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  2. Miller–Rabin primality test - Wikipedia

    en.wikipedia.org/wiki/MillerRabin_primality_test

    The MillerRabin primality test or RabinMiller 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 ...

  3. Strong pseudoprime - Wikipedia

    en.wikipedia.org/wiki/Strong_pseudoprime

    A strong pseudoprime is a composite number that passes the MillerRabin 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 ...

  4. Primality test - Wikipedia

    en.wikipedia.org/wiki/Primality_test

    The MillerRabin primality test and Solovay–Strassen primality test are more sophisticated variants, which detect all composites (once again, this means: for every composite number n, at least 3/4 (MillerRabin) or 1/2 (Solovay–Strassen) of numbers a are witnesses of compositeness of n). These are also compositeness tests.

  5. Generation of primes - Wikipedia

    en.wikipedia.org/wiki/Generation_of_primes

    For the large primes used in cryptography, provable primes can be generated based on variants of Pocklington primality test, [3] while probable primes can be generated with probabilistic primality tests such as the Baillie–PSW primality test or the MillerRabin primality test.

  6. Probable prime - Wikipedia

    en.wikipedia.org/wiki/Probable_prime

    Probable primality is a basis for efficient primality testing algorithms, which find application in cryptography. These algorithms are usually probabilistic in nature. The idea is that while there are composite probable primes to base a for any fixed a , we may hope there exists some fixed P <1 such that for any given composite n , if we choose ...

  7. P/poly - Wikipedia

    en.wikipedia.org/wiki/P/poly

    For example, the popular MillerRabin primality test can be formulated as a P/poly algorithm: the "advice" is a list of candidate values to test. It is possible to precompute a list of O ( n ) {\displaystyle O(n)} values such that every composite n -bit number will be certain to have a witness a in the list. [ 3 ]

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  9. Fermat's little theorem - Wikipedia

    en.wikipedia.org/wiki/Fermat's_little_theorem

    The MillerRabin 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).