### Volume 2018

#### Article 6

Published by the Department of Computer Science, The University of Chicago.

#### Finding Significant Fourier Coefficients: Clarifications, Simplifications, Applications and Limitations

Steven D. Galbraith
Department of Mathematics
University of Auckland
Auckland, New Zealand
s.galbraith AT auckland DOT ac DOT nz

Joel Laity
University of Auckland
Auckland, New Zealand
jlai052 AT aucklanduni DOT ac DOT nz

and

Barak Shani
University of Auckland
Auckland, New Zealand
barak.shani AT auckland DOT ac DOT nz

December 20, 2018

#### Abstract

Ideas from Fourier analysis have been used in cryptography for the last three decades. Akavia, Goldwasser and Safra unified some of these ideas to give a complete algorithm that finds significant Fourier coefficients of functions on any finite abelian group. Their algorithm stimulated a lot of interest in the cryptography community, especially in the context of bit security''. This paper attempts to be a friendly and comprehensive guide to the tools and results in this field. The intended readership is cryptographers who have heard about these tools and seek an understanding of their mechanics and their usefulness and limitations. A compact overview of the algorithm is presented with emphasis on the ideas behind it. We show how these ideas can be extended to a modulus-switching'' variant of the algorithm. We survey some applications of this algorithm, and explain that several results should be taken in the right context. In particular, we point out that some of the most important bit security problems are still open. Our original contributions include: a discussion of the limitations on the usefulness of these tools; an answer to an open question about the modular inversion hidden number problem.

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Submitted July 25, 2016, revised March 5, 2018, and December 13, 2018, published December 20, 2018.

Volume 2018, Article 5 Volume 2019, Article 1
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