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	AKS primality test - Wikipedia, the free encyclopedia
		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 three Indian scientists named Manindra Agrawal, Neeraj Kayal and Nitin Saxena on August 6, 2002 in a paper titled "PRIMES is in P".
		The key significance of AKS is that it was the first published algorithm to be simultaneously polynomial, deterministic, and unconditional.
		The proof of correctness for AKS consists of showing that there exists a suitably small r and suitably small set of integers A such that if the equivalence holds for all such a in A then n must be prime.
	en.wikipedia.org /wiki/AKS_primality_test (845 words)

	Primality test - Wikipedia, the free encyclopedia
		Since compositeness is an NP-problem, usual randomized primality tests never report a prime number as composite, but it is possible for a composite number to be reported as prime (for a small fraction of potential witnesses).
		The simplest probabilistic primality test is the Fermat primality test.
		The existence of the AKS primality test, which finally settled this long-standing question, means that PRIMES is in P.
	en.wikipedia.org /wiki/Primality_test (1292 words)

	AKS primality test (Site not responding. Last check: 2007-11-07)
		The AKS primality test (also known as Agrawal-Kayal-Saxena primality test and cyclotomic AKS test) is a deterministic primality-proving algorithm discovered and published by three India n scientists named Manindra Agrawal, Neeraj Kayal and Nitin Saxena in August 6, 2002 in a scientific paper titled "PRIMES is in P".
		Test Early is passionate about building software quality into the development process — this portal will be very beneficial to anyone seeking to make their code more defect-resistant.
		The tests were also used to prove that clamping rogue processes could result in an increase in the number of active concurrent users supported by a single server.
	www.serebella.com /encyclopedia/article-AKS_primality_test.html (1550 words)

	AKS primality test - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-07)
		The AKS primality test (also known as Agrawal-Kayal-Saxena primality test and cyclotomic AKS test) is a deterministic primality-proving algorithm discovered and published by three Indian scientists named Manindra Agrawal, Neeraj Kayal and Nitin Saxena in August 6, 2002 in a scientific paper titled "PRIMES is in P".
		AKS has a key difference with all previous general primality-proving algorithms: It does not require any unproven hypothesis (such as the Riemann hypothesis) to be true in order to have provable polynomial time on all inputs.
		Because of the many variants, Crandall and Papadopoulos refer to the "AKS-class" of algorithms in their scientific paper "On the implementation of AKS-class primality tests" published in March 2003.
	www.encyclopedia-online.info /AKS_primality_test (255 words)

	Lucas-Lehmer test for Mersenne primes: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-11-07)
		Lucas-Lehmer primality test The lucas-lehmer primality test is a method of testing the primality of some number n; it requires that the prime factors of n-1 be known....
		Miller-Rabin primality test The miller-rabin primality test is a primality test: an algorithm...
		AKS primality test The aks primality test (also known as agrawal-kayal-saxena primality test and cyclotomic aks test) is a deterministic primality-proving algorithm discovered and published by three indian scientists...
	www.absoluteastronomy.com /l/lucas-lehmer_test_for_mersenne_primes (496 words)

	primality test
		The simplest primality "test" is as follows: Given an input number N, we check each integer k > 1 other than N to see whether N is divisible by k.
		The simplest true primality test is as follows: Given an input number N, we check whether it is divisible by any integer between 1 and N exclusive.
		A more convenient primality test is as follows: Given an input number N, we check whether it is divisible by any integer greater than 1 and less than or equal to the square root of N.
	www.fact-library.com /primality_test.html (788 words)

	Randomized algorithm - Wikipedia, the free encyclopedia
		Historically, the study of randomized algorithms was spurred by the discovery by Miller and Rabin in 1976 that the problem of determining the primality of a number can be solved efficiently by a randomized algorithm.
		The Miller-Rabin primality test relies on a binary relation between two positive integers k and n that can be expressed by saying that k "is a witness to the compositeness of" n.
		Indeed, even though a deterministic polynomial-time primality test has since been found, it is has not replaced the older probabilistic tests in cryptographic software nor is it expected to do so for the foreseeable future.
	www.wikipedia.org /wiki/Probabilistic_algorithm (1756 words)

	Aks primality test - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-07)
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	www.sciencedaily.com /encyclopedia/aks_primality_test (155 words)

