Bent functions in the partial spread class generated by linear recurring sequences

Gadouleau, Maximilien; Mariot, Luca; Picek, Stjepan

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− and 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 d=2 are not EA-equivalent to any Maiorana–McFarland or Desarguesian partial spread function.

Citation

Gadouleau, M., Mariot, L., & Picek, S. (2023). Bent functions in the partial spread class generated by linear recurring sequences. Designs, Codes and Cryptography, 91(1), 63-82. https://doi.org/10.1007/s10623-022-01097-1

• Journal Article Type: Article
• Acceptance Date: Jul 28, 2022
• Online Publication Date: Aug 13, 2022
• Publication Date: 2023-01
• Deposit Date: Sep 13, 2022
• Publicly Available Date: Mar 15, 2023
• Journal: Designs, Codes and Cryptography
• Print ISSN: 0925-1022
• Electronic ISSN: 1573-7586
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 91
• Issue: 1
• Pages: 63-82
• DOI: https://doi.org/10.1007/s10623-022-01097-1
• Public URL: https://durham-repository.worktribe.com/output/1194634