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Danyao Wu

Publications and source records attributed to Danyao Wu.

11 recordsLinked to original sources
arXiv2026

On the inverses of permutation polynomials of the form $h(ψ(x))φ(x)+g(ψ(x))$ over finite fields

In this paper, we investigate the compositional inverses of permutation polynomials of the form \[ F(x)=h(ψ(x))φ(x)+g(ψ(x)) \in \mathbb{F}_{q^n}[x], \] where \(ψ(x),φ(x) \in \mathbb{F}_{q^n}[x]\) are additive polynomials, \(h(x), g(x) \in \mathbb{F}_{q^n}[x]\) satisfy $ h(ψ(\mathbb{F}_{q^n})) \subseteq \mathbb{F}_q^*, $ and there exists a polynomial \(\barψ(x) \in \mathbb{F}_{q^n}[x]\) such that $ \barψ(F(x)) = ψ(x). $

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arXiv2026

The compositional inverses of the permutation polynomials of the form $x+γ\operatorname{Tr}_{q}^{q^n}(H(x))$ over $\mathbb{F}_{q^n}$

This paper focuses on computing the compositional inverses of permutation polynomials of the form $x+γ\operatorname{Tr}_{q}^{q^{n}}(H(x))$ over finite fields via the local method. We explicitly construct compositional inverses for four families of permutation polynomials of this type over $\mathbb{F}_{q^2}$, another four families over $\mathbb{F}_{q^3}$, and one general family over $\mathbb{F}_{q^n}$. The closed-form inverse expressions derived in this work supplement the theory of trace permutation polynomials.

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arXiv2025

Permutation Polynomials of the form $L(X)+γTr_q^{q^3}(h(X))$ over finite fields with even characteristic

Permutation polynomials over finite fields have extensive applications in various areas. Particularly, permutation polynomials with simple forms are of great interest. In recent papers, several classes of permutation polynomials of the form $L(X)+Tr_q^{q^3}(h(X))$ have been constructed. This paper further investigates permutation polynomials of such form over $\mathbb{F}_{q^3}$. Unlike previous studies, we transform the problem of constructing univariate permutation polynomials over finite fields into that of constructing corresponding multivariate permutations over $\mathbb{F}_{q}$-vector spaces. Through this approach, we completely characterize a class of permutation polynomials of the form $L(X)+γTr_q^{q^3}(c_1X+c_2X^2+c_3X^3+c_4X^{q+2})$ over $\mathbb{F}_{q^3}$, where $q=2^m$, $L(X)=X^q+aX$ and $a,c_1,c_2,c_3,c_4,γ\in\mathbb{F}_q$ with $a^2+a+1\neq0$. Furthermore, using a similar method, we generalize several results from a recent work by Jiang, Li and Qu (2026).

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arXiv2025

New permutation polynomials over $\mathbb{F}_{q^2}$

In this paper, we propose a new method to obtain new permutation polynomials over $\mathbb{F}_{q^2}$. Using this method, we extend many known permutation polynomials, which take the form $\sum_i(x^q-x+δ)^{s_i}+L(x)$, where $L(x)$ is a $q$-polynomial over $\mathbb{F}_q$ and $δ\in\mathbb{F}_{q^2}$. We also present an alternative approach for constructing permutation polynomials of the form $x+γTr_q^{q^d}(x^{q+1}+x^{2q+2})$ for the cases where $q=2^m$, $2\nmid d$ and $ Tr_q^{q^d}(x)=x+x^q+\dots+x^{q^{d-1}}$.

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arXiv2025

Algebraic Structure of Permutational Polynomials over $\mathbb{F}_{q^n}$ \uppercase\expandafter{\romannumeral2}

It is well known that there exists a significant equivalence between the vector space $\mathbb{F}_{q}^n$ and the finite fields $\mathbb{F}_{q^n}$, and many scholars often view them as the same in most contexts. However, the precise connections between them still remain mysterious. In this paper, we first show their connections from an algebraic perspective, and then propose a more general algebraic framework theorem. Furthermore, as an application of this generalized algebraic structure, we give some classes of permutation polynomials over $\mathbb{F}_{q^2}$.

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arXiv2024

The compositional inverses of three classes of permutation polynomials over finite fields

Recently, P. Yuan presented a local method to find permutation polynomials and their compositional inverses over finite fields. The work of P. Yuan inspires us to compute the compositional inverses of three classes of the permutation polynomials: (a) the permutation polynomials of the form $ax^q+bx+(x^q-x)^k$ over $\mathbb{F}_{q^2},$ where $a+b \in \mathbb{F}_q^*$ or $a^q=b;$ (b) the permutation polynomials of the forms $f(x)=-x+x^{(q^2+1)/2}+x^{(q^3+q)/2} $ and $f(x)+x$ over $\mathbb{F}_{q^3};$ (c) the permutation polynomial of the form $A^{m}(x)+L(x)$ over $\mathbb{F}_{q^n},$ where ${\rm Im}(A(x))$ is a vector space with dimension $1$ over $\mathbb{F}_{q}$ and $L(x)$ is not a linearized permutation polynomial.

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arXiv2024

The compositional inverses of permutation polynomials from trace functions over finite fields

In this paper, we present the compositional inverses of several classes permutation polynomials of the form $\sum_{i=1}^kb_i\left({\rm Tr}_m^{mn}(x)^{t_i}+δ\right)^{s_i}+f_1(x)$, where $1\leq i \leq k,$ $s_i$ are positive integers, $b_i \in \mathbb{F}_{p^m},$ $p$ is a prime and $f_1(x)$ is a polynomial over $\mathbb{F}_{p^{mn}}$ satisfying the following conditions: (i) ${\rm Tr}_m^{mn}(x) \circ f_1(x)=φ(x) \circ {\rm Tr}_m^{mn}(x),$ where $φ(x)$ is a polynomial over $\mathbb{F}_{p^m};$ (ii) For any $a \in \mathbb{F}_{p^m},$ $f_1(x)$ is injective on ${\rm Tr}_m^{mn}(a)^{-1}.$

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arXiv2024

Permutation polynomials over finite fields by the local criterion

In this paper, we further investigate the local criterion and present a class of permutation polynomials and their compositional inverses over $ \mathbb{F}_{q^2}$. Additionally, we demonstrate that linearized polynomial over $\mathbb{F}_{q^n}$ is a local permutation polynomial with respect to all linear transformations from $\mathbb{F}_{q^n}$ to $\mathbb{F}_q ,$ and that every permutation polynomial is a local permutation polynomial with respect to certain mappings.

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arXiv2022

Permutation trinomials over $\mathbb{F}_{2^m}$: a corrected version

Permutation polynomials are an interesting subject of mathematics and have applications in other areas of mathematics and engineering. In this paper, we determine all permutation trinomials over $\mathbb{F}_{2^m}$ in Zieve's paper. We prove a conjecture proposed by Gupta and Sharma and obtain some new permutation trinomials over $\mathbb{F}_{2^m}$. Finally, we show that some classes of permutation trinomials with parameters are QM equivalent to some known permutation trinomials.

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