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Computing pth roots in extended finite fields of prime characteristic p ≥ 2

Electronics Letters, 2016
Direct computation of pth roots in extended finite fields of characteristic p ≥ 2 is introduced, wherein the reduction polynomial is irreducible and can be even random. Proposed method works in any case of p ≥ 2 and finite field extension. This method is
M. Repka
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Irregularities in the distribution of irreducible polynomials [PDF]

Proceedings of the American Mathematical Society, 1993
We prove that there exist monic polynomials f f over GF ⁡ ( q ) \operatorname {GF} (q) for which f + g f + g is reducible for all g ∈ GF ⁡ ( q ) [ x ] g \in ...
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Deciding Irreducibility/Indecomposability of Feedback Shift Registers Is NP-Hard

IET Information Security
Feedback shift registers (FSRs) are used as a fundamental component in electronics and confidential communication. A FSR f is said to be reducible if all the output sequences of another FSR g can also be generated by f and the FSR g costs less memory ...
Lin Wang
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Irreducibility of polynomials

Journal of Algebra, 1965
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Criteria for the Irreducibility of Polynomials

The Annals of Mathematics, 1933
Es sei \(f(x)\) ein Polynom \(n\)-ten Grades mit ganzen rationalen Koeffizienten, das an den \(m\) voneinander verschiedenen ganzen rationalen Stellen \(a_1, a_2, \dots, a_m\) den Wert \(+1\) annimmt, wobei \(4 < m\leq n\) ist. Verf. zeigt, daß \(f(x)\) nur in Faktoren der Form \[ g(x) = \prod_{\nu=1}^m (x-a_\nu) h(x) \pm 1 \] zerfallen kann.
Dorwart, H. L., Ore, Øystein
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Polynomials Irreducible by Eisenstein's Criterion

Applicable Algebra in Engineering, Communication and Computing, 2003
Let \(0 ...
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Irreducible polynomials and barker sequences

ACM Communications in Computer Algebra, 2007
A Barker sequence is a finite sequence a o , ..., a n -1 , each term ±1, for which every sum Σ i a i a i ...
Peter B. Borwein, Erich L. Kaltofen, Michael J. Mossinghoff   +2 more
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Irreducibility of Polynomials

The American Mathematical Monthly, 1935
(1935). Irreducibility of Polynomials. The American Mathematical Monthly: Vol. 42, No. 6, pp. 369-381.
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Finding Irreducible and Primitive Polynomials

Applicable Algebra in Engineering, Communication and Computing, 1992
The paper presents new fast constructions of irreducible and primitive polynomials. It contains the following main results: 1. For any \(N \in \mathbb{N}\) one can construct an irreducible polynomial of degree \(n = N + O (N \exp (-( \log \log N)^{1/2-\varepsilon}))\) over \(GF(p)\) in time \((p \log N)^{O(1)}\). 2. For sufficiently large \(Q\) one can
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Computational Aspects of Irreducible Polynomials

Computational Mathematics and Mathematical Physics, 2020
In this paper, an upper bound for the height of polynomial divisors of a given polynomial belonging to \(\mathbb Z[x]\) is computed, which is better than an already known bound. The author also gives constructive examples of a class of irreducible polynomials over discrete valued fields using Newton's polygon.
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