Permutation polynomials of the form x+γTr(H(x))

2025/07/01 by Yangcheng Li, Xuan Pang, Li, Yangcheng +4
Computer Science · Engineering · Mathematics · #11T06 #11T55 #Advanced Combinatorial Mathematics #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2507.00781

openalex publication_date 2025/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a polynomial \( H(x) \) over \(\mathbbF_qⁿ\), we study permutation polynomials of the form \( x + γTr(H(x)) \) over \(\mathbbF_qⁿ\). Let P_H=\γ∈ \mathbbF_qⁿ : x+γTr(H(x))~is a permutation polynomial\. We present some properties of the set \(P_H\), particularly its relationship with linear translators. Moreover, we obtain an effective upper bound for the cardinality of the set \(P_H\) and show that the upper bound can reach up to qⁿ - q^{n - 1}. Furthermore, we prove that when the cardinality of the set \(P_H\) reaches this upper bound, the function \(Tr(H(x))\) must be an \(\mathbbF_q\)-linear function. Finally, we study two classes of functions H(x) over \(\mathbbF_q²\) and determine the corresponding sets P_H. The sizes of these sets P_H are all relatively small, even only including the trivial case.

Citations

• Algebraic Structure of Permutational Polynomials over \mathbbF_qⁿ
• Local Method for Compositional Inverses of Permutational Polynomials
• Compositional Inverses of AGW-PPs
• Binary Cyclic Codes from Explicit Polynomials over \gf(2^m)
• Further Results on Permutation Polynomials over Finite Fields
• Constructing permutations of finite fields via linear translators
• A method for obtaining digital signatures and public-key cryptosystems

Related

• Some permutation polynomials via linear translators
• Permutation polynomials of the form x+c*Tr(x^k)
• New permutation polynomials over \mathbbF_q²
• Permutation polynomials from a linearized decomposition
• New results on permutation polynomials over finite fields