Description of the proposed Rutgers mathematics course:

The Mathematics of Communications: keeping secrets

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

        For thousands of 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.
        This course will discuss some of the mathematical and social issues related to cryptography. Historically most cryptographic investigation was done by governments and these efforts were rarely publicized. They 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 public cryptographic work 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 ATM (automatic teller machine), 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, and the difficulty and propriety of preserving intellectual property rights over a collection of bits.

Prerequisites The math background needed for this course is good knowledge of Algebra 2, and some knowledge of analytic geometry. We'll also need some acquaintance with graphing calculators, since we will write some simple programs on current graphing calculators which will help encrypt and decrypt messages. Knowing how to write such programs will not be needed. Some involvement with the web will be used. Such access is now easy at Rutgers due to the increasing presence of terminals on campus. Courses containing some material similar to what's below are now being given at non-specialized high schools, so the material is accessible to students having the mathematical preparation suggested here. We will certainly need to present the material carefully since our target student population will be those without technical majors or serious knowledge of mathematics.

Mathematical goals Students will learn about modular addition and finite fields. They will 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 (e.g., methods for factoring large numbers and alternative methods for exponentiating) will be explored. In addition, parts of probability (such as Bayes' Theorem) will be discussed. Depending on time and choice of topics, a bit of group theory may also be included. The question "What is an algorithm?" will be explored, along with descriptions of various cryptographic protocols. If time allows, there will be brief mention of the P versus NP question, which has been characterized as the fundamental problem in theoretical computer science. The mathematical topics will rarely be taught systematically, but will be supplied on a "just in time" basis ("infused into the curriculum") as some of the topics listed below need them. Traditional lecturing will be supplemented and, where possible, replaced by various active learning strategies. We hope to have writing assignments which will explore both social and mathematical issues.

Student interest and instructors The topics below are controversial and current and reach informally into the lives of students immediately (buying rock music over the Internet, sending "secret" messages between friends, ...). The topics are also intertwined with many serious public policy issues. Understanding the topics and the decisions related to them will be a part of 21st century society. There's a great deal of current academic and industrial activity on cryptography and communications, and the number of people available to teach such a course at Rutgers is large. Interested and capable individuals abound in the Math and Computer Science Departments, and many are also found at DIMACS, an NSF-sponsored national center for discrete mathematics and theoretical computer science. There are many possible guest speakers for a course of this type working locally at such businesses as AT&T Research, Lucent, Bellcore, NEC, etc. Many of these industrial scientists have shown their abilities to speak at an appropriate level during the three week long summer meeting on Cryptography and Network Security (Dimacs Research and Education Institute, summer, 1997). Over 60 talks were given, most to an audience whose members had a quite varied mathematical background, ranging from high school teachers to academic researchers.

Some possible topics

We present a list together with brief descriptions of topics which could be covered in a course of this nature. We currently plan to present this course for the first time during the fall '99 semester to a class of 20 to 30 students.

Contributed by and last modified 10/11/98.