Miller–Rabin primality test

The Miller–Rabin primality test or Rabin–Miller primality test is a probabilistic primality test: an algorithm which determines whether a given number is likely to be prime, similar to the Fermat primality test and the Solovay–Strassen primality test.

It is of historical significance in the search for a polynomial-time deterministic primality test. Its probabilistic variant remains widely used in practice, as one of the simplest and fastest tests known.

Gary L. Miller discovered the test in 1976. Miller's version of the test is deterministic, but its correctness relies on the unproven extended Riemann hypothesis. Michael O. Rabin modified it to obtain an unconditional probabilistic algorithm in 1980.

Mathematical concepts

Similarly to the Fermat and Solovay–Strassen tests, the Miller–Rabin primality test checks whether a specific property, which is known to hold for prime values, holds for the number under testing.

Strong probable primes

The property is the following. For a given odd integer

n

>

2

{\displaystyle n>2}

, let’s write

n

−

1

{\displaystyle n-1}

as

2

s

d

{\displaystyle 2^{s}d}

where

s

{\displaystyle s}

is a positive integer and

d

{\displaystyle d}

is an odd positive integer. Let’s consider an integer

a

{\displaystyle a}

, called a base, which is coprime to

n

{\displaystyle n}

.