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

Gadouleau, Maximilien and Mariot, Luca and Picek, Stjepan (2022) 'Bent functions in the partial spread class generated by linear recurring sequences.', Designs, Codes and Cryptography .

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.

Item Type:Article
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Available under License - Creative Commons Attribution 4.0.
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Status:Peer-reviewed
Publisher Web site:https://doi.org/10.1007/s10623-022-01097-1
Publisher statement:This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Date accepted:28 July 2022
Date deposited:13 September 2022
Date of first online publication:13 August 2022
Date first made open access:13 September 2022

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