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Erschienen in:
2020 | OriginalPaper | Buchkapitel
The Randomized Slicer for CVPP: Sharper, Faster, Smaller, Batchier
verfasst von : Léo Ducas, Thijs Laarhoven, Wessel P. J. van Woerden
Erschienen in: Public-Key Cryptography – PKC 2020
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Abstract
Following the recent line of work on solving the closest vector problem with preprocessing (CVPP) using approximate Voronoi cells, we improve upon previous results in the following ways:
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We derive sharp asymptotic bounds on the success probability of the randomized slicer, by modelling the behaviour of the algorithm as a random walk on the coset of the lattice of the target vector. We thereby solve the open question left by Doulgerakis–Laarhoven–De Weger [PQCrypto 2019] and Laarhoven [MathCrypt 2019].
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We obtain better trade-offs for CVPP and its generalisations (strictly, in certain regimes), both with and without nearest neighbour searching, as a direct result of the above sharp bounds on the success probabilities.
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We show how to reduce the memory requirement of the slicer, and in particular the corresponding nearest neighbour data structures, using ideas similar to those proposed by Becker–Gama–Joux [Cryptology ePrint Archive, 2015]. Using \(2^{0.185d + o(d)}\) memory, we can solve a single CVPP instance in \(2^{0.264d + o(d)}\) time.
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We further improve on the per-instance time complexities in certain memory regimes, when we are given a sufficiently large batch of CVPP problem instances for the same lattice. Using \(2^{0.208d + o(d)}\) memory, we can heuristically solve CVPP instances in \(2^{0.234d + o(d)}\) amortized time, for batches of size at least \(2^{0.058d + o(d)}\).
Our random walk model for analysing arbitrary-step transition probabilities in complex step-wise algorithms may be of independent interest, both for deriving analytic bounds through convexity arguments, and for computing optimal paths numerically with a shortest path algorithm. As a side result we apply the same random walk model to graph-based nearest neighbour searching, where we improve upon results of Laarhoven [SOCG 2018] by deriving sharp bounds on the success probability of the corresponding greedy search procedure.