1.41. Consider the affine cipher with key k = (k1, k2) whose encryption and decryption functions are given by e_k(m) = k1 m + k2 (mod p), d_k(c) = k'₁ (c-k2) (mod p)

(a) Let p = 541 and let the key be k = (34, 71). Encrypt the message m = 204. Decrypt the ciphertext c = 431.

(b) Assuming that p is public knowledge, explain why the affine cipher is vulnerable to a chosen plaintext attack. (See Property 4 on page 38.) How many plaintext/ ciphertext pairs are likely to be needed in order to recover the private key?

(c) Alice and Bob decide to use the prime p = 601 for their affine cipher. The value of p is public knowledge, and Eve intercepts the ciphertexts c1 = 324 and c2 = 381 and also manages to find out that the corresponding plaintexts are m1 = 387 and m2 = 491. Determine the private key and then use it to encrypt the message m3 = 173.

(d) Suppose now that p is not public knowledge. Is the affine cipher still vulnerable to a known plaintext attack? If so, how many plaintext/ciphertext pairs are likely to be needed in order to recover the private key?