[Paper Review] $\mathcal{P}\mathcal{S}$ bent functions constructed from finite pre-quasifield spreads

Baofeng Wu|arXiv (Cornell University)|Aug 15, 2013

Coding theory and cryptography3 references3 citations

TL;DR

This paper proposes a novel method to explicitly construct new subclasses of $π\mathcal{S}$ bent functions using finite pre-quasifield spreads from finite geometry. By computing the compositional inverses of parametric permutation polynomials derived from the Dempwolff-Müller, Knuth, and Kantor pre-quasifields, the authors construct three new explicitly representable subclasses—$π\mathcal{S}_{\text{D-M}}$, $π\mathcal{S}_{\text{Knu}}$, and $π\mathcal{S}_{\text{Kan}}$—marking the first explicit constructions of $π\mathcal{S}$ bent functions beyond the $π\mathcal{S}_{\text{ap}}$ class in over 30 years.

ABSTRACT

Bent functions are of great importance in both mathematics and information science. The $\mathcal{P}\mathcal{S}$ class of bent functions was introduced by Dillon in 1974, but functions belonging to this class that can be explicitly represented are only the $\mathcal{P}\mathcal{S}_{ ext{ap}}$ functions, which were also constructed by Dillon after his introduction of the $\mathcal{P}\mathcal{S}$ class. In this paper, a technique of using finite pre-quasifield spread from finite geometry to construct $\mathcal{P}\mathcal{S}$ bent functions is proposed. The constructed functions are in similar styles with the $\mathcal{P}\mathcal{S}_{ ext{ap}}$ functions. To explicitly represent them in bivariate forms, the main task is to compute compositional inverses of certain parametric permutation polynomials over finite fields of characteristic 2. Concentrated on the Dempwolff-Müller pre-quasifield, the Knuth pre-semifield and the Kantor pre-semifield, three new subclasses of the $\mathcal{P}\mathcal{S}$ class are obtained. They are the only sub-classes that can be explicitly constructed more than 30 years after the $\mathcal{P}\mathcal{S}_{ ext{ap}}$ subclass was introduced.

Motivation & Objective

• To address the longstanding open problem of explicitly representing $π\mathcal{S}$ bent functions beyond the $π\mathcal{S}_{\text{ap}}$ subclass.
• To develop a general technique for constructing $π\mathcal{S}$ bent functions using finite pre-quasifield spreads from finite geometry.
• To compute compositional inverses of parametric permutation polynomials over characteristic-2 finite fields to enable explicit bivariate representations.
• To identify and characterize new subclasses of the $π\mathcal{S}$ class with explicit polynomial forms.
• To extend the known class of explicitly constructible $π\mathcal{S}$ bent functions after a 30-year stagnation in progress.

Proposed method

• Leverage finite pre-quasifield spreads as the underlying algebraic structure to generate partial spreads for bent function construction.
• Define parametric permutation polynomials based on the multiplication operation of a pre-quasifield, which serve as the core component of the construction.
• Compute the compositional inverse of these parametric permutation polynomials over $ζ_{2^{m}}$ to enable explicit bivariate representation of the resulting bent functions.
• Utilize known pre-quasifields—Dempwolff-Müller, Knuth pre-semifield, and Kantor pre-semifield—whose multiplication operations allow tractable computation of compositional inverses.
• Derive explicit formulas for the compositional inverse of the linearized polynomial $L_a(z) = az^2 + \text{tr}(az)$ in characteristic 2, depending on $\text{tr}(a)$.
• Apply the inverse polynomial to define the bent function via the $π\mathcal{S}$ construction principle, resulting in new explicitly representable subclasses.

Experimental results

Research questions

• RQ1 How can $π\mathcal{S}$ bent functions be explicitly represented beyond the $π\mathcal{S}_{\text{ap}}$ class, which has remained the only known explicit subclass for over 30 years?
• RQ2 Can finite pre-quasifield spreads be systematically used to construct new subclasses of $π\mathcal{S}$ bent functions with explicit bivariate polynomial forms?
• RQ3 What are the compositional inverses of parametric permutation polynomials derived from specific pre-quasifields such as Dempwolff-Müller, Knuth, and Kantor?
• RQ4 Are there new, previously unknown subclasses of the $π\mathcal{S}$ class that can be constructed using this method and explicitly described over finite fields of characteristic 2?
• RQ5 Can the proposed technique be generalized to other pre-quasifields or extended to odd characteristic fields?

Key findings

• Three new subclasses of the $π\mathcal{S}$ class—$π\mathcal{S}_{\text{D-M}}$, $π\mathcal{S}_{\text{Knu}}$, and $π\mathcal{S}_{\text{Kan}}$—are explicitly constructed using the Dempwolff-Müller, Knuth pre-semifield, and Kantor pre-semifield, respectively.
• The compositional inverse of the parametric permutation polynomial $L_a(z) = az^2 + \text{tr}(az)$ is explicitly computed for both cases $\text{tr}(a) = 0$ and $\text{tr}(a) = 1$, enabling the construction of bent functions in bivariate form.
• The inverse polynomial $L_a^{-1}(z)$ is derived as $\left(\frac{z}{a}\right)^{1/2} + \frac{\text{tr}(az)}{a^{1/2}}$ when $\text{tr}(a) = 0$, and as a more complex expression involving sums of powers and trace terms when $\text{tr}(a) = 1$, both valid modulo $x^{2^m} + x$.
• The bent function $F(x,y) = x \diamond y$ is explicitly defined via the inverse operation, with $\diamond$ defined as $L_x^{-1}(y)$, yielding a bivariate representation over $\mathbb{F}_{2^m}$.
• The construction method is general and can be extended to other pre-quasifields such as the generalized Kantor or Albert pre-quasifields to generate further new subclasses.
• The results resolve a 30-year gap in explicit constructions of $π\mathcal{S}$ bent functions, significantly expanding the known set of explicitly representable bent functions in this class.