Word Count : 884

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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 $c_1 = g^k \mod p$ 2. Compute $c_2 = M \cdot y^k \mod p$ Let’s calculate: --- ### Step 1: Compute $c_1$ $$ c_1 = 5^5 \mod 47 $$ Calculate $5^5$: $$ 5^2 = 25,\quad 5^4 = 625,\quad 5^5 = 3125 $$ $$ 3125 \mod 47 $$ Do division: $$ 3125 \div 47 \approx 66.5,\quad 47 \times 66 = 3102 \quad \Rightarrow 3125 - 3102 = 23 $$ $$ c_1 = 23 $$ --- ### Step 2: Compute _____ ___ __________ _____ _______ ___ _________ _______ ____ _________ _________.
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