2022/08/13 by Maximilien Gadouleau, Luca Mariot, Stjepan Picek · 2 citations
Computer Science · Engineering · #Coding theory and cryptography #Cryptographic Implementations and Security #graph theory and CDMA systems
paper · pdf · doi:10.1007/s10623-022-01097-1
openalex created_date 2022/08/13 · openalex publication_date 2022/08/13 · openalex updated_date 2026/08/01
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 are 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 PS- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>PS</mml:mi> <mml:msup> <mml:mrow> <mml:mrow/> </mml:mrow> <mml:mo>-</mml:mo> </mml:msup> </mml:mrow> </mml:math> and PS+ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>PS</mml:mi> <mml:msup> <mml:mrow> <mml:mrow/> </mml:mrow> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> </mml:math> bent functions of n=8 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>8</mml:mn> </mml:mrow> </mml:math> variables generated by our construction and compute their 2-ranks. The results show that many of these functions defined by polynomials of degree d=2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> are not EA-equivalent to any Maiorana–McFarland or Desarguesian partial spread function.