The RSA Public Key Cryptosystem

Abstract

The RSA (Rivest, Shamir, Adleman) cipher algorithm has captured the imagination of many mathematicians by its elegance and basic simplicity ever since it was introduced in 1978. Numerous descriptions of the algorithm have been published. Readers with a knowledge of a little basic number theory will find the original paper [RSA] by the inventors of the algorithm, Ronald L. Rivest, Adi Shamir, and Leonard M. Adleman, quite readable. Perhaps the most famous description is Martin Gardner’s expository article [G], which is written for readers of Scientific American. Martin E. Hellman [H] wrote another good Scientific American article describing the RSA algorithm and the knapsack cipher algorithm. The goal of this paper is to lead the reader who has some mathematical maturity but no knowledge of number theory, say a first year calculus student, a clever high school student, or an interested engineer, through the basic results needed to understand the RSA algorithm. The prerequisites are only a knowledge of the elementary school arithmetic of the integers, high school algebra, some familiarity with the notions of sets and of functions, and, most importantly, a real desire to understand how the RSA algorithm works. We begin with a discussion of general crypto systems and the differences between classical systems and public key systems. Then the treatment will give an informal but fairly rigorous introduction to the division algorithm, divisibility properties, greatest common divisors, the Euclidean algorithm, modular arithmetic, repeated squaring algorithm for b a (mod m), time estimates for these algorithms, Euler’s totient function, Euler’s Theorem, and, as a corollary, Fermat’s Little Theorem.