Question

b) i) Bob uses ElGamal cyrptosystem with parametersequationand the secret value x = 3. What values will he make public?

ii) Alice wants to send Bob the messageequationShe chooses k = 5. How will she compute the cipher text? What information does she send to Bob?

iii) Explain how Bob will decrypt the message

29 Apr 2025
Answer :
Word Count : 884
Let's solve this ElGamal cryptosystem problem numerically, step by step. --- ### Given: * $p = 47$ (a prime number) * $g = 5$ (a primitive root mod 47) * Bob's private key: $x = 3$ --- ## Part (i): What values will Bob make public? In ElGamal, Bob's public key is: $$ \text{Public Key: } (p, g, y) \quad \text{where} \quad y = g^x \mod p $$ Here: $$ y = 5^3 \mod 47 = 125 \mod 47 $$ Let’s compute: $$ 125 \div 47 = 2 \text{ with remainder } 125 - 94 = 31 $$ So, $$ y = 31 $$ --- ✅ Bob’s public key: $$ (p, g, y) = (47, 5, 31) $$ Private key: $x = 3$ --- ## Part (ii): Alice encrypts the message $\mathcal{M} = 15$ using $k = 5$ ElGamal encryption steps: 1. Compute $c_1 = g^k \mod ______ ____ _________ _________ ______ ___ ___.
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