Permutation polynomials over finite fields from low-degree rational functions

2025-03-26

AI Summary

This work addresses the construction of permutation polynomials (PPs) over finite fields $mathbb{F}_{q^2}$ of even characteristic. We propose a novel method based on low-degree permutation rational functions over the subfield $mathbb{F}_q$. Through systematic algebraic construction and transformation, we derive two new classes of permutation binomials and six new classes of permutation pentanomials over $mathbb{F}_{q^2}$, and rigorously verify their permutation property in even characteristic. Furthermore, employing quasi-multiplicative equivalence analysis, we prove that all eight new classes are inequivalent to any previously known PPs. This significantly expands the known spectrum of permutation polynomials over even-characteristic finite fields and provides new algebraic tools for applications in cryptography and coding theory.

Technology Category

• Machine Learning: Feature Construction/Reformulation
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Applications of cryptography
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

This paper considers permutation polynomials over the finite field $F_{q^2}$ in even characteristic by utilizing low-degree permutation rational functions over $F_q$. As a result, we obtain two classes of permutation binomials and six classes of permutation pentanomials over $F_{q^2}$. Additionally, we show that the obtained binomials and pentanomials are quasi-multiplicative inequivalent to the known ones in the literature.

Problem

• Constructs permutation polynomials over finite fields
• Uses low-degree rational functions in even characteristic
• Introduces new inequivalent binomials and pentanomials

Innovation

• Utilizing low-degree rational functions
• Constructing permutation binomials and pentanomials
• Quasi-multiplicative inequivalent to known polynomials