[Paper Review] New constructions of quaternary bent functions

Baofeng Wu, Dongdai Lin|arXiv (Cornell University)|Sep 1, 2013

Coding theory and cryptography13 references3 citations

TL;DR

This paper presents a new construction of quaternary bent functions using quadratic forms over Galois rings of characteristic 4, generalizing prior work by Li, Tang, and Helleseth. By characterizing specific coefficient sets (QBF-sets) via gcd conditions on polynomials, the authors derive new classes of quaternary bent functions and, through connections to binary bent and semi-bent functions, obtain new classes of quadratic binary bent and semi-bent functions in polynomial form.

ABSTRACT

In this paper, a new construction of quaternary bent functions from quaternary quadratic forms over Galois rings of characteristic 4 is proposed. Based on this construction, several new classes of quaternary bent functions are obtained, and as a consequence, several new classes of quadratic binary bent and semi-bent functions in polynomial forms are derived. This work generalizes the recent work of N. Li, X. Tang and T. Helleseth.

Motivation & Objective

• To generalize recent constructions of quaternary bent functions based on trace forms over Galois rings of characteristic 4.
• To identify sufficient conditions under which a class of quaternary functions defined via trace forms are bent, using polynomial gcd conditions.
• To derive new classes of quadratic bent and semi-bent binary functions in polynomial form from the constructed quaternary bent functions.
• To extend the known families of bent and semi-bent functions by leveraging structural properties of Galois rings and Teichmüller sets.

Proposed method

• Define quaternary bent functions as trace forms over Galois rings GR(4,n), specifically Q(x) = Tr₁ⁿ(αx + 2∑cᵢβx^{1+2^{eki}}) for x in the Teichmüller set.
• Use the isomorphism between the Teichmüller set and 𝔽₂ⁿ to relate quaternary functions to binary functions via 2-adic expansion of the trace function.
• Establish a necessary and sufficient condition for bentness: gcd(c(xᵏ), xᵐ−1) = 1, where c(x) is a binary polynomial derived from the coefficients {cᵢ}.
• Characterize QBF-sets (coefficient sets satisfying the gcd condition) using algebraic number theory and properties of cyclotomic polynomials.
• Apply the connection between quaternary bent functions and binary bent/semi-bent functions (via Stănică et al.) to derive polynomial-form binary functions.
• Verify that the resulting binary functions are quadratic and bent or semi-bent depending on the parity of n.

Experimental results

Research questions

• RQ1 Under what conditions on the coefficients {cᵢ} is the quaternary function Q(x) = Tr₁ⁿ(αx + 2∑cᵢβx^{1+2^{eki}}) bent over GR(4,n)?
• RQ2 How can the gcd condition gcd(c(xᵏ), xᵐ−1) = 1 be characterized algebraically for specific coefficient sets?
• RQ3 What new classes of quadratic bent and semi-bent binary functions in polynomial form can be derived from the constructed quaternary bent functions?
• RQ4 In what way does this construction generalize the recent work of Li, Tang, and Helleseth on quaternary bent functions?
• RQ5 How do the structural properties of Galois rings and Teichmüller sets facilitate the construction of new bent functions?

Key findings

• The paper establishes a new sufficient condition for quaternary bent functions: gcd(c(xᵏ), xᵐ−1) = 1, where c(x) is a binary polynomial constructed from the coefficients {cᵢ}.
• Several new classes of quaternary bent functions are explicitly constructed by identifying QBF-sets satisfying the gcd condition, generalizing prior constructions.
• For even n, the derived binary function f_Q(x) is a quadratic bent function in polynomial form; for odd n, it is a quadratic semi-bent function.
• The resulting binary functions are of the form f_Q(x) = p(αx) + ∑cᵢ tr₁ⁿ(βx^{1+2^{eki}}), with p(x) defined by trace sums over specific exponents.
• The construction yields new families of quadratic bent and semi-bent functions that are not covered by earlier methods based on quadratic forms over finite fields.
• The method provides a systematic way to generate new bent and semi-bent functions by selecting coefficient sets that satisfy the gcd condition, with explicit examples derived from Propositions 1–10.