On the difference between permutation polynomials

Anbar, Nurdagül; Odžak, Almasa; Patel, Vandita; Quoos, Luciane; Somoza, Anna; Topuzoğlu, Alev

doi:10.1016/j.ffa.2017.09.009

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Cited by 8 publications

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“…One of the main approaches to show that P (x) is not a permutation uses the theory of curves and their number of rational points, for instance see [1,2]. The approach can be summarized as follows.…”

Section: Introductionmentioning

confidence: 99%

Curves over Finite Fields and Permutations of the Form x k

Anbar¹

2019

Turk J Math

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Section: Introductionmentioning

confidence: 99%

Curves over Finite Fields and Permutations of the Form x k

Anbar¹

2019

Turk J Math

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“…The theory of algebraic curves is one of the main tools to show that P(X) is not a permutation of certain finite fields, for instance, see [2]. The well-known approach can be summarized as follows:…”

Section: Introductionmentioning

confidence: 99%

On a special type of permutation rational functions

Anbar

2022

AAECC

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Let p be a prime and n be a positive integer. We consider rational functions f b (X) = X + 1∕(X p − X + b) over p n with Tr(b) ≠ 0 . In Hou and Sze (Finite Fields Appl 68, Paper No. 10175, 2020), it is shown that f b (X) is not a permutation for p > 3 and n ≥ 5 , while it is for p = 2, 3 and n ≥ 1 . It is conjectured that f b (X) is also not a permutation for p > 3 and n = 3, 4 , which was recently proved sufficiently large primes in Bartoli and Hou (Finite Fields Appl 76, Paper No. 101904, 2021). In this note, we give a new proof for the fact that f b (X) is not a permutation for p > 3 and n ≥ 5 . With this proof, we also show the existence of many elements b ∈ p n for which f b (X) is not a permutation for n = 3, 4.

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“…The theory of curves is one of the main tools to show that P (X) is not a permutation of certain finite fields, see for instance [2,6]. The usual approach can be summarized as follows.…”

Section: Introductionmentioning

confidence: 99%

Permutations polynomials of the form G(X)k − L(X) and curves over finite fields

Anbar

Kaşıkcı

2021

Cryptogr. Commun.

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For a positive integer k and a linearized polynomial L(X), polynomials of the form P (X) = G(X) k − L(X) ∈ F q n [X] are investigated. It is shown that when L has a non-trivial kernel and G is a permutation of F q n , then P (X) cannot be a permutation if gcd(k, q n − 1) > 1. Further, necessary conditions for P (X) to be a permutation of F q n are given for the case that G(X) is an arbitrary linearized polynomial. The method uses plane curves, which are obtained via the multiplicative and the additive structure of F q n , and their number of rational affine points.

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On the difference between permutation polynomials

Cited by 8 publications

References 11 publications

Curves over Finite Fields and Permutations of the Form x k

Curves over Finite Fields and Permutations of the Form x k

On a special type of permutation rational functions

Permutations polynomials of the form G(X)k − L(X) and curves over finite fields

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