Improved Analysis of Kannan’s Shortest Lattice Vector Algorithm

Hanrot, Guillaume; Stehlé, Damien

doi:10.1007/978-3-540-74143-5_10

Cited by 102 publications

(130 citation statements)

References 26 publications

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“…For dimension 64 we changed α (the subtree dimension) from the usual n − 11 to α = n − 14, as this leads to lower timings in high dimensions. First, one can notice that both algorithms run much faster when using stronger pre-processing, a fact that was already mentioned in [HS07]. Second, we see that the speedup of the GPU version goes down to 13% in the best case (dimension 62).…”

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confidence: 60%

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Parallel Shortest Lattice Vector Enumeration on Graphics Cards

Hermans

Schneider

Buchmann

et al. 2010

Lecture Notes in Computer Science

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Abstract. In this paper we present an algorithm for parallel exhaustive search for short vectors in lattices. This algorithm can be applied to a wide range of parallel computing systems. To illustrate the algorithm, it was implemented on graphics cards using CUDA, a programming framework for NVIDIA graphics cards. We gain large speedups compared to previous serial CPU implementations. Our implementation is almost 5 times faster in high lattice dimensions.Exhaustive search is one of the main building blocks for lattice basis reduction in cryptanalysis. Our work results in an advance in practical lattice reduction.

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confidence: 60%

“…In [HS07], improved complexity bounds for Kannan's algorithm are presented. This paper also suggests some better preprocessing of lattice bases, i.e., the authors suggest to BKZ reduce a basis before running enumeration.…”

Section: Lattice Basis Reductionmentioning

confidence: 99%

Parallel Shortest Lattice Vector Enumeration on Graphics Cards

Hermans

Schneider

Buchmann

et al. 2010

Lecture Notes in Computer Science

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“…The first class, developed by Kannan [21] and refined by many others [19,17,35], is based on combining strong basis reduction with exhaustive enumeration inside Euclidean balls. The fastest current algorithm in this class solves SVP in O(n n/(2e) ) time while using poly(n) space [17].…”

Section: Introductionmentioning

confidence: 99%

Solving the Shortest Vector Problem in 2 ⁿ Time Using Discrete Gaussian Sampling

Aggarwal

Dadush

Regev

et al. 2015

Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing

123

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We give a randomized 2 n+o(n) -time and space algorithm for solving the Shortest Vector Problem (SVP) on n-dimensional Euclidean lattices. This improves on the previous fastest algorithm: the deterministic O(4 n )-time and O(2 n )-space algorithm of Micciancio and Voulgaris (STOC 2010, SIAM J. Comp. 2013.In fact, we give a conceptually simple algorithm that solves the (in our opinion, even more interesting) problem of discrete Gaussian sampling (DGS). More specifically, we show how to sample 2 n/2 vectors from the discrete Gaussian distribution at any parameter in 2 n+o(n) time and space. (Prior work only solved DGS for very large parameters.) Our SVP result then follows from a natural reduction from SVP to DGS.In addition, we give a more refined algorithm for DGS above the so-called smoothing parameter of the lattice, which can generate 2 n/2 discrete Gaussian samples in just 2 n/2+o(n) time and space. Among other things, this implies a 2 n/2+o(n) -time and space algorithm for 1.93-approximate decision SVP.

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“…As a consequence of Theorems 2 and 4, the cost of Kannan's algorithm [19] can be decreased from Poly(n, log B)·d d 2e (1+o (1)) (see [14]…”

Section: Then the Number Of Loop Iterations Is Lower Than The Number mentioning

confidence: 99%

Rigorous and Efficient Short Lattice Vectors Enumeration

Pujol

Stehlé

2008

Advances in Cryptology - ASIACRYPT 2008

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Abstract. The Kannan-Fincke-Pohst enumeration algorithm for the shortest and closest lattice vector problems is the keystone of all strong lattice reduction algorithms and their implementations. In the context of the fast developing lattice-based cryptography, the practical security estimates derive from floating-point implementations of these algorithms. However, these implementations behave very unexpectedly and make these security estimates debatable. Among others, numerical stability issues seem to occur and raise doubts on what is actually computed. We give here the first results on the numerical behavior of the floatingpoint enumeration algorithm. They provide a theoretical and practical framework for the use of floating-point numbers within strong reduction algorithms, which could lead to more sensible hardness estimates.

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Improved Analysis of Kannan’s Shortest Lattice Vector Algorithm

Cited by 102 publications

References 26 publications

Parallel Shortest Lattice Vector Enumeration on Graphics Cards

Parallel Shortest Lattice Vector Enumeration on Graphics Cards

Solving the Shortest Vector Problem in 2 ⁿ Time Using Discrete Gaussian Sampling

Rigorous and Efficient Short Lattice Vectors Enumeration

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Improved Analysis of Kannan’s Shortest Lattice Vector Algorithm

Cited by 102 publications

References 26 publications

Parallel Shortest Lattice Vector Enumeration on Graphics Cards

Parallel Shortest Lattice Vector Enumeration on Graphics Cards

Solving the Shortest Vector Problem in 2 n Time Using Discrete Gaussian Sampling

Rigorous and Efficient Short Lattice Vectors Enumeration

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Product

Resources

About

Solving the Shortest Vector Problem in 2 ⁿ Time Using Discrete Gaussian Sampling