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Finding irreducible polynomials over finite fields

1986
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Proceedings of the eighteenth annual ACM symposium on Theory of computing - STOC '86
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##
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New Algorithms for Finding Irreducible Polynomials Over Finite Fields

1990
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Mathematics of Computation
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We present a new algorithm for

doi:10.2307/2008704
fatcat:swroxpnpbjhzxoywjwqxfahivm
*finding*an*irreducible**polynomial*of specified degree*over*a*finite**field*. ... We in fact prove the stronger result that the problem of*finding**irreducible**polynomials*of specified degree*over*a*finite**field*is deterministic*polynomial*-time reducible to the problem of factoring*polynomials*... Introduction In this paper we present some new algorithms for*finding**irreducible**polynomials**over**finite**fields*. ...##
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New algorithms for finding irreducible polynomials over finite fields

1988
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[Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science
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We in fact prove the stronger result that the problem of nding

doi:10.1109/sfcs.1988.21944
dblp:conf/focs/Shoup88
fatcat:giz25n4zp5gwncrgh2niu2rbgq
*irreducible**polynomials*of speci ed degree*over*a nite eld is deterministic*polynomial*time reducible to the problem of factoring*polynomials*... We present a new algorithm for nding an*irreducible**polynomial*of speci ed degree*over*a nite eld. Our algorithm is deterministic, and it runs in*polynomial*time for elds of small characteristic. ... We rst construct*irreducible**polynomials**over*F of degree q ei i for i = 1; : : :; r. We then \combine" these*polynomials*to form an*irreducible**polynomial*of degree n. ...##
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New algorithms for finding irreducible polynomials over finite fields

1990
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Mathematics of Computation
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We present a new algorithm for

doi:10.1090/s0025-5718-1990-0993933-0
fatcat:f2n4yr5imrcbjlena3t23g5owq
*finding*an*irreducible**polynomial*of specified degree*over*a*finite**field*. ... We in fact prove the stronger result that the problem of*finding**irreducible**polynomials*of specified degree*over*a*finite**field*is deterministic*polynomial*-time reducible to the problem of factoring*polynomials*... Introduction In this paper we present some new algorithms for*finding**irreducible**polynomials**over**finite**fields*. ...##
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An algorithm for finding the minimum degree of a polynomial over a finite field for a function over a vector space depending on the choice of an irreducible polynomial

Алгоритм нахождения минимальной степени полинома над конечным полем для функции над векторным пространством в зависимости от выбора неприводимого многочлена

2019
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PRIKLADNAYa DISKRETNAYa MATEMATIKA
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Алгоритм нахождения минимальной степени полинома над конечным полем для функции над векторным пространством в зависимости от выбора неприводимого многочлена

Московский государственный университет имени М.В. Ломоносова, г. Москва, Россия Рассматриваются преобразования над векторным пространством p-ичных векторов длины n, где p простое число. Каждому такому преобразованию ставится в соответствие полином над конечным полем GF(p n ). Конечное поле представляется кольцом вычетов по модулю неприводимого многочлена. В общем случае, в зависимости от выбора неприводимого многочлена, преобразованию над векторным пространством соответствуют различные полиномы

doi:10.17223/20710410/43/1
fatcat:wapl2bbx7fgunf6g7p7ccxsgea
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... над конечным полем. Предложен алгоритм поиска минимальной степени среди таких полиномов и неприводимого многочлена, при котором эта степень достигается. Ключевые слова: конечное поле, неприводимый многочлен, булевы функции, блочный шифр.##
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Modeling of Hash Functions on the Basis of Irreducible Polynomials in a Finite Fields

2018
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Proceedings of X International Scientific and Practical Conference "Electronics and Information Technologies"
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In this paper the method for constructing hash functions on the basis of

doi:10.30970/elit2018.a32
fatcat:jvxolf6mpbg7lp4z723vfl3x4u
*irreducible**polynomials*in*finite**fields*has been considered. ... The problem of searching for*irreducible**polynomials*was considered. Computer modeling of hash functions using*irreducible**polynomials*was performed. ... To construct a*field*n p F , it is necessary to*find*a*polynomial*) (x P of degree n*irreducible**over*a*field*p F . Such a*field*is represented by*polynomials**over*p F a degree not higher 1 − n . ...##
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Counting Irreducible Polynomials over Finite Fields Using the Inclusion-Exclusion Principle

2011
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Mathematics Magazine
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Why there are exactly

doi:10.4169/math.mag.84.5.369
fatcat:pt6lztnar5d7nelrbbqx4nlvje
*irreducible*monic*polynomials*of degree 30*over*the*field*of two elements? ... Then in general, the number of monic*irreducible**polynomials*of degree n*over*the*finite**field*F q is given by Gauss's formula where d runs*over*the set of all positive divisors of n including 1 and n, ... We would like to thank our students at Illinois State University and University of Western Ontario for their inspiration and insistence on penetrating mysteries of*finite**fields*. ...##
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Computing irreducible representations of finite groups

