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Factoring polynomials over finite fields
International Journal of Number Theory, 2021In this paper, we describe a new polynomial factorization algorithm over finite fields with odd characteristics. The main ingredient of the algorithm is special singular curves. The algorithm relies on the extension of the Mumford representation and Cantor’s algorithm to these special singular curves.
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Polynomials Over Finite Fields
1992The first step of modern algorithms of factorization of polynomials with integer coefficients consists in factorizing their image modulo some prime number. This is the reason why, in this chapter, we study the factorization of polynomials over finite fields. Most of the results of this theory were developed by E.R. Berlekamp.
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Irreducible Polynomials Over Finite Fields
2020In contrast to the theoretical construction of finite fields as splitting fields presented in Chap. 3, we now consider the explicit description of the extension field GF(qn) of F = GF(q) as the factor ring F[x] · (f) of the polynomial ring F[x] with respect to an irreducible polynomial f of degree n.
Dirk Hachenberger, Dieter Jungnickel
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Permutation polynomials over finite fields
Acta Mathematica Sinica, 1987The author proves that if \(q\) is odd and congruent to 1 modulo 3, then the polynomial \(f(x)=x^{1+(q-1)/3}+ax\) \((a\neq 0)\) is not a permutation polynomial over any field \(\mathbb F_{q^r}\) \((r\geq 2)\). Thereby a question raised by \textit{L. Carlitz} [Bull. Am. Math. Soc. 68, 120--122 (1962); Zbl 0217.33003)] receives a partial answer.
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Factoring Polynomials over Finite Fields
1993A polynomial of degree n over a finite field F q is an expression in an indeterminate x having the form $$f(x) = \sum\limits_{i = 0}^n {{a_i}{x^1}} $$ where n is a non-negative integer, a i ∈ F q , 0 ≤ i ≤ n and a n ≠ 0. To be more precise, f (x) is called a univariate polynomial to distinguish the more general situation where more ...
Ian F. Blake +4 more
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Factoring Polynomials over Finite Fields
1987We begin this chapter by considering another application of Mobius inversion, this time to the result of Theorem 6.1, viz. $$ {x^{{{q^n}}}} - x = \prod\limits_{{d|n}} {{V_d}} (x) $$ , where V d (x) denotes the product of all monic irreducible polynomials of degree d over GF(q). We recall (see Example 6.3) that here the underlying group is the set
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Irreducible polynomials over finite fields
1986Several methods of computing irreducible polynomials over finite fields are presented. If preprocessing, depending only on p , is allowed for free, then an irreducible polynomial of degree at least n over Zp can be computed deterministically with O(n logp), i.e. O(output size), bit operations. The estimates for the preprocessing time depend on unproven
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Permutation polynomials and their compositional inverses over finite fields by a local method
Designs, Codes, and Cryptography, 2023Danyao Wu
exaly
Polynomials Over a Finite Field
Journal of the London Mathematical Society, 1965openaire +2 more sources
On Inverses of Permutation Polynomials of Small Degree Over Finite Fields
IEEE Transactions on Information Theory, 2020Qiang Wang, Wenhong Wei
exaly