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Finding irreducible polynomials over finite fields

L M Adleman, H W Lenstra
1986 Proceedings of the eighteenth annual ACM symposium on Theory of computing - STOC '86  
doi:10.1145/12130.12166 dblp:conf/stoc/AdlemanL86 fatcat:6uzhqbj4yvhehgsd5saomsmjbq

New Algorithms for Finding Irreducible Polynomials Over Finite Fields

Victor Shoup
1990 Mathematics of Computation  
We present a new algorithm for 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.  ... 
doi:10.2307/2008704 fatcat:swroxpnpbjhzxoywjwqxfahivm

New algorithms for finding irreducible polynomials over finite fields

V. Shoup
1988 [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science  
We in fact prove the stronger result that the problem of nding 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.  ... 
doi:10.1109/sfcs.1988.21944 dblp:conf/focs/Shoup88 fatcat:giz25n4zp5gwncrgh2niu2rbgq

New algorithms for finding irreducible polynomials over finite fields

Victor Shoup
1990 Mathematics of Computation  
We present a new algorithm for 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.  ... 
doi:10.1090/s0025-5718-1990-0993933-0 fatcat:f2n4yr5imrcbjlena3t23g5owq

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

S. A. Belov
2019 PRIKLADNAYa DISKRETNAYa MATEMATIKA  
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
more » ... 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.
doi:10.17223/20710410/43/1 fatcat:wapl2bbx7fgunf6g7p7ccxsgea

Modeling of Hash Functions on the Basis of Irreducible Polynomials in a Finite Fields

G. Vostrov, O. Ponomarenko
2018 Proceedings of X International Scientific and Practical Conference "Electronics and Information Technologies"  
In this paper the method for constructing hash functions on the basis of 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 .  ... 
doi:10.30970/elit2018.a32 fatcat:jvxolf6mpbg7lp4z723vfl3x4u

Counting Irreducible Polynomials over Finite Fields Using the Inclusion-Exclusion Principle

Sunil K. Chebolu, JáN Mináč
2011 Mathematics Magazine  
Why there are exactly 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.  ... 
doi:10.4169/math.mag.84.5.369 fatcat:pt6lztnar5d7nelrbbqx4nlvje

Computing irreducible representations of finite groups

L{ászl{ó Babai, Lajos R{ónyai
1990 Mathematics of Computation  
We present a 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.  ... 
doi:10.1090/s0025-5718-1990-1035925-1 fatcat:theuji7o7bd77fribcpi34apqe

Computing Irreducible Representations of Finite Groups

Laszlo Babai, Lajos Ronyai
1990 Mathematics of Computation  
We present a 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.  ... 
doi:10.2307/2008443 fatcat:fqjtiyjmu5cgjkkwdbt3ha7pom

Counting irreducible polynomials over finite fields using the inclusion-exclusion principle [article]

Sunil K. Chebolu, Jan Minac
2011 arXiv   pre-print
Gauss discovered a beautiful formula for the number of 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.  ... 
arXiv:1001.0409v6 fatcat:nyvnax3bl5d5teyn2fi6kltvge

Deterministic irreducibility testing of polynomials over large finite fields

Erich Kaltofen
1987 Journal of symbolic computation  
We present a sequential deterministic 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.  ... 
doi:10.1016/s0747-7171(87)80055-x fatcat:m7cbqp7l4jbqvpmrnafsmb6tdu

Parity of the number of irreducible factors for composite polynomials

Ryul Kim, Wolfram Koepf
2010 Finite Fields and Their Applications  
Various results on the parity of the number of 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.  ... 
doi:10.1016/j.ffa.2009.12.002 fatcat:6a3mlyrgpvcihj5e5upug2n44m

An algorithm for determining the irreducible polynomials over finite fields [article]

Samuel H. Dalalyan
2015 arXiv   pre-print
We propose an algorithm for determining the 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  ... 
arXiv:1505.00776v1 fatcat:4rk26ykxhzdkfhxsnxfv3zpqk4

Improving the time complexity of the computation of irreducible and primitive polynomials in finite fields [chapter]

Josep Rifà, Joan Borrell
1991 Lecture Notes in Computer Science  
In this paper, we present a method to compute all the 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.  ... 
doi:10.1007/3-540-54522-0_123 fatcat:2im7ejbtnfa5lpsrp5ddjfxjxy

Constructing irreducible polynomials over finite fields

San Ling, Enver Ozdemir, Chaoping Xing
2012 Mathematics of Computation  
We describe a new method for constructing 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.  ... 
doi:10.1090/s0025-5718-2011-02567-6 fatcat:t3jyp5kj2nduffdsqezbb4hpka
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