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

Daniele Micciancio, Panagiotis Voulgaris
2013 SIAM journal on computing (Print)  
We give deterministicÕ(2 2n+o(n) )-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  ... 
doi:10.1137/100811970 fatcat:t7siwuxhv5fhzf7o2w773jcfba

A deterministic single exponential time algorithm for most lattice problems based on voronoi cell computations

Daniele Micciancio, Panagiotis Voulgaris
2010 Proceedings of the 42nd ACM symposium on Theory of computing - STOC '10  
In the process, we also provide 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  ... 
doi:10.1145/1806689.1806739 dblp:conf/stoc/MicciancioV10 fatcat:343zt2prlrejdkeyh2aog5vs7q

On compact representations of Voronoi cells of lattices [article]

Christoph Hunkenschröder and Gina Reuland and Matthias Schymura
2019 arXiv   pre-print
In 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.  ... 
arXiv:1811.08532v2 fatcat:nujnuaqvyvbulm4uda2lnnfvaq

Finding shortest lattice vectors faster using quantum search

Thijs Laarhoven, Michele Mosca, Joop van de Pol
2015 Designs, Codes and Cryptography  
By applying 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] .  ... 
doi:10.1007/s10623-015-0067-5 pmid:32226228 pmcid:PMC7089694 fatcat:s44dtucyz5c3dfqhsfxntoktwi

A Canonical Form for Positive Definite Matrices [article]

Mathieu Dutour Sikirić, Anna Haensch, John Voight, Wessel P.J. van Woerden
2020 arXiv   pre-print
We exhibit an explicit, 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) .  ... 
arXiv:2004.14022v3 fatcat:ykmmoyfa6vcc5inglqql3so5fq

Algorithms for the Shortest and Closest Lattice Vector Problems [chapter]

Guillaume Hanrot, Xavier Pujol, Damien Stehlé
2011 Lecture Notes in Computer Science  
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  ...  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.  ... 
doi:10.1007/978-3-642-20901-7_10 fatcat:dxcj7djiybg2bjurx6rveejmsu

A canonical form for positive definite matrices

Mathieu Dutour Sikirić, Anna Haensch, John Voight, Wessel P.J. van Woerden
2020 The Open Book Series  
We exhibit an explicit, 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.  ... 
doi:10.2140/obs.2020.4.179 fatcat:y4sgt7dt6jec3irv2hdwdm2pwe

Short Paths on the Voronoi Graph and Closest Vector Problem with Preprocessing [chapter]

Daniel Dadush, Nicolas Bonifas
2014 Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms  
While 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  ... 
doi:10.1137/1.9781611973730.22 dblp:conf/soda/DadushB15 fatcat:mkcrmn6atnbhbd24lxrxve5nmu

Current Issues in Sampling-Based Motion Planning [chapter]

Stephen R. Lindemann, Steven M. LaValle
2005 Springer Tracts in Advanced Robotics  
planning 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.  ... 
doi:10.1007/11008941_5 fatcat:kzhor5pvyjd5vf45pn47apsebm

Solving the Shortest Vector Problem in Lattices Faster Using Quantum Search [chapter]

Thijs Laarhoven, Michele Mosca, Joop van de Pol
2013 Lecture Notes in Computer Science  
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.  ...  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.  ... 
doi:10.1007/978-3-642-38616-9_6 fatcat:dnsy5cudj5gbvkj7fs5s3xl6yq

Deterministic Construction of an Approximate M-Ellipsoid and its Applications to Derandomizing Lattice Algorithms [chapter]

Daniel Dadush, Santosh Vempala
2012 Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms  
We give 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.  ... 
doi:10.1137/1.9781611973099.114 dblp:conf/soda/DadushV12 fatcat:h6kaxw7dvvewvj5tzfjnlylpua

A sieve algorithm based on overlattices

Anja Becker, Nicolas Gama, Antoine Joux
2014 LMS Journal of Computation and Mathematics  
Moreover, the 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.  ... 
doi:10.1112/s1461157014000229 fatcat:dvouxjxbxjhhdirt5gbdqwj6re

Simulation Frameworks for Morphogenetic Problems

Simon Tanaka
2015 Computation  
Whereas the modelling of the signalling has been discussed and used in 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] .  ... 
doi:10.3390/computation3020197 fatcat:rkekfo4znbhrrkr63nzrmbufq4

A Parallel Evolutionary Search for Shortest Vector Problem

Gholam Reza Moghissi, ICT Department, Malek-Ashtar University of Technology, Tehran, Iran, Ali Payandeh
2019 International Journal of Information Technology and Computer Science  
points over the 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] .  ... 
doi:10.5815/ijitcs.2019.08.02 fatcat:cf7tygpwxbeyfnhyf2z7udltay

Coverage and connectivity issues in wireless sensor networks: A survey

Amitabha Ghosh, Sajal K. Das
2008 Pervasive and Mobile Computing  
Sensing coverage and network connectivity are two of the 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.  ... 
doi:10.1016/j.pmcj.2008.02.001 fatcat:s3xoyljajvc2vmo3me2oc757by
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