Constructions of full diversity $D_n$-lattices for all $n$

Araujo, Robson R. de; Jorge, Grasiele C.

doi:10.1216/rmj.2020.50.1137

Cited by 3 publications

(2 citation statements)

References 15 publications

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“…Analogously to Proposition 4.5, similar results can be obtained for other classes of algebraic lattices such as D n -lattices [33,79,78,61]. In coding theory, algebraic lattice constructions have been used to obtain lattices with good properties for Gaussian and Rayleigh fading channels.…”

Section: And Ring Of Integersmentioning

confidence: 52%

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Theoretical and practical contributions to Ring-LWE cryptography

Nayara Ortiz

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Ao concluir este trabalho, agradeço a Deus, em primeiro lugar, pelas oportunidades e pessoas com quem compartilhei o período da pós-graduação.Aos meus pais, Aidê e Jorge, que me transmitiram os valores que sustentam a minha resiliência e busca pela independência pessoal através dos estudos, da perseverança e da fé. Ao meu marido, Douglas, que me proveu apoio ininterrupto, compartilhando com carinho e respeito do processo de constante aprimoramento. À minha irmã, Nayane, e ao meu cunhado, Rafael, por terem feito parte deste processo, me ouvindo e gerando momentos de descontração.Ao meu amigo e coautor, Pedro, que dividiu, desde o primeiro semestre, as dificuldades de cada estágio da nossa formação como pesquisadores no Brasil. Obrigada por ter sido meu olhar pragmático e pela atenção à minha saúde mental. Ao meu amigo e coautor, Robson, pela paciência em transmitir os seus conhecimentos de álgebra e reticulados. A sua colaboração foi essencial para a conclusão desta tese.Aos meus orientadores, Prof. Ricardo Dahab e Prof. Diego F. Aranha, pelas oportunidades, apoio e sabedoria. Aos colegas dos laboratórios LASCA e LMCAD no IC/Unicamp pela companhia no dia a dia e nas conferências, e pelas conversas e diversos cafés. Aos demais professores e funcionários do IC, pelo convívio e todo o apoio operacional.A todos que direta ou indiretamente contribuíram para a realização deste trabalho.

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Section: And Ring Of Integersmentioning

confidence: 52%

“…Details can be found in [89,92,57]. Another line for future work is the exploration of other classes of algebraic lattices for the Ring-LWE, such as D n -lattices [33,79,78,61].…”

Section: Discussionmentioning

confidence: 99%

Theoretical and practical contributions to Ring-LWE cryptography

Nayara Ortiz

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Algebraic lattices coming from $${\mathbb {Z}}$$-modules generalizing ramified prime ideals in odd prime degree cyclic number fields

de Andrade,

de Araujo,

da Nobrega Neto

et al. 2024

AAECC

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The Ring-LWE Problem in Lattice-Based Cryptography: The Case of Twisted Embeddings

Ortiz

Araujo

Aranha

et al. 2021

Entropy

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Several works have characterized weak instances of the Ring-LWE problem by exploring vulnerabilities arising from the use of algebraic structures. Although these weak instances are not addressed by worst-case hardness theorems, enabling other ring instantiations enlarges the scope of possible applications and favors the diversification of security assumptions. In this work, we extend the Ring-LWE problem in lattice-based cryptography to include algebraic lattices, realized through twisted embeddings. We define the class of problems Twisted Ring-LWE, which replaces the canonical embedding by an extended form. By doing so, we allow the Ring-LWE problem to be used over maximal real subfields of cyclotomic number fields. We prove that Twisted Ring-LWE is secure by providing a security reduction from Ring-LWE to Twisted Ring-LWE in both search and decision forms. It is also shown that the twist factor does not affect the asymptotic approximation factors in the worst-case to average-case reductions. Thus, Twisted Ring-LWE maintains the consolidated hardness guarantee of Ring-LWE and increases the existing scope of algebraic lattices that can be considered for cryptographic applications. Additionally, we expand on the results of Ducas and Durmus (Public-Key Cryptography, 2012) on spherical Gaussian distributions to the proposed class of lattices under certain restrictions. As a result, sampling from a spherical Gaussian distribution can be done directly in the respective number field while maintaining its format and standard deviation when seen in Zn via twisted embeddings.

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Constructions of full diversity $D_n$-lattices for all $n$

Cited by 3 publications

References 15 publications

Theoretical and practical contributions to Ring-LWE cryptography

Theoretical and practical contributions to Ring-LWE cryptography

Algebraic lattices coming from $${\mathbb {Z}}$$-modules generalizing ramified prime ideals in odd prime degree cyclic number fields

The Ring-LWE Problem in Lattice-Based Cryptography: The Case of Twisted Embeddings

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