Uniformity properties of Construction C

Bollauf, Maiara F.; Zamir, Ram

doi:10.1109/isit.2016.7541552

Cited by 5 publications

(3 citation statements)

References 12 publications

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“…Observe that jwe C1,C2 (xz, xt, yz, yt) = W C1 (x, y)W C2 (z, t) Therefore, by comparing (21) with (22), we obtain swe C = swe C ⊥ . This completes the proof.…”

Section: Appendix a Proof Of Theoremmentioning

confidence: 99%

On the Secrecy Gain of Formally Unimodular Construction $\text{A}_4$ Lattices

Bollauf¹,

Lin²,

Ytrehus³

2022

Preprint

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Lattice coding for the Gaussian wiretap channel is considered, where the goal is to ensure reliable communication between two authorized parties while preventing an eavesdropper from learning the transmitted messages. Recently, a measure called secrecy gain was proposed as a design criterion to quantify the secrecy-goodness of the applied lattice code. In this paper, the theta series of the so-called formally unimodular lattices obtained by Construction A4 from codes over Z4 is derived, and we provide a universal approach to determine their secrecy gains. Initial results indicate that Construction A4 lattices can achieve a higher secrecy gain than the best-known formally unimodular lattices from the literature. Furthermore, a new code construction of formally self-dual Z4-linear codes is presented.

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“…Observe that jwe C1,C2 (xz, xt, yz, yt) = W C1 (x, y)W C2 (z, t) Therefore, by comparing (21) with (22), we obtain swe C = swe C ⊥ . This completes the proof.…”

Section: Appendix a Proof Of Theoremmentioning

confidence: 99%

On the Secrecy Gain of Formally Unimodular Construction $\text{A}_4$ Lattices

Bollauf¹,

Lin²,

Ytrehus³

2022

Preprint

Self Cite

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“…We also write derivations that shows how to construct more general geometrically uniform constellations. These results appear in [11,13] and they were inspired by [18, pp. 150-156] and [24].…”

Section: Construction Cmentioning

confidence: 97%

“…The proposal of Construction C ✍ and the study of its characteristics are motivated by a coding scheme called bit-interleaved coded modulation (BICM) [47], [58] and the particular study of latticeness was inspired by [32]. The results in this chapter appear in [12,13].…”

Section: Chapter 3 Construction C ✍mentioning

confidence: 99%

Lattices for communication problems

Bollauf¹

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ResumoO estudo de códigos no contexto de reticulados e outras constelações discretas para aplicações em comunicações é um tópico de interesse na área de teoria da informação. Certas construções de reticulados, como é o caso das Construções A e D, e de outras constelações que não são reticulados, como a Construção C, são utilizadas na decodificação multi-estágio e para quantização vetorial eficiente. Isso motiva a primeira contribuição deste trabalho, que consiste em investigar características da Construção C e propor uma nova construção baseada em códigos lineares, que chamamos de Construção C ✍ , analisando suas propriedades (condições para ser reticulado, uniformidade geométrica e distância mínima) e relação com a Construção C. Problemas na área de comunicações envolvendo reticulados podem ser computacionalmente difíceis à medida que a dimensão aumenta, como é o caso de, dado um vetor no espaço real n✁dimensional, determinar o ponto do reticulado mais próximo a este. A segunda contribuição deste trabalho é a análise desse problema restrito a um sistema distribuído, ou seja, onde o vetor a ser decodificado possui cada uma de suas coordenadas disponíveis em um nó distinto desse sistema. Nessa investigação, encontramos uma solução aproximada para duas e três dimensões considerando a partição de Babai e também estudamos o custo de comunicação envolvido.Palavras-chave: Teoria dos reticulados. Códigos corretores de erros (Teoria da Informação). Teoria da informação em matemática. Sistemas distribuídos.

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On the Secrecy Gain of Formally Unimodular Construction A₄ Lattices

Bollauf

Lin

Ytrehus

2022

2022 IEEE International Symposium on Information Theory (ISIT)

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The theta series of a lattice has been extensively studied in the literature and is closely related to a critical quantity widely used in the fields of physical layer security and cryptography, known as the flatness factor, or equivalently, the smoothing parameter of a lattice. Both fields raise the fundamental question of determining the (globally) maximum theta series over a particular set of volume-one lattices, namely, the stable lattices. In this work, we present a property of unimodular lattices, a subfamily of stable lattices, to verify that the integer lattice Z n achieves the largest possible value of theta series over the set of unimodular lattices. Such a result moves towards proving the conjecture recently stated by Regev and Stephens-Davidowitz: any unimodular lattice, except for those lattices isomorphic to Z n , has a strictly smaller theta series than that of Z n . Our techniques are mainly based on studying the ratio of the theta series of a unimodular lattice to the theta series of Z n , called the secrecy ratio. We relate the Regev and Stephens-Davidowitz conjecture with another conjecture for unimodular lattices, known in the literature as the Belfiore-Solé conjecture. The latter assumes that the secrecy ratio of any unimodular lattice has a symmetry point, which is exactly where the global minimum of the secrecy ratio is achieved.Our technical contributions are three-fold.

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Uniformity properties of Construction C

Cited by 5 publications

References 12 publications

On the Secrecy Gain of Formally Unimodular Construction $\text{A}_4$ Lattices

On the Secrecy Gain of Formally Unimodular Construction $\text{A}_4$ Lattices

Lattices for communication problems

On the Secrecy Gain of Formally Unimodular Construction A₄ Lattices

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Uniformity properties of Construction C

Cited by 5 publications

References 12 publications

On the Secrecy Gain of Formally Unimodular Construction $\text{A}_4$ Lattices

On the Secrecy Gain of Formally Unimodular Construction $\text{A}_4$ Lattices

Lattices for communication problems

On the Secrecy Gain of Formally Unimodular Construction A4 Lattices

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On the Secrecy Gain of Formally Unimodular Construction A₄ Lattices