Algebra for Cryptologists by Alko R. Meijer

Author:Alko R. Meijer
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

Proof

because X and Y are statistically independent. Therefore

because H(Y | K) ≤ H(Y ).

This result is usually quoted as saying that for perfect secrecy the key must be at least as long as the message. In fact it says (more precisely) that the entropy of the key space must be at least as great as the entropy of the message space. This is the principle behind the one time pad, which, while providing perfect secrecy, is highly impractical in all but the applications requiring very high security, such as, in the 1960s, the hotline used for fax transmissions between Moscow and Washington.

But the ambition of every cipher designer is to achieve a situation in which the ciphertext is for practical purposes indistinguishable from a truly random stream of characters (bits, usually), so that the entropy of the plaintext, given the ciphertext, is, again for practical purposes, equal to the number of bits in the message.

Exercises

1.Suppose that the message space consists of all n-digit decimal integers, and that encipherment is by a Caesar cipher in which the key K is a single decimal digit, with addition modulo 10. (For example: if n = 3, X = 450, and K = 7 then Y = (4+7) mod 10 (5 + 7) mod 10 (0+ 7) mod 10 = 127.)

Under the assumption that all values of X and all values of K are equally likely, find H(X | C) and H(K | C).

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