material from the text, while others are quite challenging and are introduc-
tions to more advanced topics. These problems are meant to supplement
the text and to allow students of different levels and interests to explore
the material in different ways. Instructors may contact the authors (either
directly or through the AMS webpage) to request a complete solution key.
• Chapter 1 is a brief introduction to the history of cryptography.
There is not much mathematics here. The purpose is to provide
the exciting historical importance and background of cryptography,
introduce the terminology, and describe some of the problems and
• Chapter 2 deals with classical methods of encryption. For the most
part we postpone the attacks and vulnerabilities of these meth-
ods for later chapters, concentrating instead on describing popular
methods to encrypt and decrypt messages. Many of these methods
involve procedures to replace the letters of a message with other
letters. The main mathematical tool used here is modular arith-
metic. This is a generalization of addition on a clock (if it’s 10
o’clock now, then in five hours it’s 3 o’clock), and this turns out
to be a very convenient language for cryptography. The final sec-
tion on the Hill cipher requires some basic linear algebra, but this
section may safely be skipped or assigned as optional reading.
• Chapter 3 describes one of the most important encryption methods
ever, the Enigma. It was used by the Germans in World War II and
thought by them to be unbreakable due to the enormous number of
possibilities provided. Fortunately for the Allies, through espionage
and small mistakes by some operators, the Enigma was successfully
broken. The analysis of the Enigma is a great introduction to
some of the basic combinatorial functions and problems. We use
these to completely analyze the Enigma’s complexity, and we end
with a brief discussion of Ultra, the Allied program that broke the
• Chapters 4 and 5 are devoted to attacks on the classical ciphers.
The most powerful of these is frequency analysis. We further de-
velop the theory of modular arithmetic, generalizing a bit more
operations on a clock. We end with a discussion of one-time pads.
When used correctly, these offer perfect security; however, they re-
quire the correspondents to meet and securely exchange a secret.
Exchanging a secret via insecure channels is one of the central prob-
lems of the subject, and that is the topic of Chapters 7 and 8.
• In Chapter 6 we begin our study of modern encryption methods.
Several mathematical tools are developed, in particular binary ex-
pansions (which are similar to the more familiar decimal or base 10
expansions) and recurrence relations (which you may know from the
Fibonacci numbers, which satisfy the recursion Fn+2 = Fn+1 +Fn).