Geometric and Analytic Number Theory

Hlawka, Edmund; Schoißengeier, Johannes; Taschner, Rudolf

doi:10.1007/978-3-642-75306-0

Cited by 53 publications

(22 citation statements)

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“…APPENDIX A BLICHFELDT'S PRINCIPLE AND MINKOWSKI-HLAWKA THEOREM Theorem 9 (Blichfeldt's Principle [16]): Let f be a Riemann integrable function with bounded support. If Λ n is a lattice with fundamental region P n then…”

Section: Discussionmentioning

confidence: 99%

Joint Physical Layer Coding and Network Coding for Bidirectional Relaying

Wilson

Narayanan

Pfister

et al. 2010

IEEE Trans. Inform. Theory

344

346

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We consider the problem of two transmitters wishing to exchange information through a relay in the middle. The channels between the transmitters and the relay are assumed to be synchronized, average power constrained additive white Gaussian noise channels with a real input with signal-to-noise ratio (SNR) of snr. An upper bound on the capacity is 1 2 log(1 + snr) bits per transmitter per use of the medium-access phase and broadcast phase of the bi-directional relay channel. We show that using lattice codes and lattice decoding, we can obtain a rate of 1 2 log( 1 2 + snr) bits per transmitter, which is essentially optimal at high SNR. The main idea is to decode the sum of the codewords modulo a lattice at the relay followed by a broadcast phase which performs Slepian-Wolf coding with structured codes. For asymptotically low SNR, joint decoding of the two transmissions at the relay (MAC channel) is shown to be optimal. We also show that if the two transmitters use identical lattices with minimum angle decoding, we can achieve the same rate of 1 2 log( 1 2 + snr). The proposed scheme can be thought of as a joint physical layer, network layer code which outperforms other recently proposed analog network coding schemes.

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Section: Discussionmentioning

confidence: 99%

Joint Physical Layer Coding and Network Coding for Bidirectional Relaying

Wilson

Narayanan

Pfister

et al. 2010

IEEE Trans. Inform. Theory

344

346

View full text Add to dashboard Cite

show abstract

“…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%

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

Steinfeld

Pieprzyk

Wang

2007

IEEE Trans. Inform. Theory

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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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“…; a N in the design variable space during Stage 1 of the PSI method. Other uniformly distributed sequences and nets (Faure, 1982;Halton, 1960;Hammersley, 1960;Hlawka and Taschner, 1991;Kuipers and Niederreiter, 1974;Niederreiter, 1990;Statnikov and Matusov, 1996) can be also successfully used in the PSI method. However, prior to solving a specific problem, one cannot say with certainty which uniformly distributed sequences are the most suitable.…”

Section: Number Generators For Systematic Search In the Design Variabmentioning

confidence: 97%

Visualization approaches for the prototype improvement problem

Statnikov

Anıl

Bordetsky

et al. 2008

Multi Criteria Decision Anal

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One of the basic engineering optimization problems is improvement of the prototype. This problem is often encountered by industrial and academic organizations that produce and design various objects (e.g. motor vehicles, machine tools, ships, and aircrafts). This paper presents an approach for improving the prototype by constructing the feasible and Pareto sets while performing multicriteria analysis. We introduce visualization methods that facilitate construction of the feasible and Pareto sets. Using these techniques developed on the basis of Parameter Space Investigation method, an expert can correctly state and solve the problem under consideration in a series of dialogues with the computer. Finally, we present a case study of improving the ship prototype.

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Geometric and Analytic Number Theory

Cited by 53 publications

References 0 publications

Joint Physical Layer Coding and Network Coding for Bidirectional Relaying

Joint Physical Layer Coding and Network Coding for Bidirectional Relaying

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

Visualization approaches for the prototype improvement problem

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