	[No title]
		Primality tests are algorithms that are used to directly determine whether a specified number is prime or composite without resorting to factoring.
		With the Miller-Rabin test, for example, the probability of error may be reduced to an arbitrarily small number by increasing the number of iterations performed.
		The problem is that this test has the potential for marking a composite number as prime, although the probability is very small for the number of iterations that are used.
	bass.gmu.edu /courses/ECE636/project/S05_draft_reports_word/AKS_robert_paul.doc (3605 words)

	Contrast primality tests (Site not responding. Last check: 2007-11-07)
		To test n for primality (to see if it is prime) just divide by all of the primes less than the square root of n.
		This is the basis of all modern primality tests whether they are as simple as the test above or something as elaborate such as the methods using elliptic curves or number fields.
		AKS also showed that if Sophie Germain primes have the expected distribution (and they certainly should!), then the exponent 12 in the time estimate can be reduced to 6, which means that in practice, we expect the exponents to be the same in this method and the (probabilistic) ECPP method.
	www.csam.iit.edu /~cs549/cs549/project/Contrastprimalitytests.htm (5193 words)

	NationMaster - Encyclopedia: Manindra Agrawal (Site not responding. Last check: 2007-11-07)
		He co-created the AKS primality test with Neeraj Kayal and Nitin Saxena, and won the 2002 Clay Research Award, and the 2006 Godel Prize (along with his co-authors).
		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 three Indian scientists named Manindra Agrawal, Neeraj Kayal and Nitin Saxena on August 6, 2002 in a scientific paper titled PRIMES is in P...
		Dr Manindra Agrawal is conferred with the Distinguished Alumnus Award of IIT Kanpur for his outstanding contributions in Complexity Theory and by developing a Polynomial Time Algorithm for Primality Testing.
	www.nationmaster.com /encyclopedia/Manindra-Agrawal (648 words)

	Citations: Primality Testing and Abelian Varieties Over Finite Fields - Adleman, Huang (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		A significant step towards this result was achieved by Goldwasser and Kilian [35] It is interesting to note that the primality tests used in practice [20] 64] appear to have super polynomial running time and hat the algorithm of....
		There exist special purpose primality tests for numbers of certain special forms (for instance for Mersenne numbers which are of the form 2 q Gamma 1 where q is a prime [48]....
		Their algorithm, called the Abelian variety test is a randomized algorithm (Las Vegas type) and provably runs in expected polynomial time O(log n) 6) Although their algorithm is totally non practical, it....
	citeseer.ist.psu.edu /context/151533/0 (2587 words)

	How to find primes and prove primality (merged version)
		The third chapter cover the classical primality tests that have been used to prove primality for 99.99% of the numbers on the largest known prime list.
		It has been proven ([Monier80] and [Rabin80]) that the strong probable primality test is wrong no more than 1/4th of the time (3 out of 4 numbers which pass it will be prime).
		The third chapter (these pages) cover the classical primality tests that have been used to prove primality for 99.99% of the numbers on the largest known prime list.
	primes.utm.edu /prove/merged.html (6842 words)

	Finding Primes & Proving Primality (Site not responding. Last check: 2007-11-07)
		The other simple way of testing if a number, N, is prime is to try and divide N by all thee numbers (excluding 1) less than N. If one of those numbers divides N we know it must be composite and if not it must be prime.
		It can be used to test for primality of p by picking random a’s and see if the equality holds.
		This test is also known as the Agrawal-Kayal-Saxena primality test, named after the three scientists that discovered it in 2002.
	www.bath.ac.uk /~jch21/Primality.htm (572 words)

	Primality test - QuickSeek Encyclopedia (Site not responding. Last check: 2007-11-07)
		Rather than testing all m up to n-1, we need only test m up to $\sqrt{n}$ : if n is composite then it can be factored into two values, at least one of which is less than or equal to $\sqrt{n}$ .
		We first test if n is divisible by 2 or 3, then we run through all the numbers of form 6k ± 1 $\leq$ $\sqrt{n}$ .
		The first deterministic primality test significantly faster than the naïve methods was the cyclotomy test; its runtime can be proven to be O((log n)
	primalitytest.quickseek.com (1204 words)

	Indice
		B.C. Higgins - The Rabin-Miller probabilistic primality test
		- An exposition of the AKS primality test (2002)
		Klappenecker - An introduction to the AKS primality test (2002)
	www.geocities.com /CapeCanaveral/Launchpad/2833/indicearticoli.html (651 words)

