Signature functions for algebraic numbers (original) (raw)
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Testing polynomials which are easy to compute (Extended Abstract)
Proceedings of the twelfth annual ACM symposium on Theory of computing - STOC '80, 1980
We exploit the fact that the set of all polynomials P~ ~[Xl,..,x n] of degree Kd which can be evaluated with ~v nonscalar steps can be embedded into a Zariski-closed affine set W(d,n,v),dim W(d,n,v)~(v+1 +n) 2 and deg W(d,n,v)~(2vd) (v+14n) 2 As a consequence we prove that for u:= 2v(d+1) 2 and s:= 6(v+1+n) 2 there exist a1,..,~s~ [u]n = {1,2,..,u}n such that for all polynomials P~ W(d,n,v) :P(~ I) = p(2) =... = p(~s) = O implies PHO. This means that ~1,...,~s is a correct test sequence for a zero test on all polynomials in W(d,n,v). Moreover, "almost every" sequence al,..,aSg [u]n is such a correct test sequence for W(d,n,v). The existence of correct test sequences al,..,~se [~n is established by a counting argument without constructing a correct test sequence• We even show that it is beyond the known methods to establish (i.e. to construct and to prove correctness) of such a short correct test sequence for W(d,n,v). We prove that given such a short, correct test sequence for W(d,n,v) we can efficiently construct a multivariate polynomial P~[Xl,..,x ~ with deg(P) = d and small integer coefficients such that P~ W(d,n,v). For v>n log d lower bounds of this type are beyond our present methods in algebraic complexity theory.
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.
Testing low-degree polynomials over prime fields
Random Structures and Algorithms, 2009
We present an efficient randomized algorithm to test if a given function f : F n p → F p (where p is a prime) is a low-degree polynomial. This gives a local test for Generalized Reed-Muller codes over prime fields. For a given integer t and a given real ǫ > 0, the algorithm queries f at
A New Method for Testing Whether a Number Is Prime
The present paper describes a new method for the decomposition of integer numbers into their prime factors. To know if a number is prime, the classic and oldest method is to perform a series of Euclidean divisions by the prime factors in increasing order. In this article, I report a new and alternative method to the classic one. It is based on the fact that an even number that precedes a prime number must in no case have a potential prime factor that lacks a unit. This even number with one unit before the number to be decomposed can then be used to prove its primality or not by looking if it has a factor missing one unit. To achieve this, we divide the even by the prime factors (p) in ascending order and follow the decimal part. If the latter is equal to a ratio of (p-1)/p (p is any prime factor), then the number to be decomposed is not prime and the prime number giving the ratio is its prime factor. This method has the potential to have applications in computer science and to lead to a new algorithm for decomposing numbers or further improve the performance of those existing.
Efficient quasi-deterministic primality test improving AKS
preprint, 2003
We combine ideas from the seminal paper of Agrawal, Kayal and Saxena [AKS] as improved by Lenstra [Le3] with the particular case sharpening of Berrizbeitia and introduce the cyclotomy of rings setting [Le2, BvdH, Mi3] for the latter. Thus we deduce a new variant of the AKS algorithm which: (i) has running time O`log 4+o(1) (n)´; (ii) works on all prime candidates n > e e 2e ; and (iii) is "quasi deterministic", in the sense that it is deterministic under the assumption that some roots of unity can be found in polynomial time, while failing to do so would raise an explicit contradiction to the GRH. The bottleneck of the algorithm are the space requirements.
On the reduction in multiplicative complexity achieved by the polynomial residue number system
IEEE Transactions on Signal Processing, 1992
The polynomial residue number system (PRNS) is known to reduce the complexity of polynomial m iltiplication from O(N2) to O (N). A new interpretation of this complexity reduction is given in the context of associative algt,bras over a finite field. The new point of view provides a clearer understanding of the Chinese remainder theorem.
On the genericity of the modular polynomial GCD algorithm
1999
In this paper we st,udy the generic setting of the modular GCD algorithm. We develop the algorithm for multivariate polynomials over Euclidean domains which have a spc:&l kind of remainder function. Details for the parameterixation and generic Maple code are given. Applying this grncric algorithm to a GCD problem in Z/(~)[t][z] where 1~ is small yields an improved asymptotic performance over t.he usual approach, and a very practical algorithm for polynomials over small finite fields. 'This nxderial is bawd on work support,ed in part, by t.hr National Science Yound;rLion under C:rmlt X0. C!CR-9712267 (Erich Kalt.ofen) and on work support,ed by NSEHC: of C!anacla (Michael Mormgan 1. Pcrnlission 1.0 make digit,al or hard copies of ,211 or part. of this work for personal or classroo~n use is granted wit.hout fee providctl that copies are not. made or dist.rilmtcd for prolit or commercial aclvant.age, arid that. ropies bear this ndice and the full cit.ation 011 the first. page. To c'opy othrrwise, to republish, to post on sc?rvers i)r to rcdist.ribute t.0 lists. requires prior specific permission and/or a fee. ISSAC' 99.
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)).
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 ...