# Chapter 20 Information Theory

In this chapter we introduce the theoretical concepts behind the security of a cryptosystem. The basic question is the following: If Eve observes a piece of ciphertext, does she gain any new information about the encryption key that she did not already have? To address this issue, we need a mathematical definition of information. This involves probability and the use of a very important measure called entropy.

Many of the ideas in this chapter originated with Claude Shannon in the 1940s.

Before we start, let’s consider an example. Roll a standard six-sided die. Let $A$ be the event that the number of dots is odd, and let $B$ be the event that the number of dots is at least 3. If someone tells you that the roll belongs to the event $A\cap B$ , then you know that there are only two possibilities for what the roll is. In this sense, $A\cap B$ tells you more about the value of the roll than just the event $A$, or just the event $B$. In this sense, the information contained in the event $A\cap B$ is larger than the information just in $A$ or just in $B$.

The idea of information is closely linked with the idea of uncertainty. Going back to the example of the die, if you are told that the event $A\cap B$ happened, you become less uncertain about what the value of the roll was than if you are simply told that event $A$ occurred. Thus the information increased while the uncertainty decreased. Entropy provides a measure of the increase in information or the decrease in uncertainty provided by the outcome of an experiment.