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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 a paper titled "PRIMES is in P". The algorithm determines whether a number is prime or composite within polynomial time. The authors received the 2006 Gödel Prize and the 2006 Fulkerson Prize for this work. ==Importance== AKS is the first primality-proving algorithm to be simultaneously ''general'', ''polynomial'', ''deterministic'', and ''unconditional''. 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 expressed as 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, the Miller 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. For 64-bit inputs, the Baillie–PSW 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「AKS primality test」の詳細全文を読む スポンサード リンク
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