Miller–Rabin Probabilistic Primality Test

Synonyms

Miller-Rabin test

Related Concepts

Fermat Primality Test; Fermat’s Little Theorem; Modular Arithmetic; Primality Test; Prime Number

Definition

The Miller–Rabin probabilistic primality test is a probabilistic algorithm for testing whether a number is a prime number using modular exponentiation, Fermat’s little theorem, and the fact that the only square roots of 1 modulo a prime are ± 1.

Background

The Miller–Rabin test was described initially by Miller [2]. Rabin provided further analysis [3], hence the name.

Theory and Applications

One property of primes is that any number whose square is congruent to 1 modulo a prime p must itself be congruent to 1 or − 1. This is not true of composite numbers. If a number n is the product of k distinct odd prime powers, then there will be ${2}^{k}$ distinct “square roots” of 1 modulo n. For example, there are four square roots of 1 modulo 77. The roots must be either 1 or − 1 modulo 7, and either 1 or − 1 modulo 11, since 7 and 11 divide...