Cyclic Groups and Cryptography

Abstract

This chapter presents another public-key cryptographic method used world-wide, the Diffie-Hellman key exchange. Section 8.5 introduced an efficient algorithm for finding $h = g^{m}$ where g is an element of a finite group, for example when the group is the group of units modulo a prime p and m is a large exponent. The reverse problem, given g and h, find m, is called the discrete logarithm problem, and is a hard problem. The security of Diffie-Hellman key exchange and the closely related ElGamal cryptosystem is based on that fact. Needed for Diffie-Hellman are cyclic groups of large order. We find many such groups by proving the Primitive Root Theorem, which states that for every prime number p, the group of units of $\mathbb {Z}_{p}$ is a cyclic group. The chapter concludes with two methods that can be more efficient than constructing log tables for solving the discrete logarithm problem; one method involves use of the Chinese Remainder Theorem.