Slide Reduction, Revisited—Filling the Gaps in SVP Approximation

“…The most basic approach for solving both SVP and CVP exactly is enumeration, which requires n O (n) -time and poly-space (see, e.g., [13]). The classical approach for approximating SVP is known as lattice reduction, which is to find good reduced bases consisting of reasonably short and almost orthogonal vectors: it was revived with the celebrated LLL algorithm [23] and continued with blockwise algorithms [1,10,29,32,33]. Both enumeration and lattice reduction are still very active in recent years (see, e.g., [1,27,29]).…”

Section: Introductionmentioning

confidence: 99%

“…. , b n ) (see, e.g., [1,8,10,12,23,29,32]). More precisely, some classical lattice reduction algorithms follow the paradigm below (see [24,App. A] for the proof).…”

Section: Introductionmentioning

confidence: 99%

“…It can be checked that the LLL-reduction [23], BKZ-reduction [32,33], semi k-reduction [32] and slide reduction (achieving the best time/quality trade-off among known lattice reduction algorithms) [1,10] follow the paradigm. This means that at least in theory, the local ratios ∥b * ik+1 ∥/∥b * ik+k +1 ∥ might play a more important role on the output quality of lattice reduction than the local Hermite factors…”

Section: Introductionmentioning

confidence: 99%

“…In mathematics, SVP is tightly related to the study of Hermite's constant γ n , namely γ n = max λ 1 (L) 2 over all n-rank lattices L of unit co-volume. This fundamental parameter in the geometry of numbers is typically used in the study of both enumeration algorithms and lattice reduction algorithms, e.g., to measure the running time [13,17] and output quality [1,10,12,29]…”

Section: Introductionmentioning

confidence: 99%

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On the Smallest Ratio Problem of Lattice Bases

2021

Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation

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Let (b 1 , . . . , b n ) be a lattice basis with Gram-Schmidt orthogonalization (b * 1 , . . . , b * n ), the ratios ∥b 1 ∥/∥b * i ∥ for i = 1, . . . , n do arise in the analysis of many lattice algorithms and are somehow related to their performances. In this paper, we study the problem of minimizing the ratio ∥b 1 ∥/∥b * n ∥ over all bases (b 1 , . . . , b n ) of a given n-rank lattice. We first prove that there exists a basis (b 1 , . . . , b n ) for any nrank lattice L such that ∥b 1 ∥ = min v∈L\{0} ∥v∥, ∥b 1 ∥/∥b * i ∥ ≤ i and ∥b i ∥/∥b * i ∥ ≤ i 1.5 for 1 ≤ i ≤ n. This leads us to introduce a new NP-hard computational problem, namely the smallest ratio problem (SRP): given an n-rank lattice L, find a basis) is a basis of L} and a new lattice constant µ n = max µ n (L) over all n-rank lattices L: both the minimum and maximum are justified. Some properties of µ n (L) and µ n are investigated. We also present an exact algorithm and an approximation algorithm for SRP. This is the first sound study of SRP. Our work is a tiny step towards solving an open problem proposed by Dadush-Regev-Stephens-Davidowitz (CCC '14) for tackling the closest vector problem with preprocessing, i.e., whether there exists a basis (b 1 , . . . , b n ) for any n-rank lattice s.t. max 1≤i ≤j ≤n ∥b * i ∥/b * j ∥ ≤ poly(n).

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