A quantitative version of the Beurling-Helson theorem

Konyagin, Sergeĭ; Shkredov, Ilya D.

doi:10.1007/s10688-015-0093-0

Cited by 9 publications

(13 citation statements)

References 15 publications

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“…Our aim is to show a converse to this fact: if ϕ s is a rational function and if conditions 1 and 2 both hold then ϕ s must be affine. This result is closely related to the Beurling-Helson Theorem [7] (see [27,30] for related results in Z p ) and the work of Green and Konyagin [23] on the Fourier transform of balanced functions.…”

Section: Another Generalisation Would Be Fourier Analysis In Other Grmentioning

confidence: 62%

Untitled

Galbraith¹,

Laity²,

Shani³

2018

Chicago J. of Theoretical Comp. Sci.

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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.

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Section: Another Generalisation Would Be Fourier Analysis In Other Grmentioning

confidence: 62%

Untitled

Galbraith¹,

Laity²,

Shani³

2018

Chicago J. of Theoretical Comp. Sci.

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show abstract

“…, converges in M p (R d ) (recall that M p is a Banach space). At the same time from (12) it follows that this sequence converges to g in L ∞ (R d ). It remains to recall that · L ∞ = · M 2 ≤ · Mp .…”

Section: Proofs Of the Theoremsmentioning

confidence: 67%

“…He also conjectured ( [7], [8]) that the condition e inϕ A(T) = o(log |n|) already implies linearity of ϕ. The first result in this direction was obtained in [17]; further strengthening is obtained in [12], however the o(log |n|) -conjecture remains unproved.…”

Section: Remarks and Open Problemsmentioning

confidence: 99%