Methods of making messages unintelligible to adversaries have been important throughout history. In this chapter we shall cover some of the older cryptosystems that were primarily used before the advent of the computer. These cryptosystems are too weak to be of much use today, especially with computers at our disposal, but they give good illustrations of several of the important ideas of cryptology.

First, for these simple cryptosystems, we make some conventions.

• plaintext will be written in lowercase letters and CIPHERTEXT will be written in capital letters (except in the computer problems).

• The letters of the alphabet are assigned numbers as follows:

  $$\begin{array}{c}\begin{array}{cccccccccccccccc}a& b& c& d& e& f& g& h& i& j& k& l& m& n& o& p\\ 0& 1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12& 13& 14& 15\end{array}\\ \begin{array}{cccccccccc}q& r& s& t& u& v& w& x& y& z\\ 16& 17& 18& 19& 20& 21& 22& 23& 24& 25\end{array}\end{array}$$

  Note that we start with $a=0$, so $z$ is letter number 25. Because many people are accustomed to $a$ being 1 and $z$ being 26, the present convention can be annoying, but it is standard for the elementary cryptosystems that we’ll consider.

• Spaces and punctuation are omitted. This is even more annoying, but it is almost always possible to replace the spaces in the plaintext after decrypting. If spaces were left in, there would be two choices. They could be left as spaces; but this yields so much information on the structure of the message that decryption becomes easier. Or they could be encrypted; but then they would dominate frequency counts (unless the message averages at least eight letters per word), again simplifying decryption.

Note: In this chapter, we’ll be using some concepts from number theory, especially modular arithmetic. If you are not familiar with congruences, you should read the first three sections of Chapter 3 before proceeding.