Bent Functions in the Partial Spread Class Generated by Linear Recurring Sequences

TL;DR

This paper introduces a new method for constructing partial spread bent functions using linear recurring sequences, characterizes the parameters for such constructions, and analyzes their properties and equivalences.

Contribution

It provides a novel LRS-based construction of partial spread bent functions, characterizes when such constructions are possible, and compares their properties to existing classes.

Findings

Construction coincides with Desarguesian partial spread for degree 1

Many degree 2 functions are not EA-equivalent to known classes

Successful enumeration and analysis of bent functions for 8 variables

Abstract

We present a construction of partial spread bent functions using subspaces generated by linear recurring sequences (LRS). We first show that the kernels of the linear mappings defined by two LRS have a trivial intersection if and only if their feedback polynomials are relatively prime. Then, we characterize the appropriate parameters for a family of pairwise coprime polynomials to generate a partial spread required for the support of a bent function, showing that such families exist if and only if the degrees of the underlying polynomials is either $1$ or $2$. We then count the resulting sets of polynomials and prove that for degree $1$, our LRS construction coincides with the Desarguesian partial spread. Finally, we perform a computer search of all $\mathcal{PS}^-$ and $\mathcal{PS}^+$ bent functions of $n=8$ variables generated by our construction and compute their 2-ranks. The results show that many of these functions defined by polynomials of degree $b=2$ are not EA-equivalent to any Maiorana-McFarland or Desarguesian partial spread function.