Geometry of numbers

Gruber, Peter M.

doi:10.1007/978-3-0348-5765-9_7

Cited by 48 publications

(46 citation statements)

References 212 publications

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“…We may use classical terminology to reexpress the import of this equation in terms taken from the geometry of numbers [GL87]. Using this terminology, we may write the content of (2.2) as follows: In passing, it is pertinent to mention that many other criteria are in use to characterise efficient cubature rules, and that some, like the enhanced degree in (1.2) above, are based on exact evaluation of specified sets of Fourier coefficients.…”

Section: Underlying Theorymentioning

confidence: 99%

Three- and four-dimensional 𝐾-optimal lattice rules of moderate trigonometric degree

Cools

Lyness

2001

Math. Comp.

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Abstract. A systematic search for optimal lattice rules of specified trigonometric degree d over the hypercube [0, 1) s has been undertaken. The search is restricted to a population K(s, δ) of lattice rules Q(Λ). This includes those where the dual lattice Λ ⊥ may be generated by s points h for each of which |h| = δ = d + 1. The underlying theory, which suggests that such a restriction might be helpful, is presented. The general character of the search is described, and, for s = 3, d ≤ 29 and s = 4, d ≤ 23, a list of K-optimal rules is given. It is not known whether these are also optimal rules in the general sense; this matter is discussed.

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Section: Underlying Theorymentioning

confidence: 99%

Three- and four-dimensional 𝐾-optimal lattice rules of moderate trigonometric degree

Cools

Lyness

2001

Math. Comp.

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“…In this paper, we will call lattice any integer lattice, that is, any subgroup of (Z n , +) for some n. Background on lattice theory can be found in several textbooks, such as [16,35]. For lattice-based cryptanalysis, we refer to [18].…”

Section: Lattice-based Methods For Noisy Polynomial Interpolationmentioning

confidence: 99%

Noisy Polynomial Interpolation and Noisy Chinese Remaindering

Bleichenbacher¹,

Nguyêñ

2000

Advances in Cryptology — EUROCRYPT 2000

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Abstract. The noisy polynomial interpolation problem is a new intractability assumption introduced last year in oblivious polynomial evaluation. It also appeared independently in password identification schemes, due to its connection with secret sharing schemes based on Lagrange's polynomial interpolation. This paper presents new algorithms to solve the noisy polynomial interpolation problem. In particular, we prove a reduction from noisy polynomial interpolation to the lattice shortest vector problem, when the parameters satisfy a certain condition that we make explicit. Standard lattice reduction techniques appear to solve many instances of the problem. It follows that noisy polynomial interpolation is much easier than expected. We therefore suggest simple modifications to several cryptographic schemes recently proposed, in order to change the intractability assumption. We also discuss analogous methods for the related noisy Chinese remaindering problem arising from the well-known analogy between polynomials and integers.

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“…Here we collect several known results that we use about lattices, which can be found in [8,10,7]. Let {b 1 , .…”

Section: Latticesmentioning

confidence: 99%

“…We will use the following point-counting variant of Minkowski's 'first theorem', which is due to Blichfeldt and van der Corput(see [8]). …”

Section: Latticesmentioning

confidence: 99%

Lattice-Based Threshold-Changeability for Standard Shamir Secret-Sharing Schemes

Steinfeld

Wang

Pieprzyk

2004

Advances in Cryptology - ASIACRYPT 2004

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Abstract. We consider the problem of increasing the threshold parameter of a secret-sharing scheme after the setup (share distribution) phase, without further communication between the dealer and the shareholders. Previous solutions to this problem require one to start off with a non-standard scheme designed specifically for this purpose, or to have communication between shareholders. In contrast, we show how to increase the threshold parameter of the standard Shamir secret-sharing scheme without communication between the shareholders. Our technique can thus be applied to existing Shamir schemes even if they were set up without consideration to future threshold increases.Our method is a new positive cryptographic application for lattice reduction algorithms, inspired by recent work on lattice-based list decoding of Reed-Solomon codes with noise bounded in the Lee norm. We use fundamental results from the theory of lattices (Geometry of Numbers) to prove quantitative statements about the information-theoretic security of our construction. These lattice-based security proof techniques may be of independent interest.

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Geometry of numbers

Cited by 48 publications

References 212 publications

Three- and four-dimensional 𝐾-optimal lattice rules of moderate trigonometric degree

Three- and four-dimensional 𝐾-optimal lattice rules of moderate trigonometric degree

Noisy Polynomial Interpolation and Noisy Chinese Remaindering

Lattice-Based Threshold-Changeability for Standard Shamir Secret-Sharing Schemes

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