On sets of large Fourier transform under changes in domain

Laity, Joel; Shani, Barak

doi:10.1016/j.acha.2017.08.002

Cited by 3 publications

(6 citation statements)

References 6 publications

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“…In Section 4 we outline our recent work [28] on applying modulus switching to this subject (namely to re-cast a function on Z p to a function on Z 2 n for the nearest power of 2 to p). These ideas are very similar to the approach taken in Shor's (period-finding) algorithm [42].…”

Section: Roadmapmentioning

confidence: 99%

“…Specifically, significant coefficients are "preserved" even when the time domain representation of the function is extended (by "preserved" we mean that there is a clear relation between the significant coefficients of both functions). We refer to Laity and Shani [28] for the technical details.…”

Section: Simplifications: Modulus Switchingmentioning

confidence: 99%

“…Thus, if f (α) is a significant coefficient for f , then one expects that for β = Nα/p , the coefficient f (β ) is significant for f . The work of Laity and Shani [28] made these arguments to a precise theorem. Specifically, if f (x) is a concentrated function on Z p then f (x) is a concentrated function on Z 2 n .…”

Section: Simplifications: Modulus Switchingmentioning

confidence: 99%

“…Instead one can simply modulus-switch to a power of two and apply the SFT algorithm for the group Z 2 n . This is addressed in [28,Section 6.1]. Since the algorithms for Z 2 n have been optimised significantly (see [17,35]) we believe that the resulting algorithms will be no less efficient than applying the AGS algorithm directly.…”

Section: Simplifications: Modulus Switchingmentioning

confidence: 99%

“…[28, Lemma 6.2]). Let k ∈ N and 0 ≤ i < k. Define bit i :Z 2 k → {−1, 1} by bit i (x) = (−1) x i where x = ∑ k−1 j=0x j 2 j and x j ∈ {0, 1}.…”

mentioning

confidence: 99%

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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: Roadmapmentioning

confidence: 99%

Section: Simplifications: Modulus Switchingmentioning

confidence: 99%

Section: Simplifications: Modulus Switchingmentioning

confidence: 99%

Section: Simplifications: Modulus Switchingmentioning

confidence: 99%

“…[28, Lemma 6.2]). Let k ∈ N and 0 ≤ i < k. Define bit i :Z 2 k → {−1, 1} by bit i (x) = (−1) x i where x = ∑ k−1 j=0x j 2 j and x j ∈ {0, 1}.…”

mentioning

confidence: 99%

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Galbraith¹,

Laity²,

Shani³

2018

Chicago J. of Theoretical Comp. Sci.

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The security of all private-key bits in isogeny-based schemes

Shani

2020

Discrete Applied Mathematics

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

Galbraith,

Laity,

Shani

2016

Preprint

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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 "modulusswitching" 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.

show abstract

On sets of large Fourier transform under changes in domain

Cited by 3 publications

References 6 publications

Untitled

Untitled

The security of all private-key bits in isogeny-based schemes

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

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