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

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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.

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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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  1. Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenberger Str. 69, 4040, Linz, Austria

    Richard Hofer & Arne Winterhof

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  1. Richard Hofer
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  2. Arne Winterhof
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Correspondence to Arne Winterhof.

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Hofer, R., Winterhof, A. r-th order nonlinearity, correlation measure and least significant bit of the discrete logarithm. Cryptogr. Commun. 11, 993–997 (2019). https://doi.org/10.1007/s12095-018-0344-z

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  • DOI: https://doi.org/10.1007/s12095-018-0344-z

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