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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 ...
The Miller–Rabin and the Solovay–Strassen primality tests are simple and are much faster than other general primality tests. One method of improving efficiency further in some cases is the Frobenius pseudoprimality test ; a round of this test takes about three times as long as a round of Miller–Rabin, but achieves a probability bound ...
A strong pseudoprime is a composite number that passes the Miller–Rabin primality test. All prime numbers pass this test, but a small fraction of composites also pass, making them " pseudoprimes ". Unlike the Fermat pseudoprimes , for which there exist numbers that are pseudoprimes to all coprime bases (the Carmichael numbers ), there are no ...
This leads to probabilistic algorithms such as the Solovay–Strassen primality test, the Baillie–PSW primality test, and the Miller–Rabin primality test, which produce what are known as industrial-grade primes. Industrial-grade primes are integers for which primality has not been "certified" (i.e. rigorously proven), but have undergone a ...
The BigInteger class in standard versions of Java and in open-source implementations like OpenJDK has a method called isProbablePrime. This method does one or more Miller–Rabin tests with random bases. If n, the number being tested, has 100 bits or more, this method also does a non-strong Lucas test that checks whether U n+1 is 0 (mod n).
If you make, say, 25 iterations of the Miller-Rabin tests, the algorithm as such is correct with probability smaller than 10 −15. This is orders of magnitude less than the probability that the computation will be corrupted by hardware errors, software bugs, mistyping the input, some user's death due to heart attack during the computation, or ...
In practice however, the cost of doing many iterations and other differences leads to worse performance for Miller–Rabin. [ clarification needed ] The most efficient deterministic primality test for any n -digit number, the AKS primality test , requires Õ(n 6 ) bit operations in its best known variant and is extremely slow even for ...
Hence, the probability of failure is at most 2 −k (compare this with the probability of failure for the Miller–Rabin primality test, which is at most 4 −k). For purposes of cryptography the more bases a we test, i.e. if we pick a sufficiently large value of k, the better the accuracy of test.