The mathematician Claude Shannon first formalized the notion of perfect secrecy and showed that certain cryptosystems realized it. We do not cover all of his theory, but address the part that is directly relevant to our study of cryptography and that relies on math that is within the scope of the text.
In this chapter, we discuss how to tell whether a cryptosystem is perfectly secure. In Chapter 7, we discuss some more ways to use perfect secrecy. It should become clear to the reader why modular arithmetic is used instead of ordinary arithmetic for much of cryptography.
As we will see, perfect secrecy of a cryptosystem and unique decryptability are mathematical cousins. A cryptosystem may be uniquely decryptable and not perfectly secure, or vice versa. However, the math that goes into determining perfect secrecy is very similar to the math that goes into determining unique decryptability.
What does an eavesdropper learn from seeing a cyphertext?
A cryptosystem is perfectly secure if an eavesdropper learns nothing about the plaintext from seeing the cyphertext. To understand what cryptosystems are secure, therefore, we consider what it means to learn something.
For this purpose, we consider a very simple scenario. Alice sends Bob an encrypted message, and Eve intercepts the cyphertext. (For now, we ignore the possibility that Bob may respond using the same cryptosystem and even the same key.) To understand what Eve has learned from seeing the cyphertext, we consider her knowledge of the plaintext before (her a priori knowledge) and after she sees the cyphertext (her a posteriori knowledge).