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r-th order nonlinearity, correlation measure and least significant bit of the discrete logarithm

  • Richard Hofer
  • Arne WinterhofEmail author
Article
  • 15 Downloads

Abstract

Each finite binary sequence (sh) is associated with a Boolean function B. The correlation measure of order k and the r-th order nonlinearity are figures of merit for the unpredictability of (sh) and B, respectively. We estimate the r-th order nonlinearity of B in terms of the correlation measure of order 2r of (sh). We apply our result to Boolean functions associated with the Legendre sequence, that is, the binary sequence describing the least significant bit of the discrete logarithms in the finite field \(\mathbb {F}_{p}\) of p elements, where p > 2 is a prime.

Keywords

r-th order nonlinearity Correlation measure of order k Discrete logarithm Legendre sequence Boolean functions Pseudorandom sequences 

Mathematics Subject Classification (2010)

06E30 11K45 94C10 

Notes

Acknowledgements

The authors are partially supported by the Austrian Science Fund FWF Projects F 5511-N26 which is part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications” and P 30405-N32. They also like to thank the anonymous referees for their very helpful comments.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Johann Radon Institute for Computational and Applied MathematicsAustrian Academy of SciencesLinzAustria

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