryption system with a large prime number $p=137$ and the primitive root $g=3$. Alice chooses a private key $a=5$ and computes her public key as $A=g^a \pmod{p}$. She wants to send the message "87" to Bob, who has the public key $B=21$. 

What is the ciphertext that Alice sends to Bob? Use the ElGamal encryption method and show all steps of the computation.### Response: To encrypt the message "87" using the ElGamal encryption method, Alice needs to follow these steps:

1. Compute her public key $A$.
2. Choose a random ephemeral key $k$.
3. Compute the first part of the ciphertext $C_1$.
4. Compute the second part of the ciphertext $C_2$.

Let's go through these steps one by one:

1. Compute Alice's public key $A$:

$A = g^a \pmod{p} = 3^5 \pmod{137} = 243 \pmod{137} = 106$

So, Alice's public key is $A = 106$.

2. Choose a random ephemeral key $k$:

For this example, let's choose $k = 7$. In practice, $k$ should be a large random number.

3. Compute the first part of the ciphertext $C_1$:

$C_1 = g^k \pmod{p} = 3^7 \pmod{137} = 2187 \pmod{137} = 46$

4. Compute the second part of the ciphertext $C_2$:

First, compute the shared secret $s = B^k \pmod{p} = 21^7 \pmod{137} = 1801088541 \pmod{137} = 29$

Then, compute $C_2 = (M \times s) \pmod{p} = (87 \times 29) \pmod{137} = 2523 \pmod{137} = 99$

So, the ciphertext that Alice sends to Bob is the pair $(C_1, C_2) = (46, 99)$.