Designs, Codes and Cryptography

, Volume 73, Issue 2, pp 299–318 | Cite as

On the arithmetic Walsh coefficients of Boolean functions



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.


Arithmetic Walsh transform Boolean function Poisson summation formula Resilience 

Mathematics Subject Classification

11E95 94A55 94A60 94C10 



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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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.LAGA, Department of MathematicsUniversity of Paris 8 (and Paris 13 and CNRS)Saint–Denis Cedex 02France
  2. 2.Department of Computer ScienceUniversity of KentuckyLexingtonUSA

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