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On the arithmetic Walsh coefficients of Boolean functions

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Abstract

We generalize to the arithmetic Walsh transform (AWT) some results which were previously known for the Walsh–Hadamard transform of Boolean functions. We first generalize the classical Poisson summation formula to the AWT. We then define a generalized notion of resilience with respect to an arbitrary statistical measure of Boolean functions. We apply the Poisson summation formula to obtain a condition equivalent to resilience for one such statistical measure. Last, we show that the AWT of a large class of Boolean functions can be expressed in terms of the AWT of a Boolean function of algebraic degree at most three in a larger number of variables.

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Notes

  1. Since we sometimes treat Boolean values as integers, it is helpful to indicate when we want to reduce an expression involving such values modulo 2.

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Acknowledgments

This material is based upon work supported by the National Science Foundation under Grant No. CCF-0514660. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

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Correspondence to Andrew Klapper.

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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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Carlet, C., Klapper, A. On the arithmetic Walsh coefficients of Boolean functions. Des. Codes Cryptogr. 73, 299–318 (2014). https://doi.org/10.1007/s10623-013-9915-3

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