1990
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Mathematics of Computation
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We present a

doi:10.1090/s0025-5718-1990-1035925-1
fatcat:theuji7o7bd77fribcpi34apqe
*polynomial*-time algorithm to*find*a complete set of nonequivalent*irreducible*representations*over*the*field*of complex numbers of a*finite*group given by its multiplication table. ... We also consider the problem of decomposing a given representation 'V of the*finite*group G*over*an algebraic number*field*F into absolutely*irreducible*constituents. ...*Find*the*irreducible*constituents*over*Q of a representation G -► GL(n, Q) of a*finite*group G in*polynomial*time. ...##
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Computing Irreducible Representations of Finite Groups

1990
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Mathematics of Computation
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We present a

doi:10.2307/2008443
fatcat:fqjtiyjmu5cgjkkwdbt3ha7pom
*polynomial*-time algorithm to*find*a complete set of nonequivalent*irreducible*representations*over*the*field*of complex numbers of a*finite*group given by its multiplication table. ... We also consider the problem of decomposing a given representation 'V of the*finite*group G*over*an algebraic number*field*F into absolutely*irreducible*constituents. ...*Find*the*irreducible*constituents*over*Q of a representation G -► GL(n, Q) of a*finite*group G in*polynomial*time. ...##
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Counting irreducible polynomials over finite fields using the inclusion-exclusion principle
[article]

2011
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arXiv
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pre-print

Gauss discovered a beautiful formula for the number of

arXiv:1001.0409v6
fatcat:nyvnax3bl5d5teyn2fi6kltvge
*irreducible**polynomials*of a given degree*over*a*finite**field*. ... Assuming just a few elementary facts in*field*theory and the exclusion-inclusion formula, we show how one see the shape of this formula and its proof instantly. ... We would like to thank our students at Illinois State University and University of Western Ontario for their inspiration and insistence on penetrating mysteries of*finite**fields*. ...##
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Deterministic irreducibility testing of polynomials over large finite fields

1987
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Journal of symbolic computation
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We present a sequential deterministic

doi:10.1016/s0747-7171(87)80055-x
fatcat:m7cbqp7l4jbqvpmrnafsmb6tdu
*polynomial*-time algorithm for testing dense multivariate*polynomials**over*a large*finite**field*for*irreducibility*. ... Our deterministic solution is based on our algorithm for absolute*irreducibility*testing combined with Berlekamp's algorithm. ... Here we present an algorithm that tests dense multivariate*polynomials**over*large*finite**fields*for*irreducibility*in deterministic*polynomial*time. ...##
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Parity of the number of irreducible factors for composite polynomials

2010
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Finite Fields and Their Applications
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Various results on the parity of the number of

doi:10.1016/j.ffa.2009.12.002
fatcat:6a3mlyrgpvcihj5e5upug2n44m
*irreducible*factors of given*polynomials**over**finite**fields*have been obtained in the recent literature. ... We apply this to obtain some results concerning the parity of the number of*irreducible*factors for several special types of*polynomials**over**finite**fields*. ... We apply Theorem 8 to trinomials*over*F 2 to get the following. has an even number of*irreducible*factors*over*F 2 in the following cases: (1) n − k = 1 and n is odd, (2) n − k 2 and n is even. ...##
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An algorithm for determining the irreducible polynomials over finite fields
[article]

2015
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arXiv
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pre-print

We propose an algorithm for determining the

arXiv:1505.00776v1
fatcat:4rk26ykxhzdkfhxsnxfv3zpqk4
*irreducible**polynomials**over**finite**fields*, based on the use of the companion matrix of*polynomials*and the generalized Jordan normal form of square matrices ... Introduction and the main results The problem of*finding*the*irreducible**polynomials**over**finite**fields*together with the related topic of the*irreducible*factorization of*polynomials*is one of central ... If we know one primitive unitary*irreducible**over*a*finite**field*F q*polynomial*f (t) of a degree d, then we can*find*all unitary*irreducible**over*F q*polynomials*g(t) of a degree d ′ , dividing d, by ...##
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Improving the time complexity of the computation of irreducible and primitive polynomials in finite fields
[chapter]

1991
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Lecture Notes in Computer Science
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In this paper, we present a method to compute all the

doi:10.1007/3-540-54522-0_123
fatcat:2im7ejbtnfa5lpsrp5ddjfxjxy
*irreducible*and primitive*polynomials*of degree m*over*a*finite**field*. ... Our algorithm is especially well-suited for applications using large*finite**fields*. ... In this paper, we propose a method to compute*irreducible*and primitive*polynomials**over*a*finite**field*. ...##
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Constructing irreducible polynomials over finite fields

2012
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Mathematics of Computation
*

We describe a new method for constructing

doi:10.1090/s0025-5718-2011-02567-6
fatcat:t3jyp5kj2nduffdsqezbb4hpka
*irreducible**polynomials*modulo a prime number p. The method mainly relies on Chebotarev's density theorem. ... In the current state of the art,*finding*the Hilbert class*polynomial*is a necessary step for constructing elliptic curves*over**finite**fields*with a desired endomorphism ring. ... For example, one of the efficient methods described in [12] constructs an*irreducible**polynomial*through the factorization of some special*polynomials**over**finite**fields*. ...
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