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

TL;DR

This paper provides a comprehensive overview of algorithms for finding significant Fourier coefficients in functions over finite abelian groups, discussing their applications, limitations, and extensions within cryptography.

Contribution

It offers a clear explanation of the Fourier coefficient algorithm, extends it to modulus-switching, and addresses open questions and limitations in the field.

Findings

The Fourier coefficient algorithm can be extended to modulus-switching.

Several cryptographic applications of the algorithm are surveyed.

Limitations and open problems in bit security are highlighted.

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 manuscript 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.