Miller-Rabin prime test

The Miller-Rabin prime test is a probabilistic test based on the following

Theorem 1.

Let N3 be an odd numberMathworldPlanetmathPlanetmath and N-1=2tn, n odd. If and only if for every aZ with gcd(a,N)=1 we have

an1mod N  𝑜𝑟 (1a)
a2sn-1mod N (1b)

for some s{0,,t-1}, then N is prime. If N is not prime, then let A be the set of all a satisfying the above conditions. We have


where φ is Euler’s Phi-function.

A composite numberMathworldPlanetmath N satisfying conditions (1) for some a is called a strong in the basis a. Note that this theorem states that there are no such things as Carmichael numbersMathworldPlanetmath for strong pseudoprimes (i.e. composite numbers that are strong pseudoprimes for all values of a). From this theorem we can construct the following algorithmMathworldPlanetmath:

  1. 1.

    Choose a random 2aN-1.

  2. 2.

    If gcd(a,N)1, we have a non-trivial divisorMathworldPlanetmathPlanetmath of N, so N is not prime.

  3. 3.

    If gcd(a,N)=1, check if the conditions (1) are satisfied. If not, N is not prime, otherwise the probability that N is not prime is less than 14.

In general the probability, that the test falsely reports N as a prime is far less than 14. Of course the algorithm can be applied several times in order to reduce the probability, that N is not a prime but reported as one.

When comparing this test with the related Solovay-Strassen test one sees that this test is superior is several ways:

  • There is no need to calculate the Jacobi symbolMathworldPlanetmath.

  • The amount of false witnesses, i.e. numbers a that report N as a prime when it is not, is at most N4 instead of N2.

  • One can even show that N which passes Miller-Rabin for some a also passes Solovay-Strassen for that a, so Miller-Rabin is always better.

Note that when the test returns that N is composite, then this is indeed the case, so it is really a compositeness test rather than a test for primality.

Title Miller-Rabin prime test
Canonical name MillerRabinPrimeTest
Date of creation 2013-03-22 14:54:18
Last modified on 2013-03-22 14:54:18
Owner mathwizard (128)
Last modified by mathwizard (128)
Numerical id 9
Author mathwizard (128)
Entry type Algorithm
Classification msc 11Y11
Synonym Rabin prime test
Related topic SolovayStrassenTest
Related topic FermatCompositenessTest
Defines strong pseudoprime