Covering cubes and the closest vector problem

Eisenbrand, Friedrich; Hähnle, Nicolai; Niemeier, Martin

doi:10.1145/1998196.1998264

Cited by 25 publications

(24 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The previously discussed solvers are used for the Euclidean norm. Recent results for general norms are presented in [7,13,14,12]. The paper of Hanrot, Pujol, Stehlé [21] gives a good survey and deep analysis about SVP and CVP solvers.…”

Section: The Shortest Lattice Vector Problem (Svp) and The Closest Lamentioning

confidence: 99%

On the complexity of quasiconvex integer minimization problem

et al. 2018

View full text Add to dashboard Cite

In this paper, we consider the class of quasiconvex functions and its proper subclass of conic functions. The integer minimization problem of these functions is considered, assuming that the optimized function is defined by the comparison oracle. We will show that there is no a polynomial algorithm on log R to optimize quasiconvex functions in the ball of radius R using only the comparison oracle. On the other hand, if the optimized function is conic, then we show that there is a polynomial on log R algorithm (the dimension is fixed). We also present an exponential on the dimension lower bound for the oracle complexity of the conic function integer optimization problem. Additionally, we give examples of known problems that can be polynomially reduced to the minimization problem of functions in our classes.

show abstract

Section: The Shortest Lattice Vector Problem (Svp) and The Closest Lamentioning

confidence: 99%

On the complexity of quasiconvex integer minimization problem

et al. 2018

View full text Add to dashboard Cite

show abstract

“…6]: For a lattice L ⊆ R m and a target t ∈ R m , we define L ′ as the lattice spanned by (L × {0}) ∪ (t, γ); provided that γ is large enough, if (u, x) ∈ R m × R is a shortest vector of L ′ \(R n × {0}), then x = ±γ and the solution to CVP(L, t) is ±u. More recently, Eisenbrand et al [18] built upon [9] to decrease the cost dependence in ε.…”

Section: Solving Cvpmentioning

confidence: 99%

Algorithms for the Shortest and Closest Lattice Vector Problems

Hanrot

Pujol

Stehlé

2011

Lecture Notes in Computer Science

101

View full text Add to dashboard Cite

Abstract. We present the state of the art solvers of the Shortest and Closest Lattice Vector Problems in the Euclidean norm. We recall the three main families of algorithms for these problems, namely the algorithm by Micciancio and Voulgaris based on the Voronoi cell [STOC'10], the Monte-Carlo algorithms derived from the Ajtai, Kumar and Sivakumar algorithm [STOC'01] and the enumeration algorithms originally elaborated by Kannan [STOC'83] and Fincke and Pohst [EUROCAL'83]. We concentrate on the theoretical worst-case complexity bounds, but also consider some practical facets of these algorithms.

show abstract

“…Starting with the seminal work of [1], algorithms for solving these problems either exactly or approximately have been studied intensely. These algorithms have found applications in various fields, such as factoring polynomials over rationals [1], integer programming [2][3][4][5], cryptanalysis [6][7][8], checking the solvability by radicals [9], and solving low-density subsetsum problems [10]. More recently, many powerful cryptographic primitives have been constructed whose security is based on the worst-case hardness of these or related lattice problems [11][12][13][14][15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

Faster Provable Sieving Algorithms for the Shortest Vector Problem and the Closest Vector Problem on Lattices in ℓp Norm

Mukhopadhyay

2021

Algorithms

View full text Add to dashboard Cite

In this work, we give provable sieving algorithms for the Shortest Vector Problem (SVP) and the Closest Vector Problem (CVP) on lattices in ℓp norm (1≤p≤∞). The running time we obtain is better than existing provable sieving algorithms. We give a new linear sieving procedure that works for all ℓp norm (1≤p≤∞). The main idea is to divide the space into hypercubes such that each vector can be mapped efficiently to a sub-region. We achieve a time complexity of 22.751n+o(n), which is much less than the 23.849n+o(n) complexity of the previous best algorithm. We also introduce a mixed sieving procedure, where a point is mapped to a hypercube within a ball and then a quadratic sieve is performed within each hypercube. This improves the running time, especially in the ℓ2 norm, where we achieve a time complexity of 22.25n+o(n), while the List Sieve Birthday algorithm has a running time of 22.465n+o(n). We adopt our sieving techniques to approximation algorithms for SVP and CVP in ℓp norm (1≤p≤∞) and show that our algorithm has a running time of 22.001n+o(n), while previous algorithms have a time complexity of 23.169n+o(n).

show abstract

Covering cubes and the closest vector problem

Cited by 25 publications

References 12 publications

On the complexity of quasiconvex integer minimization problem

On the complexity of quasiconvex integer minimization problem

Algorithms for the Shortest and Closest Lattice Vector Problems

Faster Provable Sieving Algorithms for the Shortest Vector Problem and the Closest Vector Problem on Lattices in ℓp Norm

Contact Info

Product

Resources

About