Data Protection and Security

I

Introduction To Cryptography

I.IV

One-Time Pad

Random variable

A random variable x has a probability distribution p(x), which is the probability that X = x. For two random variables x and y, the distribution p(x,y) gives the probability that X = x and Y = y. The probability that X = x given that Y = y is the conditional probability, and is written p(x|y).

Note that

p(x,y) = p(x|y) * p(y)

And

p(x,y) = p(y|x) * p(x)

The above equation can be rewritten as Bayes' Theorem

p(x|y) = p(x) * p(y|x) / p(y)

A cryptosystem is unconditionally secure (has perfect secrecy) if

P_p(x|y) = P_p(x) for all x P, y C.

What the above formula simply says is that for all plaintext/ciphertext pairs, the posteriori probabilities of being particular plaintexts are equal to the a priori probabilities independently of the values of either plaintext or ciphertext. In other words, ciphertext gives no additional information to determine the plaintext.