Introduction to Cryptography with Coding Theory, 3rd Edition

20.4 Perfect Secrecy

Intuitively, the one-time pad provides perfect secrecy. In Section 4.4, we gave a mathematical meaning to this statement. In the present section, we repeat some of the arguments of that section and phrase some of the ideas in terms of entropy.

Suppose we have a cipher system with possible plaintexts $\mathit{P}$, ciphertexts $\mathit{C}$, and keys $\mathit{K}$. Each plaintext in $\mathit{P}$ has a certain probability of occurring; some are more likely than others. The choice of a key in $\mathit{K}$ is always assumed to be independent of the choice of plaintext. The possible ciphertexts in $\mathit{C}$ have various probabilities, depending on the probabilities for $\mathit{P}$ and $\mathit{K}$.

If Eve intercepts a ciphertext, how much information does she obtain for the key? In other words, what is $H(K|C)$? Initially, ...