Bent Functions in the Partial Spread Class Generated by Linear Recurring Sequences
This work addresses a specific problem in cryptography and coding theory by generating new bent functions, but it is incremental as it builds on existing partial spread classes.
The paper tackles constructing partial spread bent functions using linear recurring sequences, showing that such constructions exist only for polynomial degrees 1 or 2 and that many resulting functions for degree 2 are not equivalent to known classes, as verified by a computer search for n=8 variables.
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.