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{{Short description|Algorithm checking for prime numbers}}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".JOURNAL, Manindra, Agrawal, Neeraj, Kayal, Nitin, Saxena,weblink PRIMES is in P, Annals of Mathematics, 160, 2004, 2, 781–793, 10.4007/annals.2004.160.781, 3597229, free, The algorithm was the first one which is able to determine in polynomial time, whether a given number is prime or composite and this without relying on mathematical conjectures such as the generalized Riemann hypothesis. The proof is also notable for not relying on the field of analysis.JOURNAL, Andrew, Granville,weblink It is easy to determine whether a given integer is prime, Bull. Amer. Math. Soc., 42, 2005, 3–38, 10.1090/S0273-0979-04-01037-7, free, In 2006 the authors received both the Gödel Prize and Fulkerson Prize for their work.

Importance

AKS is the first primality-proving algorithm to be simultaneously general, polynomial-time, deterministic, and unconditionally correct. Previous algorithms had been developed for centuries and achieved three of these properties at most, but not all four.

• The AKS algorithm can be used to verify the primality of any general number given. Many fast primality tests are known that work only for numbers with certain properties. For example, the Lucas–Lehmer test works only for Mersenne numbers, while Pépin's test can be applied to Fermat numbers only.
• The maximum running time of the algorithm can be bounded by a polynomial over the number of digits in the target number. ECPP and APR conclusively prove or disprove that a given number is prime, but are not known to have polynomial time bounds for all inputs.
• The algorithm is guaranteed to distinguish deterministically whether the target number is prime or composite. Randomized tests, such as Miller–Rabin and Baillie–PSW, can test any given number for primality in polynomial time, but are known to produce only a probabilistic result.
• The correctness of AKS is not conditional on any subsidiary unproven hypothesis. In contrast, Miller's version of the Miller–Rabin test is fully deterministic and runs in polynomial time over all inputs, but its correctness depends on the truth of the yet-unproven generalized Riemann hypothesis.

While the algorithm is of immense theoretical importance, it is not used in practice, rendering it a galactic algorithm. For 64-bit inputs, the Baillie–PSW test is deterministic and runs many orders of magnitude faster. For larger inputs, the performance of the (also unconditionally correct) ECPP and APR tests is far superior to AKS. Additionally, ECPP can output a primality certificate that allows independent and rapid verification of the results, which is not possible with the AKS algorithm.

Concepts

The AKS primality test is based upon the following theorem: Given an integer nge 2 and integer a coprime to n, n is prime if and only if the polynomial congruence relation{{NumBlk|:|(X + a)^{n} equiv X^{n} + a pmod{n}|{{EquationRef|1}}}}holds within the polynomial ring (mathbb Z/nmathbb Z)[X]. Note that X denotes the indeterminate which generates this polynomial ring.This theorem is a generalization to polynomials of Fermat's little theorem. In one direction it can easily be proven using the binomial theorem together with the following property of the binomial coefficient: