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A Deterministic Single Exponential Time Algorithm for Most Lattice Problems Based on Voronoi Cell Computations

2013
*
SIAM journal on computing (Print)
*

We give deterministicÕ(2 2n+o(n) )-

doi:10.1137/100811970
fatcat:t7siwuxhv5fhzf7o2w773jcfba
*time**algorithms*to solve all the*most*important*computational**problems**on*point*lattices*in NP, including the Shortest Vector*Problem*(SVP), Closest Vector*Problem*(CVP ... In the process, we also give*algorithms**for*several other*lattice**problems*, including*computing*the kissing number of*a**lattice*, and*computing*the set of all*Voronoi*relevant vectors. ... We give*a**single**exponential**time**algorithm*that*on*input*a**lattice*Λ,*a*list V (of size at*most*2 n+1 ) containing all*Voronoi*relevant vectors of Λ, and*a*target point t,*computes**a**lattice*point closest ...##
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A deterministic single exponential time algorithm for most lattice problems based on voronoi cell computations

2010
*
Proceedings of the 42nd ACM symposium on Theory of computing - STOC '10
*

In the process, we also provide

doi:10.1145/1806689.1806739
dblp:conf/stoc/MicciancioV10
fatcat:343zt2prlrejdkeyh2aog5vs7q
*deterministic**single**exponential**time**algorithms**for*various other classic*computational**problems*in*lattices*, like*computing*the kissing number, and*computing*the list ... We give deterministicÕ(2 2n )-*time*andÕ(2 n )-space*algorithms*to solve all the*most*important*computational**problems**on*point*lattices*in NP, including the Shortest Vector*Problem*(SVP), Closest Vector ... It would be nice to extend our*algorithm*to yield*a**single**exponential**time*solution to the covering radius*problem*, or equivalently, the*problem*of*computing*the diameter of the*Voronoi**cell*of*a**lattice*...##
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On compact representations of Voronoi cells of lattices
[article]

2019
*
arXiv
*
pre-print

In

arXiv:1811.08532v2
fatcat:nujnuaqvyvbulm4uda2lnnfvaq
*a*seminal work, Micciancio Voulgaris (2010) described*a**deterministic**single*-*exponential**time**algorithm**for*the Closest Vector*Problem*(CVP)*on**lattices*. ... It is*based**on*the*computation*of the*Voronoi**cell*of the given*lattice*and thus may need*exponential*space as well. ... We thank Daniel Dadush and Frank Vallentin*for*helpful remarks and suggestions. ...##
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Finding shortest lattice vectors faster using quantum search

2015
*
Designs, Codes and Cryptography
*

By applying

doi:10.1007/s10623-015-0067-5
pmid:32226228
pmcid:PMC7089694
fatcat:s44dtucyz5c3dfqhsfxntoktwi
*a*quantum search*algorithm*to various heuristic and provable sieve*algorithms*from the literature, we obtain improved asymptotic quantum results*for*solving the shortest vector*problem**on**lattices*... These quantum complexities will be an important guide*for*the selection of parameters*for*post-quantum cryptosystems*based**on*the hardness of the shortest vector*problem*. ...*Computing*the*Voronoi**cell*In 2010, Micciancio and Voulgaris presented*a**deterministic**algorithm**for*solving SVP*based**on*constructing the*Voronoi**cell*of the*lattice*[58] . ...##
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A Canonical Form for Positive Definite Matrices
[article]

2020
*
arXiv
*
pre-print

We exhibit an explicit,

arXiv:2004.14022v3
fatcat:ykmmoyfa6vcc5inglqql3so5fq
*deterministic**algorithm**for*finding*a*canonical form*for**a*positive definite matrix under unimodular integral transformations. ... The*algorithm*runs in*a*number of arithmetic operations that is*exponential*in the dimension n, but it is practical and more efficient than canonical forms*based**on*Minkowski reduction. ... In*a*more theoretical direction, Haviv-Regev [13] proposed*algorithms**based**on*the Shortest Vector*Problem*and an isolation lemma*for*these purposes as well, with*a**time*complexity of n O(n) . ...##
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Algorithms for the Shortest and Closest Lattice Vector Problems
[chapter]

2011
*
Lecture Notes in Computer Science
*

We recall the three main families of

doi:10.1007/978-3-642-20901-7_10
fatcat:dxcj7djiybg2bjurx6rveejmsu
*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 ... We present the state of the art solvers of the Shortest and Closest*Lattice*Vector*Problems*in the Euclidean norm. ... Acknowledgments We thank Panagiotis Voulgaris*for*very helpful discussions*on*the*Voronoi*-*based*SVP/CVP solver. We also thank the anonymous reviewer*for*her/his comments. ...##
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A canonical form for positive definite matrices

2020
*
The Open Book Series
*

We exhibit an explicit,

doi:10.2140/obs.2020.4.179
fatcat:y4sgt7dt6jec3irv2hdwdm2pwe
*deterministic**algorithm**for*finding*a*canonical form*for**a*positive definite matrix under unimodular integral transformations. ... The*algorithm*runs in*a*number of arithmetic operations that is*exponential*in the dimension n, but it is practical and more efficient than canonical forms*based**on*Minkowski reduction. ... advances were made during*a*visit to the Simons Institute*for*the Theory of*Computing*. ...##
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Short Paths on the Voronoi Graph and Closest Vector Problem with Preprocessing
[chapter]

