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The simplest probabilistic primality test is the Fermat primality test (actually a compositeness test). It works as follows: Given an integer n, choose some integer a coprime to n and calculate a n − 1 modulo n. If the result is different from 1, then n is composite. If it is 1, then n may be prime.
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.
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 false statement that all numbers that pass the Fermat primality test for base 2 are prime is called the Chinese hypothesis. The smallest base-2 Fermat pseudoprime is 341. It is not a prime, since it equals 11·31, but it satisfies Fermat's little theorem: 2 340 ≡ 1 (mod 341) and thus passes the Fermat primality test for the base 2.
The AKS primality test (also known as Agrawal–Kayal–Saxena primality test and cyclotomic AKS test) is a deterministic primality-proving algorithm created and published by Manindra Agrawal, Neeraj Kayal, and Nitin Saxena, computer scientists at the Indian Institute of Technology Kanpur, on August 6, 2002, in an article titled "PRIMES is in P". [1]
For base 4, see OEIS: A020230, and for base 6 to 100, see OEIS: A020232 to OEIS: A020326. By testing the above conditions to several bases, one gets somewhat more powerful primality tests than by using one base alone. For example, there are only 13 numbers less than 25·10 9 that are strong pseudoprimes to bases 2, 3, and 5 simultaneously.
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 ...
For example, the popular Miller–Rabin 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 ]