New classes of permutation trinomials over finite fields with even characteristic
Kirpa Garg, Sartaj Ul Hasan, Chandan Kumar Vishwakarma
TL;DR
The paper advances the study of permutation trinomials over even-characteristic finite fields by introducing three new classes of the form $f(X)=X^r\big(X^{\alpha(q-1)}+X^{\beta(q-1)}+1\big)$ with $\alpha>6$ and $r\ge7$, over $\mathbb{F}_{2^{2m}}$, and proves a nonexistence result for a specific parameter set when $m>3$. It uses the Wan–Lidl criterion, CRT, and algebraic-geometry techniques (via absolute irreducibility and the Hasse–Weil bound) to establish permutation properties and nonpermutation outcomes, and combines these with quasi-multiplicative (QM) equivalence analysis to show that the new classes are QM-inequivalent to known examples and to each other. The authors also prove a QM-equivalence conjecture proposed by Harsh, making the QM landscape of quadratic-extensions permutation trinomials more complete. Computational tools (e.g., SageMath) and classical algebraic geometry underpin the arguments, underscoring the interplay between number theory, algebraic geometry, and finite-field permutations. Overall, the work broadens the parameter regime for permutation trinomial constructions and clarifies their QM relationships within the existing literature, with potential implications for cryptography and coding theory where structured permutation polynomials are valuable.
Abstract
The construction of permutation trinomials of the form $X^r(X^{α(2^m-1)}+X^{β(2^m-1)} + 1)$ over $\F_{2^{2m}}$ where $α> β$ and $r$ are positive integers, is an active area of research. To date, many classes of permutation trinomials with $α\leq 6$ have been introduced in the literature. Here, we present three new classes of permutation trinomials with $α>6$ and $r \geq 7$ over $\F_{2^{2m}}$. Additionally, we prove the nonexistence of a class of permutation trinomials over $\F_{2^{2m}}$ of the same type for $r=9$, $α=7$, and $β=3$ when $m > 3$. Moreover, we show that the newly obtained classes are quasi-multiplicative inequivalent to both the existing permutation trinomials and to one another. Furthermore, we provide a proof for the recent conjecture on the quasi-multiplicative equivalence of two classes of permutation trinomials, as proposed by Yadav, Gupta, Singh, and Yadav (Finite Fields Appl. 96:102414, 2024).