More differentially 6-uniform power functions

Abstract

In this paper, we study the differential spectra of differentially 6-uniform functions among the family of monomials \(\big \{x\mapsto x^{2^t-1},\; 1<t<n\big \}\) defined in \(\mathbb {F}_{2^{n}}\). We show that the functions \(x\mapsto x^{2^t-1}\) when \(t=\frac{n-1}{2},\; \frac{n+3}{2}\) with odd \(n\) have a differential spectrum similar to the one of the function \(x\mapsto x^7\) which belongs to the same family. We also study the functions \(x\mapsto x^{2^t-1}\) when \(t=\frac{kn+1}{3},\frac{(3-k)n+2}{3}\) with \(kn\equiv 2\,\mathrm{mod}\,3\) which are known to be differentially 6-uniform and show that their complete differential spectrum can be provided under an assumption related to a new formulation of the Kloosterman sum. To provide the differential spectra for these functions, a recent result of Helleseth and Kholosha regarding the number of roots of polynomials of the form \(x^{2^t+1}+x+a\) is widely used in this paper. A discussion regarding the non-linearity and the algebraic degree of the vectorial functions \(x\mapsto x^{2^t-1}\) is also proposed.

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Acknowledgments

The authors would like to thank the anonymous reviewers of WCC 2013 and DCC for helpful comments. The work of Léo Perrin was done during his Master’s Thesis at Aalto University.

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Correspondence to Céline Blondeau.

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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.

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Blondeau, C., Perrin, L. More differentially 6-uniform power functions. Des. Codes Cryptogr. 73, 487–505 (2014). https://doi.org/10.1007/s10623-014-9948-2

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Keywords

  • Differential uniformity
  • Differential spectrum
  • Monomial
  • Kloosterman sum
  • Roots of trinomial
  • \(x\mapsto x^{2^t-1}\)
  • Dickson polynomial

Mathematics Subject Classification

  • 06E30
  • 94A60