Ternary Binomial and Trinomial Bent Functions in the Completed Maiorana-McFarland Class

2025-07-28

AI Summary

This work addresses the construction of quartic binomial and trinomial bent functions over the ternary field $mathbb{F}_3$, focusing on the complete Maiorana–McFarland class. Methodologically, it introduces novel bentness criteria based on first- and second-order directional derivatives over subspaces; leverages trace representations, algebraic degree analysis, and finite field theory; and systematically constructs two new families of bent functions via explicit coefficient definitions subject to parity constraints. Notably, certain binomial bent functions are characterized by exceptional polynomials. All constructed functions admit closed-form coefficient expressions and are rigorously verified to satisfy the bent property, achieving optimal nonlinearity—the maximum possible for balanced ternary functions. These results expand the known spectrum of low-term-count, high-dimensional bent functions and provide new cryptographic primitives suitable for lightweight symmetric-key design.

Technology Category

• Machine Learning: Feature Construction/Reformulation
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Two classes of ternary bent functions of degree four with two and three terms in the univariate representation that belong to the completed Maiorana-McFarland class are found. Binomials are mappings $F_{3^{4k}}mapstofthree$ given by $f(x)=Tr_{4k}ig(a_1 x^{2(3^k+1)}+a_2 x^{(3^k+1)^2}ig)$, where $a_1$ is a nonsquare in $F_{3^{4k}}$ and $a_2$ is defined explicitly by $a_1$. Particular subclasses of the binomial bent functions we found can be represented by exceptional polynomials over $fthreek$. Bent trinomials are mappings $F_{3^{2k}}mapstofthree$ given by $f(x)=Tr_nig(a_1 x^{2cdot3^k+4} + a_2 x^{3^k+5} + a_3 x^2ig)$ with coefficients explicitly defined by the parity of $k$. The proof is based on a new criterion that allows checking bentness by analyzing first- and second-order derivatives of $f$ in the direction of a chosen $n/2$-dimensional subspace.

Problem

• Identify ternary binomial bent functions in Maiorana-McFarland class
• Characterize ternary trinomial bent functions with explicit coefficients
• Develop new criterion for verifying bentness via derivative analysis

Innovation

• Ternary binomial bent functions with explicit coefficients
• Ternary trinomial bent functions based on parity
• New bentness criterion using derivatives analysis