Efficient (Ideal) Lattice Sieving Using Cross-Polytope LSH

Becker, Anja; Laarhoven, Thijs

doi:10.1007/978-3-319-31517-1_1

Cited by 25 publications

(18 citation statements)

References 56 publications

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“…The application of nearest neighbor searching techniques to sieving has previously been described in [BGJ15,BL15,Laa15,LdW15].…”

Section: θ(mentioning

confidence: 99%

“…A recent class of algorithms for solving SVP is lattice sieving [AKS01, NV08,MV10], which are algorithms running in time and space 2 O(n) . Heuristic sieving algorithms are currently the fastest algorithms known for solving SVP in high dimensions, and various recent work has shown how these algorithms can be sped up with NNS techniques [BGJ15,BL15,Laa15,LdW15]. The fastest heuristic algorithms to date for solving SVP in high dimensions are based on spherical LSH [AR15a,LdW15] and cross-polytope LSH [AIL + 15, BL15] and achieve time complexities of 2 0.298n+o(n) .…”

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confidence: 99%

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New directions in nearest neighbor searching with applications to lattice sieving

Becker¹,

Ducas²,

Gama³

et al. 2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

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202

203

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To solve the approximate nearest neighbor search problem (NNS) on the sphere, we propose a method using locality-sensitive filters (LSF), with the property that nearby vectors have a higher probability of surviving the same filter than vectors which are far apart. We instantiate the filters using spherical caps of height 1 − α, where a vector survives a filter if it is contained in the corresponding spherical cap, and where ideally each filter has an independent, uniformly random direction.For small α, these filters are very similar to the spherical locality-sensitive hash (LSH) family previously studied by Andoni et al. For larger α bounded away from 0, these filters potentially achieve a superior performance, provided we have access to an efficient oracle for finding relevant filters. Whereas existing LSH schemes are limited by a performance parameter of ρ ≥ 1/(2c 2 − 1) to solve approximate NNS with approximation factor c, with spherical LSF we potentially achieve smaller asymptotic values of ρ, depending on the density of the data set. For sparse data sets where the dimension is super-logarithmic in the size of the data set, we asymptotically obtain ρ = 1/(2c 2 − 1), while for a logarithmic dimensionality with density constant κ we obtain asymptotics of ρ ∼ 1/(4κc 2 ). To instantiate the filters and prove the existence of an efficient decoding oracle, we replace the independent filters by filters taken from certain structured random product codes. We show that the additional structure in these concatenation codes allows us to decode efficiently using techniques similar to lattice enumeration, and we can find the relevant filters with low overhead, while at the same time not significantly changing the collision *

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“…The application of nearest neighbor searching techniques to sieving has previously been described in [BGJ15,BL15,Laa15,LdW15].…”

Section: θ(mentioning

confidence: 99%

mentioning

confidence: 99%

New directions in nearest neighbor searching with applications to lattice sieving

Becker¹,

Ducas²,

Gama³

et al. 2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

Self Cite

202

203

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“…Although this work focuses on applying angular LSH to sieving, more generally this work could be considered the first to succeed in applying LSH to lattice algorithms. Various recent followup works have already further investigated the use of different LSH methods [7,8] and other nearest neighbor search methods [9,11,38] in the context of lattice sieving [11][12][13]30,37], and an open problem is whether other lattice algorithms (e.g. provable sieving algorithms, the Voronoi cell algorithm [39]) can benefit from related techniques as well.…”

Section: Introductionmentioning

confidence: 99%

Sieving for Shortest Vectors in Lattices Using Angular Locality-Sensitive Hashing

Laarhoven

2015

Lecture Notes in Computer Science

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100

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Abstract. By replacing the brute-force list search in sieving algorithms with Charikar's angular localitysensitive hashing (LSH) method, we get both theoretical and practical speedups for solving the shortest vector problem (SVP) on lattices. Combining angular LSH with a variant of Nguyen and Vidick's heuristic sieve algorithm, we obtain heuristic time and space complexities for solving SVP in dimension n of 2 0.3366n+o(n) and 2 0.2075n+o(n) respectively, while combining the same ideas with Micciancio and Voulgaris' GaussSieve algorithm leads to a practical algorithm with (conjectured) time and space complexities bounded by 2 0.3366n+o(n) , leading to the best complexities for solving SVP in high dimensions to date. Experiments show that in moderate dimensions the GaussSieve-based HashSieve algorithm already outperforms the GaussSieve, and the practical increase in the space complexity is smaller than the asymptotic bounds suggest, and can be further reduced with probing. Extrapolating to higher dimensions, we estimate that a fully optimized and parallelized implementation of the GaussSieve-based HashSieve algorithm might need a few core years to solve SVP in dimension 130 or even 140.

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“…The fastest (heuristic) algorithms for solving SVP in high dimensions are based on lattice sieving, originally described in [2] and later improved in e.g. [35,38,32,42,44,21,10,24,11,9]. Given a basis of a lattice L, lattice sieving attempts to solve SVP by first generating a long list L ⊂ L of relatively long lattice vectors, and then iteratively combining vectors in L to form shorter and shorter lattice vectors until a shortest lattice vector is found.…”

Section: Preliminariesmentioning

confidence: 99%

Faster Sieving for Shortest Lattice Vectors Using Spherical Locality-Sensitive Hashing

Laarhoven

Weger

2015

Lecture Notes in Computer Science

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To overcome the large memory requirement of classical lattice sieving algorithms for solving hard lattice problems, Bai-Laarhoven-Stehlé [ANTS 2016] studied tuple lattice sieving, where tuples instead of pairs of lattice vectors are combined to form shorter vectors. Herold-Kirshanova [PKC 2017] recently improved upon their results for arbitrary tuple sizes, for example showing that a triple sieve can solve the shortest vector problem (SVP) in dimension d in time 2 0.3717d+o(d) , using a technique similar to locality-sensitive hashing for finding nearest neighbors.In this work, we generalize the spherical locality-sensitive filters of Becker-Ducas-Gama-Laarhoven [SODA 2016] to obtain space-time tradeoffs for near neighbor searching on dense data sets, and we apply these techniques to tuple lattice sieving to obtain even better time complexities. For instance, our triple sieve heuristically solves SVP in time 2 0.3588d+o(d) . For practical sieves based on Micciancio-Voulgaris' GaussSieve [SODA 2010], this shows that a triple sieve uses less space and less time than the current best near-linear space double sieve.

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Efficient (Ideal) Lattice Sieving Using Cross-Polytope LSH

Cited by 25 publications

References 56 publications

New directions in nearest neighbor searching with applications to lattice sieving

New directions in nearest neighbor searching with applications to lattice sieving

Sieving for Shortest Vectors in Lattices Using Angular Locality-Sensitive Hashing

Faster Sieving for Shortest Lattice Vectors Using Spherical Locality-Sensitive Hashing

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