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Designs, Codes and Cryptography

, Volume 73, Issue 2, pp 487–505 | Cite as

More differentially 6-uniform power functions

  • Céline BlondeauEmail author
  • Léo Perrin
Article

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.

Keywords

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

Mathematics Subject Classification

06E30 94A60 

Notes

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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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Information and Computer Science, School of ScienceAalto UniversityEspooFinland
  2. 2.University of LuxembourgLuxembourg cityLuxembourg

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