Let n be a positive integer. A Latin square of order n is an n times n array L of the integers 1, 2, ..., n such that every one of the n integers occurs exactly once in each row and each column of L. An example of a Latin square of order 3 is as follows: (1 2 3 3 1 2 2 3 1) Given any Latin square L ot order n, we can define the following cryptosystem: Take P = C = K = {1, 2, ..., n}. For 1 lessthanorequalto i lessthanorequalto n, the encryption function Ej is defined to be E_i, (j) = E(i, j), (Hence each row of L gives rise to one encryption function.) Give a complete proof that this Latin square cryptosystem achieves perfect secrecy