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A primality test is an algorithm for determining whether an input number is prime.Among other fields of mathematics, it is used for cryptography.Unlike integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not.
For example, take n = 71. Then n − 1 = 70 and the prime factors of 70 are 2, 5 and 7.We randomly select an a=17 < n.Now we compute: (). For all integers a it is known that
Fermat's little theorem states that if p is prime and a is not divisible by p, then a p − 1 ≡ 1 ( mod p ) . {\displaystyle a^{p-1}\equiv 1{\pmod {p}}.} If one wants to test whether p is prime, then we can pick random integers a not divisible by p and see whether the congruence holds.
inputs: n, a value to test for primality k, a parameter that determines the accuracy of the test output: composite if n is composite, otherwise probably prime repeat k times: choose a randomly in the range [2,n − 1] if x = 0 or / then return composite return probably prime. Using fast algorithms for modular exponentiation, the running time of ...
Using repeated squaring, the running time of this algorithm is O(k n 3), for an n-digit number, and k is the number of rounds performed; thus this is an efficient, polynomial-time algorithm. FFT-based multiplication, for example the Schönhage–Strassen algorithm, can decrease the running time to O(k n 2 log n log log n) = Õ(k n 2).
For example, if a = 2 and p = 7, then 2 7 = 128, and 128 − 2 = 126 = 7 × 18 is an integer multiple of 7. If a is not divisible by p, that is, if a is coprime to p, then Fermat's little theorem is equivalent to the statement that a p − 1 − 1 is an integer multiple of p, or in symbols: [1] [2] ().
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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]