Superencipherment for substitution ciphers, Recall that for the shift cipher, we have M = C = K = Z/26Z (the integers modulo 26). Encryption is given by EK(M) M + K (mod 26) (modular addition of message and key).

Give a formal mathematical proof that superencipherment (i.e. multiple encryption) for the shift cipher results again in a shift cipher. Using induction on the number of encipherments What is the key of the multiple encipherment?

how do I formally explain why encryption under a Vigen`ere cipher using a key word w1 of length m, followed by encryption under a second Vigen`ere cipher using another key word w2 of length n, is again a Vigen`ere cipher under some new key word w. How is w obtained, and what is its length?