All mathematicians know the feeling that happens when we are at a cocktail party and someone asks us to describe our research and "what it is good for." I'm not sure how ring theorists or topologists handle this situation, but most number theorists I know start talking about cryptography as quickly as possible. Cryptography is a subject that students (and civilians) find very exciting and it can serve as a great hook to pull students into the world of more abstract mathematics. In recent years there has been a proliferation of books about cryptography which are at the level of advanced undergraduates or which combine cryptography with the study of number theory or abstract algebra as part of a larger course. Anne L. Young's new book Mathematical Ciphers from Caesar to RSA is structured differently from most of these books: it is truly a book about cryptography and codes, which introduces exactly the amount of mathematics that one needs to discuss a handful of ciphers and not a drop more.
As the title promises, Young begins by introducing Caesar's cipher, which simply shifts each letter in the message one is trying to encode a fixed number of letters (for example, A becomes D, B becomes E, etc). She first introduces this in a historical and practical way and then starts to discuss it in terms of modular arithmetic. This interpretation is completely natural to mathematicians, but requires a little bit of background in number theory which Young describes quite well. The next few chapters discuss other ciphers which can be described in terms of modular arithmetic, including multiplication and multiplication-shift ciphers. She also discusses how one can use number theory to break these ciphers, giving a nice description of both the strengths and the limitations of these substitution ciphers.
Young introduces a series of increasingly complicated ciphers, introducing the relevant number theory as needed to discuss exponential ciphers (in which the encoded text is the plaintext raised to a fixed power mod n for a fixed n) and more general substitution ciphers, some of which are not based in number theory at all. The book builds up to a discussion of public key cryptography in general and the most famous example of a public key cipher, the RSA cipher, and it concludes with brief discussions of digital signatures and the security and implementations of the RSA cipher.
Young's book needs to be taken for what it is. It is not a thorough or exhaustive treatment of either number theory or cryptography, and if that is what a reader is looking for then they would be best served elsewhere (and Young herself provides a comprehensive list of Further Readings). However, Young has written a nice textbook which I can easily imagine being used by beginning undergraduates or even high school students to learn both a little bit of abstract mathematics and also one answer to the question "what is this good for?"
Darren Glass is an assistant professor of mathematics at Gettysburg College whose mathematical interests tend towards number theory, algebraic geometry, and cryptography. He can be reached at firstname.lastname@example.org.