Let n be a positive integer. A Latin square of order n is an n times n array L o
ID: 3799232 • Letter: L
Question
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.Explanation / Answer
solution--
There are two ways to discuss to discuss the security of cryptosystem -- computational security and unconditional security.
If you want a cryptosystem to have perfect secrecy then these two conditions should be satisfied--
p P, c C, PrP,C (p/c) = PrP (p)
p P, c C, PrP,C (p/c) = Pr(p).Pr(c/p) Pr(c)
Each key is used with equal probability, so knowing j, there is only one key that encrypt j to a L(i,j) among the n
keys (Each number appears once in a column). Concerning Pr(c), each L(i,j) appears n times in the square among
the n 2 possible cases. Pr(p/c) = Pr(p). 1 n n n2 = Pr(p) In conclusion, the Latin Square Cryptosystem achieves
perfect secrecy provided that every key is used with equal probability.
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