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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 ...
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 Miller–Rabin primality test uses the following extension of Fermat's little theorem: [14] If p is an odd prime and p − 1 = 2 s d with s > 0 and d odd > 0, then for every a coprime to p, either a d ≡ 1 (mod p) or there exists r such that 0 ≤ r < s and a 2 r d ≡ −1 (mod p).
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).
Using fast algorithms for modular exponentiation and multiprecision multiplication, the running time of this algorithm is O(k log 2 n log log n) = Õ(k log 2 n), where k is the number of times we test a random a, and n is the value we want to test for primality; see Miller–Rabin primality test for details.
A prime sieve or prime number sieve is a fast type of algorithm for finding primes. There are many prime sieves. The simple sieve of Eratosthenes (250s BCE), the sieve of Sundaram (1934), the still faster but more complicated sieve of Atkin [1] (2003), sieve of Pritchard (1979), and various wheel sieves [2] are most common.
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.