Cryptography: the mathematics of communications

Keeping secrets

Three may keep a secret, if two of them are dead.
Benjamin Franklin

For several thousand years people have tried to communicate secretly and securely. Cryptography is the field of mathematics dedicated to exploring schemes to conceal messages and to verifying the difficulty of "breaking" these schemes: that is, revealing the hidden message without the consent or knowledge of those communicating.

Historically most cryptographic investigation was done by governments and their efforts were publicized quite rarely. The efforts were massive: the largest single employers of mathematically trained people in the United States and the former Soviet Union have been the government agencies with cryptographic responsibilities. But there's been an enormous increase in the public work done in cryptography in the last quarter century, and in the accompanying controversies. This increase has been caused by the easy availability of computers and their interconnections (via the Internet and the web) and by the development of new ideas, such as public key cryptography, which allow for secure communication between parties who have made no previous commitment to each other. Every person who has used an automatic teller machine (ATM), made a phone call using a portable phone, or had their health records transmitted among caregivers or insurers should be concerned about secure communication. Social issues include the conflict between the right to privacy and the desire of some government agencies to have assured access to certain communications. Some of the topics relating to cryptography may be introduced in various high school math courses.

The mathematics
Students need a good algebra background for many of the topics listed below. Some of the topics involving "civil liberties" questions may, however, provide opportunities for fertile interaction with social studies classes. Analytic geometry is useful for many of the mathematical topics. The abilitiy to think "outside the lines" is an additional hard-to-assess requirement. What mathematical topics may students learn while studying cryptography? They may learn about modular addition and finite fields. They may learn the basics of combinatorics, such as various ways to count. Additional topics of discrete mathematics, including basic graph theory, may be studied. Appropriate topics from number theory, such as some methods for factoring large numbers and alternative methods for exponentiating, are used. In addition, some parts of probability (such as Bayes' Theorem) can be introduced. Depending on time and choice of topics, a bit of group theory may also be included. The question "What is an algorithm?" is essential in much of mathematical cryptography, along with good descriptions of various cryptographic protocols. Therefore reading and writing skills combined with appropriate mathematical sophistication are needed. It is also possible to mention the P versus NP controversy in a cryptography course. This question has been characterized as the fundamental problem in theoretical computer science.

Some topics
Here are brief descriptions of some topics which could be addressed in some high school courses:

There are many cryptography links on the web. Here's a list of links containing probably more than any one person needs to know about cryptography. But we welcome additional links!