More classes of permutation quadrinomials from Niho exponents in characteristic two

Zheng, Lan; Liu, Baixiang; Kan, Haibin; Peng, Jie; Tang, Deng

doi:10.1016/j.ffa.2021.101962

Cited by 16 publications

(9 citation statements)

References 29 publications

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“…Next we show that the sufficient conditions in [46,Thm. 1.3] are both necessary and sufficient, as was conjectured in [46, pp.…”

Section: Resolution Of Eight Conjectures and Open Problemsmentioning

confidence: 89%

Determination of a class of permutation quadrinomials

Ding¹,

Zieve²

2022

Preprint

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We determine all permutation polynomials over F q 2 of the form X r A(X q−1 ) where, for some Q which is a power of the characteristic of Fq, we have r ≡ Q + 1 (mod q + 1) and all terms of A(X) have degrees in {0, 1, Q, Q+1}. We then use this classification to resolve eight conjectures and open problems from the literature. Our proof makes a novel use of geometric techniques in a situation where they previously did not seem applicable, namely to understand the arithmetic of highdegree rational functions over small finite fields, despite the fact that in this situation the Weil bounds do not provide useful information.

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“…Next we show that the sufficient conditions in [46,Thm. 1.3] are both necessary and sufficient, as was conjectured in [46, pp.…”

Section: Resolution Of Eight Conjectures and Open Problemsmentioning

confidence: 89%

Determination of a class of permutation quadrinomials

Ding¹,

Zieve²

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Next, we show that the sufficient conditions in [54, Thm. 1.3] are both necessary and sufficient, as was conjectured in [54, pp.…”

Section: Resolution Of Eight Conjectures and Open Problemsmentioning

confidence: 99%

“…Next, we show that the sufficient conditions in [54, Thm. 1.3] are both necessary and sufficient, as was conjectured in [54, pp. 5 and 20]: Corollary Let

k

and

\ell

be positive integers, and write

m:=2^{\operatorname{ord}_2(\gcd (k,\ell ))}

.…”

Section: Resolution Of Eight Conjectures and Open Problemsmentioning

confidence: 99%

See 1 more Smart Citation

Determination of a class of permutation quadrinomials

Ding

Zieve

2023

Proceedings of London Math Soc

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We determine all permutation polynomials over 𝔽 𝑞 2 of the form 𝑋 𝑟 𝐴(𝑋 𝑞−1 ) where, for some 𝑄 that is a power of the characteristic of 𝔽 𝑞 , we have 𝑟 ≡ 𝑄 + 1 (mod 𝑞 + 1) and all terms of 𝐴(𝑋) have degrees in {0, 1, 𝑄, 𝑄 + 1}. We use this classification to resolve eight conjectures and open problems from the literature, and we list 77 recent results from the literature that follow immediately from the simplest special cases of our result. Our proof makes a novel use of geometric techniques in a situation where they previously did not seem applicable, namely to understand the arithmetic of high-degree rational functions over small finite fields, despite the fact that in this situation the Weil bounds do not provide useful information.

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“…If the mapping f : c → f (c) is a bijection on F 2 n , then f (x) is called a permutation polynomial over F 2 n . Due to its important applications in cryptography, coding theory and combinatorial designs (see [8], [13], [14] for instance), permutation polynomials have been intensively studied in the past. The study can be generalized from the case of univariate to bivariate.…”

Section: Introductionmentioning

confidence: 99%