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Improved analysis of the reduction from BDD to uSVP.

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Názov: Improved analysis of the reduction from BDD to uSVP.
Autori: Zhao, Chunhuan1 (AUTHOR) zhao-ch13@mails.tsinghua.edu.cn, Zheng, Zhongxiang2 (AUTHOR)
Zdroj: Information Processing Letters. Aug2019, Vol. 148, p28-32. 5p.
Predmety: LATTICE constants, MATHEMATICAL equivalence, EMBEDDINGS (Mathematics)
Abstrakt: In ICALP 2016, Bai, Stehlé and Wen showed that for polynomially bounded parameter γ , the Bounded Distance Decoding (BDD) problem with parameter 1 / (2 γ) can be probabilistically reduced to the unique Shortest Vector Problem (uSVP) with parameter γ. Their reduction is based on the lattice sparsification and embedding techniques. In this paper, we improve their analysis for specific values of the parameter and prove that BDD 1 / (2 γ) can be reduced to uSVP γ 1 with a slightly larger gap. The improvement follows from a careful study of the radius of a sphere which contains at most polynomial number of vectors, then a proper choice of the parameter in the embedding lattice is used to obtain a unique lattice with a larger gap. Our result shows that the equivalence between BDD 1 / (2 γ) and uSVP γ does not hold under the assumption that uSVP γ does not reduce to uSVP γ ′ for any two constants γ < γ ′. • It is proven that BDD can be reduced to an easier uSVP for certain parameters. • Our result shows that the equivalence between BDD 1 / (2 γ) and uSVP γ does not hold under certain assumption. • The maximum radius of sphere which at most poly(n) vectors are packed is investigated in a general way. [ABSTRACT FROM AUTHOR]
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Abstrakt:In ICALP 2016, Bai, Stehlé and Wen showed that for polynomially bounded parameter γ , the Bounded Distance Decoding (BDD) problem with parameter 1 / (2 γ) can be probabilistically reduced to the unique Shortest Vector Problem (uSVP) with parameter γ. Their reduction is based on the lattice sparsification and embedding techniques. In this paper, we improve their analysis for specific values of the parameter and prove that BDD 1 / (2 γ) can be reduced to uSVP γ 1 with a slightly larger gap. The improvement follows from a careful study of the radius of a sphere which contains at most polynomial number of vectors, then a proper choice of the parameter in the embedding lattice is used to obtain a unique lattice with a larger gap. Our result shows that the equivalence between BDD 1 / (2 γ) and uSVP γ does not hold under the assumption that uSVP γ does not reduce to uSVP γ ′ for any two constants γ < γ ′. • It is proven that BDD can be reduced to an easier uSVP for certain parameters. • Our result shows that the equivalence between BDD 1 / (2 γ) and uSVP γ does not hold under certain assumption. • The maximum radius of sphere which at most poly(n) vectors are packed is investigated in a general way. [ABSTRACT FROM AUTHOR]
ISSN:00200190
DOI:10.1016/j.ipl.2019.04.004