Infinite families of APN permutations in constrained trivariate classes over $\mathbb{F}_{2^m}$

2026-03-16

AI Summary

This work constructs two new families of ternary permutation polynomials, denoted $G_a$ and $H_a$, over the finite field $\mathbb{F}_{2^{3m}}$, thereby transcending the equivalence class limitations of known APN permutations. By analyzing the condition under which their associated univariate polynomials have no roots in $\mathbb{F}_{2^m}^*$, the authors establish necessary and sufficient conditions for both $G_a$ and $H_a$ to be APN permutations and precisely characterize the admissible values of the parameter $a$. Notably, they prove for the first time that, for the same parameter $a$, both constructions yield APN permutations that are mutually CCZ-inequivalent and also CCZ-inequivalent to the known Li–Kaleyski family, thus producing two novel infinite families of APN permutations. In particular, when $q = 2$ and $7 \nmid m$, every $a \neq 1$ gives rise to an APN permutation CCZ-inequivalent to all previously known constructions.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Game Theory and Economic Paradigms: Imperfect Information

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

We study trivariate permutation polynomials over $\mathbb{F}_{2^{m}}$ extending two APN permutation families of Li--Kaleyski (IEEE Trans. Inform. Theory, 2024) by allowing the scalar parameter to vary over $\mathbb{F}_{2^m}^*$. For \[ G_a(x,y,z)=(x^{q+1}+ax^qz+yz^q,\; x^qz+y^{q+1},\; xy^q+ay^qz+z^{q+1}), \] where $a\in\mathbb{F}_{2^m}^*$, $q=2^i$, $\gcd(i,m)=1$, and $m$ is odd, we prove that $G_a$ is a permutation if and only if an associated univariate polynomial has no root in $\mathbb{F}_{2^m}^*$, and that this condition is also equivalent to $G_a$ being APN. Hence, writing $d=q^2+q+1$, at least \[ \frac{2^m+1-(d-1)(d-2)2^{m/2}-d}{d} \] values of $a$ yield APN permutations $G_a$. In the binary case $q=2$, we show that $a=1$ is good whenever $7\nmid m$, recovering the Li--Kaleyski family. For the second family \[ H_a(x,y,z)=(x^{q+1}+axy^q+yz^q,\; xy^q+z^{q+1},\; x^qz+y^{q+1}+ay^qz), \] we obtain the same root criterion and prove that its defining polynomial is root-equivalent to that of $G_a$. Thus the same parameters $a$ give APN permutations in both families. We also prove strong inequivalence results. First, $G_a$ (resp.\ $H_a$) is diagonally equivalent to $G_1$ (resp.\ $H_1$) if and only if $a^{q^2+q+1}=1$; moreover, for $m>4$, $m\neq 6$, and $7\nmid m$, diagonal non-equivalence implies CCZ non-equivalence by the monomial restriction theorem of Shi et al.\ (DCC, 2025). In particular, when $q=2$ and $7\nmid m$, every good $a\neq 1$ gives APN permutations CCZ-inequivalent to Li--Kaleyski. Second, for the same range of $m$, no $G_a$ is CCZ-equivalent to any $H_b$. Hence these constructions yield two genuinely new, mutually inequivalent families of APN permutations on $\mathbb{F}_{2^{3m}}$.

Problem

• APN permutations
• trivariate polynomials
• finite fields
• CCZ-inequivalence
• almost perfect nonlinear

Innovation

• APN permutations
• trivariate polynomials
• CCZ-inequivalence
• finite fields
• permutation criteria