Introduction to the RSA algorithm and modular arithmetic

These notes are an introduction to the RSA algorithm, and to the mathematics needed to understand it. The RSA algorithm — the name comes from the initials of its inventors, Rivest, Shamir, and Adleman — is the foundation of modern public key cryptography. It is used for electronic commerce and many other types of secure communication over the Internet. The RSA algorithm is based on a type of mathematics known as modular arithmetic. Modular arithmetic is an interesting variation of ordinary arithmetic, but whereas everyday arithmetic is familiar to school children everywhere, modular arithmetic is a somewhat obscure subject. It’s not that modular arithmetic is particularly difficult, or confusing; one could teach it in high school, or even earlier. Conventional thinking, however, places a higher value on the ability to balance a checkbook than on the ability to communicate in code; this is probably the reason why most people have never heard of modular arithmetic. Today, the rapid proliferation of the Internet and the growing popularity of electronic financial transactions are causing a shift in attitudes. These days, even an introductory understanding of public key cryptography can be enormously useful. Consequently, today’s students deserve an opportunity to become acquainted with the methods and ideas of modular arithmetic. First, a quick word about these notes. I have tried to make the material here as down to earth, and accessible as possible. As such, the emphasis is on concrete calculations, rather than abstruse theory. My goal is to guide you through the concrete steps needed to implement and understand RSA