Efficient quasi-deterministic primality test improving AKS (original) (raw)
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Papers from the international symposium on Symbolic and algebraic computation - ISSAC '92, 1992
Rabin's algorithm is commonly used in computer algebra systems and elsewhere for primality testing. This paper presents an experience with this in the Axiom* computer algebra system. As a result of this experience, we suggest certain strengthenings of the algorithm.
ArXiv, 2019
In this set of three companion manuscripts/articles, we unveil our new results on primality testing and reveal new primality testing algorithms enabled by those results. The results have been classified (and referred to) as lemmas/corollaries/claims whenever we have complete analytic proof(s); otherwise the results are introduced as conjectures. In Part/Article 1, we start with the Baseline Primality Conjecture~(PBPC) which enables deterministic primality detection with a low complexity = O((log N)^2) ; when an explicit value of a Quadratic Non Residue (QNR) modulo-N is available (which happens to be the case for an overwhelming majority = 11/12 = 91.67% of all odd integers). We then demonstrate Primality Lemma PL-1, which reveals close connections between the state-of-the-art Miller-Rabin method and the renowned Euler-Criterion. This Lemma, together with the Baseline Primality Conjecture enables a synergistic fusion of Miller-Rabin iterations and our method(s), resulting in hybrid ...
Galois theory and primality testing
Lecture Notes in Mathematics, 1985
In this paper we show how Galois theory for rings can be applied to the problem of distinguishing prime numbers from composite numbers. It develops ideas that were first formulated in [11, Section 8; 12]. A positive integer n is prime if and only if the ring ZZ/nZZ, is a field. Many primality testing algorithms make use of extension rings A of 2Z/nE that are fields if n is prime. They depend on known properties of such fields and of the Frobenius map A-> A that sends every χ e A to its n-th power. If n is composite then usually one of these properties is found not to be satisfied, and one is finished. If one does not succeed in proving n composite in this way then the problem suggests itself how to prove that n is prime. Only after this proof has been completed one knows that the rings one works with are actually fields; in particular, this fact may not be used in the proof. It is for this reason that Galois theory for rings rather than for fields is needed. Galois theory for rings can be found in [4; 6, Chapter III]. For the convenience of the reader we prove in Section 2 all facts from this theory that we need, starting only from basic properties of tensor products, localizations, and projective modules [1; 2]. In Section 3 we restrict to finite rings and abelian Galois groups, and we treat the Artin symbol, which replaces the Frobenius map. Section 4 is devoted to a special class of extensions of Z3/nE, which we call cyclotomic extensions. These play an important role in primality testing. In Section 5 we prove a result about Gauss sums that can be viewed äs a generalization of [5, Theorem (7.8)], and we show how to Interpret this result in terms of Artin symbols. The application to primality testing occupies Section 6. We describe a test that is closely related to the methods of [3], äs generalized by Williams (see [14] for references). The second test that we describe is an improvement of the method proposed in [5]. Finally, we show how the theory presented in this paper can be used to combine the two tests. It may be expected that this combined method, once implemented, will perfonn better than any existing primality testing algorithm.
Generalized Strong Pseudoprime Tests and Applications
Journal of Symbolic Computation, 2000
We describe probabilistic primality tests applicable to integers whose prime factors are all congruent to 1 mod r where r is a positive integer; r = 2 is the Miller-Rabin test. We show that if ν rounds of our test do not find n = (r + 1) 2 composite, then n is prime with probability of error less than (2r) −ν. Applications are given, first to provide a probabilistic primality test applicable to all integers, and second, to give a test for values of cyclotomic polynomials.
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Finite Fields and Their Applications, 2012
We develop a general framework for producing deterministic primality tests based on commutative group schemes over rings of integers. Our focus is on the cases of algebraic tori and elliptic curves. The proposed general machinery provides several series of tests which include, as special cases, tests discovered by Gross and by Denomme and Savin for Mersenne and Fermat primes, primes of the form 2 2 l+1 − 2 l + 1, as well as some new ones.
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Archiv der Mathematik, 2009
We propose a deterministic primality test based on a section of a group scheme. Pépin's test and the tests of Lucas-Lehmer type are special cases of our construction, provided the group scheme is taken to be the multiplicative group and the Waterhouse-Weisfeiler group scheme, respectively. Besides, we suggest a test involving formal completions of these schemes.
A generalization of Miller's primality theorem
Proceedings of the American Mathematical Society, 2008
For any integer r we show that the notion of ω-prime to base a introduced by Berrizbeitia and Berry, 2000, leads to a primality test for numbers n congruent to 1 modulo r, which runs in polynomial time assuming the Extended Riemann Hypothesis (ERH). For r = 2 we obtain Miller's classical result.
INVESTIGATION STUDY OF FEASIBLE PRIME NUMBER TESTING ALGORITHMS
In this expository paper, we describe three primality-testing algorithms: Miller-Rabin, Fermat and AKS primality tests. The first test is very efficient, but is only capable of proving that a given number is either composite or 'very probably' prime. The second test is also probabilistic with lower probability and higher execution time. The third test is a deterministic unconditional polynomial time algorithm to prove that a given number is either prime or composite; however, it had no practical applications due to the time complexity O (log 5 (n)). Thus, the first primality test is at present one of the most widely used in practice as it run at logarithmic run time complexity O (log (n)).