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Finding irreducible polynomials over finite fields
1986
Proceedings of the eighteenth annual ACM symposium on Theory of computing - STOC '86
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New Algorithms for Finding Irreducible Polynomials Over Finite Fields
1990
Mathematics of Computation
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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
1988
[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 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
1990
Mathematics of Computation
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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
2019
PRIKLADNAYa DISKRETNAYa MATEMATIKA
Алгоритм нахождения минимальной степени полинома над конечным полем для функции над векторным пространством в зависимости от выбора неприводимого многочлена
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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.
Modeling of Hash Functions on the Basis of Irreducible Polynomials in a Finite Fields
2018
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 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
2011
Mathematics Magazine
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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
1990
Mathematics of Computation
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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
1990
Mathematics of Computation
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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]
2011
arXiv
pre-print
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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
1987
Journal of symbolic computation
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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
2010
Finite Fields and Their Applications
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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]
2015
arXiv
pre-print
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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]
1991
Lecture Notes in Computer Science
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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
2012
Mathematics of Computation
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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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