Lower Bounds on Lattice Enumeration with Extreme Pruning

Aono, Yoshinori; Nguyêñ, Phong Q.; Seito, Takenobu; Shikata, Junji

doi:10.1007/978-3-319-96881-0_21

Cited by 13 publications

(10 citation statements)

References 30 publications

(72 reference statements)

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“…These examples suggest that the algorithm is best suited for distributions D, where the weight of the sum v+w of elements v, w ∼ D concentrates at d 2 . 3 Let us start with the reformulation of the theorem.…”

Section: Different Input Distributionsmentioning

confidence: 99%

A Faster Algorithm for Finding Closest Pairs in Hamming Metric

Esser¹,

Kübler²,

Zweydinger³

2021

Preprint

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We study the Closest Pair Problem in Hamming metric, which asks to find the pair with the smallest Hamming distance in a collection of binary vectors. We give a new randomized algorithm for the problem on uniformly random input outperforming previous approaches whenever the dimension of input points is small compared to the dataset size. For moderate to large dimensions, our algorithm matches the time complexity of the previously best-known locality sensitive hashing based algorithms. Technically our algorithm follows similar design principles as Dubiner (IEEE Trans. Inf. Theory 2010) and May-Ozerov (Eurocrypt 2015). Besides improving the time complexity in the aforementioned areas, we significantly simplify the analysis of these previous works. We give a modular analysis, which allows us to investigate the performance of the algorithm also on non-uniform input distributions. Furthermore, we give a proof of concept implementation of our algorithm which performs well in comparison to a quadratic search baseline. This is the first step towards answering an open question raised by May and Ozerov regarding the practicability of algorithms following these design principles.

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Section: Different Input Distributionsmentioning

confidence: 99%

A Faster Algorithm for Finding Closest Pairs in Hamming Metric

Esser¹,

Kübler²,

Zweydinger³

2021

Preprint

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“…Their idea is tempting not to enumerate all the tree nodes, by discarding certain branches. (See Aono et al 2018 for a lower bound of the time complexity of pruned enumeration.) However, it decreases the success probability to find the shortest non-zero lattice vector s. For instance, one might intuitively hope that π n/2 (s) 2 s 2 /2, which is more restrictive than the inequality…”

Section: Algorithm: the Basic Schnorr-euchner Enumeration Schnorr And Euchner (1994)mentioning

confidence: 99%

A Survey of Solving SVP Algorithms and Recent Strategies for Solving the SVP Challenge

Yasuda

2020

Mathematics for Industry

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Recently, lattice-based cryptography has received attention as a candidate of post-quantum cryptography (PQC). The essential security of lattice-based cryptography is based on the hardness of classical lattice problems such as the shortest vector problem (SVP) and the closest vector problem (CVP). A number of algorithms have been proposed for solving SVP exactly or approximately, and most of them are useful also for solving CVP. In this paper, we give a survey of typical algorithms for solving SVP from a mathematical point of view. We also present recent strategies for solving the Darmstadt SVP challenge in dimensions higher than 150.

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“…We also computed the upper bound of success probability and approximated cost by the methods in [22]. To optimize the bounding function, we used the method described in [11]. Fig.…”

Section: B2 Discrete Vs Cylinder Pruning: the Case Of Lwementioning

confidence: 99%

“…6], which states that sieving is more efficient than enumeration in dimension ≥ 250 for both classical and quantum computers. But this analysis is debatable: [8] estimates the cost of sieving by a lower bound (ignoring sub-exponential terms) and that of enumeration by an upper bound (either [18, Table 4] or [17, Table 5.2]), thereby ignoring the lower bound of [18] (see [11] for improved bounds).…”

Section: Introductionmentioning

confidence: 99%

“…1 illustrates the impact of our quantum enumeration on security estimates: the red and yellow curves show √ #bases * N where N is an upper bound cost, i.e., number of nodes of enumeration with extreme pruning with probability 1/#bases. The upper bounds for HKZ/Rankin bases are computed by the method of [11]. Here, we omitted the polynomial overhead factor because small factors in quantum sieve have also never been investigated.…”

Section: Introductionmentioning

confidence: 99%

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Quantum Lattice Enumeration and Tweaking Discrete Pruning

Aono

Nguyêñ

Shen

2018

Lecture Notes in Computer Science

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Enumeration is a fundamental lattice algorithm. We show how to speed up enumeration on a quantum computer, which affects the security estimates of several lattice-based submissions to NIST: if T is the number of operations of enumeration, our quantum enumeration runs in roughly √ T operations. This applies to the two most efficient forms of enumeration known in the extreme pruning setting: cylinder pruning but also discrete pruning introduced at Eurocrypt '17. Our results are based on recent quantum tree algorithms by Montanaro and Ambainis-Kokainis. The discrete pruning case requires a crucial tweak: we modify the preprocessing so that the running time can be rigorously proved to be essentially optimal, which was the main open problem in discrete pruning. We also introduce another tweak to solve the more general problem of finding close lattice vectors.

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Lower Bounds on Lattice Enumeration with Extreme Pruning

Cited by 13 publications

References 30 publications

A Faster Algorithm for Finding Closest Pairs in Hamming Metric

A Faster Algorithm for Finding Closest Pairs in Hamming Metric

A Survey of Solving SVP Algorithms and Recent Strategies for Solving the SVP Challenge

Quantum Lattice Enumeration and Tweaking Discrete Pruning

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