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Finding Shortest Lattice Vectors in the Presence of Gaps

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Book cover Topics in Cryptology –- CT-RSA 2015 (CT-RSA 2015)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 9048))

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Abstract

The \(\lambda _i\)-gap \(\lambda _i/\lambda _1\) among the successive minima of a lattice especially its \(\lambda _2\)-gap often provides useful information for analyzing the security of lattice-based cryptographic schemes. In this paper, we mainly study the efficiency of shortest vector problem (SVP) algorithms for lattices with \(\lambda _i\)-gap. First, we prove new upper bounds for the packing density of this type of lattices. Based on these results, we discuss the efficiency of the ListSieve-Birthday algorithm proposed by Pujol and Stehlé for SVP, and obtain the conclusion that the complexity will decrease obviously as the \(\lambda _i\)-gap increases. Particularly, ListSieve-Birthday becomes faster than the current best deterministic (Voronoi cell-based) algorithm for SVP, as long as \(\lambda _2\)-gap is larger than 1.78. When \(\lambda _2\)-gap is up to 28, the time complexity is \(2^{0.9992n+o(n)}\), and the coefficient factor of \(n\) is approximately to 0.802 if \(\lambda _2\)-gap is large enough. Moreover, we provide an SVP approximation algorithm modified by the ListSieve-Birthday algorithm. This algorithm terminates sieve process earlier and relaxes the birthday search, and hence decreases the time complexity significantly.

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Authors and Affiliations

  1. State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, China

    Wei Wei

  2. Institute for Advanced Study, Tsinghua University, Beijing, China

    Wei Wei & Xiaoyun Wang

  3. Beijing Research Institute of Telemetry, Beijing, China

    Mingjie Liu

  4. Key Lab of Cryptologic Technology and Information Security, Ministry of Education, Shandong University, Jinan, China

    Xiaoyun Wang

Authors
  1. Wei Wei
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  2. Mingjie Liu
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  3. Xiaoyun Wang
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Correspondence to Xiaoyun Wang .

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Editors and Affiliations

  1. Aalto University School of Science, Espoo, Finland

    Kaisa Nyberg

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Wei, W., Liu, M., Wang, X. (2015). Finding Shortest Lattice Vectors in the Presence of Gaps. In: Nyberg, K. (eds) Topics in Cryptology –- CT-RSA 2015. CT-RSA 2015. Lecture Notes in Computer Science(), vol 9048. Springer, Cham. https://doi.org/10.1007/978-3-319-16715-2_13

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  • DOI: https://doi.org/10.1007/978-3-319-16715-2_13

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  • Online ISBN: 978-3-319-16715-2

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