A Brief History of Factoring and Primality Testing B. C. (Before Computers)

Abstract

Factoring and primality testing have become increasingly important in today's information based society, since they both have produced techniques used in the secure transmission of data. However, often lost in the modern-day shuffle of information are the contributions of the pioneers whose ideas ushered in the computer age and, as we shall see, some of whose ideas are still used today as the underpinnings of powerful algorithms for factoring and primality testing. We offer this brief history to help readers know more about these contributions and appreciate their significance. Virtually everyone who has graduated from high school knows the definition of a number, namely a p E N = {1, 2, 3, 4, . . .} such that p > 1 and if p = Em where X, m E , then either f = 1 or m = 1. (If E N and > 1 is not prime, then is called composite.) Although we cannot be certain, the concept of primality probably arose with the ancient Greeks over two and one-half millennia ago. The first recorded definition of numbers was given by Euclid around 300 BCE in his Elements. However, there is some indirect evidence that the concept of primality might have been known far earlier, for instance, to Pythagoras and his followers. The Greeks of antiquity used the term arithmetic to mean what today we would call number theory, namely the study of the properties of the natural numbers and the relationships between them. The Greeks reserved the word logistics for the study of ordinary computations using the standard operations of addition/subtraction and multiplication/division, which we now call arithmetic. The Pythagoreans introduced the term mathematics, which to them meant the study of arithmetic, astronomy, geometry, and music. This curriculum became known as the quadrivium in the Middle Ages. Although we have enjoyed the notion of a for millennia, only very recently have we developed eJficient tests for primality. This seemingly trivial task is in fact much more difficult than it appears. A primality test is an algorithm (a methodology following a set of rules to achieve a goal), the steps of which verify that given some integer n, we may conclude n is a number. A primality proof is a successful application of a primality test. Such tests are typically called true primality tests to distinguish them from probabilistic primality tests (which can only conclude that n is prime up to a specified likelihood). We will not discuss such algorithms here (see [9] for these). A concept used frequently in primality testing is the notion of a sieve. A is a process to find numbers with particular characteristics (for instance primes) by searching among all integers up to a prescribed bound, and eliminating invalid candidates until only the desired numbers remain. Eratosthenes (ca. 284-204 scE) proposed the first sieve for finding primes. The following example illustrates the Sieve of Eratosthenes.