Using ‘RSA’ public key cryptosystem, if 𝑝=3, 𝑞=11 and 𝑑=7, find the value…

Compute n and φ(n): n = p × q = 3 × 11 = 33. φ(n) = (p−1)(q−1) = 2 × 10 = 20.

Find the public exponent e:

• e must satisfy e · d ≡ 1 (mod φ(n)), so solve 7 · e ≡ 1 (mod 20).

• Try e = 3: 7 · 3 = 21 ≡ 1 (mod 20), therefore e = 3.

Encrypt the message m = 19:

• Compute 19^2 = 361 ≡ 31 (mod 33).

• Then 19^3 = 19^2 × 19 ≡ 31 × 19 = 589 ≡ 28 (mod 33).

• So the ciphertext c = 28.

Answer: Public exponent e = 3 and ciphertext = 28.