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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 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.
While there, Rabin invented the Miller–Rabin primality test, a randomized algorithm that can determine very quickly (but with a tiny probability of error) whether a number is prime. [ 10 ] [ 11 ] Rabin's method was based on previous work of Gary Miller that solved the problem deterministically with the assumption that the generalized Riemann ...
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 ]
The probability of a composite number n passing the Fermat test approaches zero for . Specifically, Kim and Pomerance showed the following: The probability that a random odd number n ≤ x is a Fermat pseudoprime to a random base 1 < b < n − 1 {\displaystyle 1<b<n-1} is less than 2.77·10 -8 for x= 10 100 , and is at most (log x) -197 <10 ...
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. They are listed in Table 7 of. [2] The smallest such number is 25326001.
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]
The algorithm can be written in pseudocode as follows: algorithm lucas_primality_test is input: n > 2, an odd integer to be tested for primality. k, a parameter that determines the accuracy of the test. output: prime if n is prime, otherwise composite or possibly composite. determine the prime factors of n−1.