2014
*
Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms
*

While

doi:10.1137/1.9781611973730.22
dblp:conf/soda/DadushB15
fatcat:mkcrmn6atnbhbd24lxrxve5nmu
*Voronoi**cell**based*CVPP*algorithms*require*exponential**time*and space*on*general*lattices*, it was recently shown in [23] that*a*variant of [28] can be implemented in polynomial*time**for**lattices*... Improving*on*the*Voronoi**cell**based*techniques of [28, 24] , we give*a*Las Vegas O(2 n ) expected*time*and space*algorithm**for*CVPP (the preprocessing version of the Closest Vector*Problem*, CVP). ... To solve CVP*on**a*target t, the idea of*Voronoi**cell**based*methods is to*compute**a*short path*on*the*Voronoi*graph G from*a*"close enough" starting vertex x ∈ L to t (usually,*a*rounded version of t under ...##
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Current Issues in Sampling-Based Motion Planning
[chapter]

2005
*
Springer Tracts in Advanced Robotics
*

planning

doi:10.1007/11008941_5
fatcat:kzhor5pvyjd5vf45pn47apsebm
*algorithms*, capable of solving challenging*problems*with many degrees of freedom. ... We then discuss*a*variety of important issues*for*sampling-*based*motion planning, including uniform and regular sampling, topological issues, and search philosophies. ... Acknowledgement We thank Pekka Isto*for*bringing Glavina's work to our attention. We are grateful*for*the funding provided in part by NSF awards 9875304, 0118146, and 0208891. ...##
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Solving the Shortest Vector Problem in Lattices Faster Using Quantum Search
[chapter]

2013
*
Lecture Notes in Computer Science
*

These quantum complexities will be an important guide

doi:10.1007/978-3-642-38616-9_6
fatcat:dnsy5cudj5gbvkj7fs5s3xl6yq
*for*the selection of parameters*for*post-quantum cryptosystems*based**on*the hardness of the shortest vector*problem*. ...*for*solving the shortest vector*problem*. ... This report is partly*a*result of fruitful discussions at the Lorentz Center Workshop*on*Post-Quantum Cryptography and Quantum*Algorithms*, Nov. 5-9, Leiden, The Netherlands. ...##
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Deterministic Construction of an Approximate M-Ellipsoid and its Applications to Derandomizing Lattice Algorithms
[chapter]

2012
*
Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms
*

We give

doi:10.1137/1.9781611973099.114
dblp:conf/soda/DadushV12
fatcat:h6kaxw7dvvewvj5tzfjnlylpua
*a**deterministic*O(log n) n -*time*and space*algorithm**for*the Shortest Vector*Problem*(SVP) of*a**lattice*under any norm, improving*on*the previous best*deterministic*n O(n) -*time**algorithms**for*... This approaches the 2 O(n) -*time*and space complexity of the randomized sieve*based*SVP*algorithms*(Arvind and Joglekar, FSTTCS 2008), first introduced by Ajtai, Kumar and Sivakumar (STOC 2001)*for*2 - ... We are deeply grateful to Grigoris Paouris and Chris Peikert*for*illuminating discussions, and to Gilles Pisier*for*his book*on*convex bodies. ...##
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A sieve algorithm based on overlattices

2014
*
LMS Journal of Computation and Mathematics
*

Moreover, the

doi:10.1112/s1461157014000229
fatcat:dvouxjxbxjhhdirt5gbdqwj6re
*algorithm*is straightforward to parallelize*on**most**computer*architectures. ... AbstractIn this paper, we present*a*heuristic*algorithm**for*solving exact, as well as approximate, shortest vector and closest vector*problems**on**lattices*. ... Probabilistic sieving*algorithms*, as well as the*deterministic**Voronoi*-*cell**algorithm*are simply*exponential*in*time*and memory. ...##
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Simulation Frameworks for Morphogenetic Problems

2015
*
Computation
*

Whereas the modelling of the signalling has been discussed and used in

doi:10.3390/computation3020197
fatcat:rkekfo4znbhrrkr63nzrmbufq4
*a*multitude of works, the realistic modelling of the tissue has only started*on**a*larger scale in the last decade. ... The*most*widely used frameworks and modelling markup languages and standards are presented. ... However, as*for*many*lattice*-*based**algorithms*, high-performance-*computing*and parallelization techniques can be applied efficiently [74] . ...##
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A Parallel Evolutionary Search for Shortest Vector Problem

2019
*
International Journal of Information Technology and Computer Science
*

points over the

doi:10.5815/ijitcs.2019.08.02
fatcat:cf7tygpwxbeyfnhyf2z7udltay
*lattice*vectors) and*a**single*main enumeration. ... The hardness assumption of approximate shortest vector*problem*(SVP) within the polynomial factor in polynomial*time*reduced to the security of many*lattice*-*based*cryptographic primitives, so solving this ...*Voronoi**cell**algorithm*is theoretically*one*of the fastest*deterministic**algorithm*with*time*complexity of ( )*for*solving SVP which needed*exponential*space of ( ) [13] . ...##
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Coverage and connectivity issues in wireless sensor networks: A survey

2008
*
Pervasive and Mobile Computing
*

Sensing coverage and network connectivity are two of the

doi:10.1016/j.pmcj.2008.02.001
fatcat:s3xoyljajvc2vmo3me2oc757by
*most*fundamental*problems*in wireless sensor networks. ... Therefore, maximizing coverage as well as maintaining network connectivity using the resource constrained nodes is*a*non-trivial*problem*. ... Acknowledgments The authors would like to thank Professor Bhaskar Krishnamachari of the Autonomous Networks Research Group at USC*for*his valuable inputs. ...
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