Let l be a positive integer. Let ? be the set of all Vigenere ciphers of key length l. Denoting ? the

Let l be a positive integer. Let ? be the set of all Vignere ciphers of key length l. Denoting ? the composition of two functions, prove that (?, ?) is a group. What is the product cipher of two Vignere ciphers with distinct key lengths? (e) S1 and S2 are two Vignere ciphers with keys of length m1 and m2 respectively, with m1 > m2. Prove that if m1 6= 0 mod m2, then S1 × S2 6= S3, where S3 is the Vignere cipher with keyword lcm(m1, m2).