	PRIMES is in P little FAQ (Site not responding. Last check: 2007-11-07)
		For the problem of testing the primality of an integer n, we will consider the number of bits needed to represent n, this is lg (n), where lg represents the logarithm base 2.
		This primality test is very inefficient however: to determine if an integer n is prime one needs to execute sqrt(n) divisions, which is exponential in lg(n) (the size of n), thus not bounded by a polynomial in the size of n.
		The test is not bulletproof for primality however, there exist integers n which are not prime and for which the equality in (1) will be satisfied for all integers a, thus if the algorithm does not determine that an integer is not prime we cannot conclude with certainty that it is a prime.
	crypto.cs.mcgill.ca /~stiglic/PRIMES_P_FAQ.html (2589 words)

	Primality Proving 4.3: A polynomial-time algorithm
		As we mentioned before, many of the primality proving methods are conjectured to be polynomial-time.
		AKS also showed that if Sophie Germain primes have the expected distribution [HL23] (and they certainly should!), then the exponent 12 in the time estimate can be reduced to 6, bringing it much closer to the the (probabilistic) ECPP method.
		It seems plausible that a variant of AKS may soon compete in practice with ECPP for 'general' primality proofs.
	www.primepages.org /prove/prove4_3.html (720 words)

	Primality Tests (Site not responding. Last check: 2007-11-07)
		(I'm testing vectors of prime lengths ~10^10) is non-trivial.
		In testing primality of very large numbers chosen at random, the chance of stumbling upon a value that fools the Fermat...
		Impressive tests on the effect of the news on prices were done by the financial empiricist Victor Niederhoffer in...
	www.ljseek.com /Primality-Tests_s4.html (485 words)

	The AKS "PRIMES in P" Algorithm Resource
		AKS 'PRIMES is in P' home page with the original paper, and the revised 'v3' paper.
		Qi Cheng's "Primality Proving via One Round in ECPP and One Iteration in AKS" hybrid building on Berrizbeitia's work for randomised O~(d^4) time (certificate finding, O(d^4) to verify).
		Similarly "Conjecture 4" means assuming the truth of the conjecture stated in section 6 of the AKS paper, and explained further in AKS's [BP01] reference.
	fatphil.org /maths/AKS (713 words)

	Learn more about Prime number in the online encyclopedia. (Site not responding. Last check: 2007-11-07)
		It is possible to quickly check whether a given large number (say, up to a few thousand digits) is prime using probabilistic primality tests.
		Historically, the largest known prime has almost always been a Mersenne prime since the dawn of electronic computers, because there exists a particularly fast primality test for numbers of this form, the Lucas-Lehmer test.
		A primality test algorithm is an algorithm which tests a number for primality.
	www.onlineencyclopedia.org /p/pr/prime_number.html (1927 words)

	Articles - Prime number (Site not responding. Last check: 2007-11-07)
		A probable prime is an integer which, by virtue of having passed a certain test, is considered to be probably prime.
		Some of these tests are not perfect: there may be some composite numbers, called pseudoprimes for the respective test, that will be declared "probably prime" no matter what witness is chosen.
		This is known as trial division; it is the simplest primality test and it quickly becomes impractical for testing large integers because the number of possible factors grows exponentially as the number of digits in the number-to-be-tested increases.
	www.izeez.com /articles/Prime_number (3689 words)

	NSDL Metadata Record -- AKS Primality Test -- from MathWorld
		While this had long been believed possible (Wagon 1991), no one had previously been able to produce an explicit polynomial time deterministic algorithm (although probabilistic algorithms were known that seem to run in polynomial time).
		This test is now known as the Agarwal-Kayal-Saxena primality test, cyclotomic AKS test, or AKS primality test.
		Crandall, R. and Papadopoulos, J. "On the Implementation of AKS-Class Primality Tests." 18 Mar 2003.
	nsdl.org /mr/696571 (304 words)

	Prime time
		Despite struggling for centuries to find a simple and efficient way to test whether a number is prime, mathematicians have been gazumped by two computer science students and their professor.
		A polynomial time algorithm, such as the AKS algorithm, is one where the complexity is some power of the input.
		And not only does it finally prove that primality testing is possible in polynomial time, but it does so using relatively simple mathematics.
	plus.maths.org /issue22/news/prime/index.html (615 words)

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