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Improved algorithms for the Shortest Vector Problem and the Closest Vector Problem in the infinity norm [article]

Divesh Aggarwal, Priyanka Mukhopadhyay
2018 arXiv   pre-print
As in [AKS02,BN09], we also extend this algorithm to obtain significantly faster algorithms for approximate versions of the shortest vector problem and the closest vector problem (CVP) in the ℓ_∞ norm.  ...  We give a new sieving procedure that runs in time linear in N, thereby significantly improving the running time of the algorithm for SVP in the ℓ_∞ norm.  ...  The focus of this work is to study the complexity of the closest vector problem and the shortest vector problem in the ℓ ∞ norm. 1.1 Prior Work. Algorithms in the Euclidean Norm.  ... 
arXiv:1801.02358v2 fatcat:3yanvzdlg5adtiw3iu4aaaklha

Improved Algorithms for the Shortest Vector Problem and the Closest Vector Problem in the Infinity Norm

Divesh Aggarwal, Priyanka Mukhopadhyay, Michael Wagner
2018 International Symposium on Algorithms and Computation  
As in [6, 11] , we also extend this algorithm to obtain significantly faster algorithms for approximate versions of the shortest vector problem and the closest vector problem (CVP) in thenorm.  ...  Ajtai, Kumar and Sivakumar [5] gave the first 2 O(n) algorithm for solving the Shortest Vector Problem (SVP) on n-dimensional Euclidean lattices.  ...  Acknowledgements We thank the anonymous referees who helped improve the draft of this paper.  ... 
doi:10.4230/lipics.isaac.2018.35 dblp:conf/isaac/AggarwalM18 fatcat:23h37gqehjemvhuufnnqqxjfi4

Hardness of Approximate Nearest Neighbor Search under L-infinity [article]

Young Kun Ko, Min Jae Song
2020 arXiv   pre-print
This shows a conditional separation between ANN under the ℓ_1/ ℓ_2 norm and the ℓ_∞ norm since there are sublinear time algorithms achieving better than 3-approximation for the ℓ_1 and ℓ_2 norm.  ...  Our first reduction shows that hardness of a special case of the Shortest Vector Problem (SVP), which captures many provably hard instances of SVP, implies a lower bound for ANN with polynomial preprocessing  ...  For finite ℓ p norms, NPhardness of exact SVP p , the Shortest Vector Problem in the ℓ p norm, was shown by [Ajt98] using randomized reductions.  ... 
arXiv:2011.06135v1 fatcat:n4ywphahqnfqbf35gri33pj774

Efficient lattice-based signature scheme

Thomas Plantard, Willy Susilo, Khin Than Win, Qiong Huang
2008 International Journal of Applied Cryptography  
In Crypto 1997, Goldreich, Goldwasser and Halevi (GGH) proposed a lattice analogue of McEliece public key cryptosystem, in which security is related to the hardness of approximating the Closest Vector  ...  In this article, we present a novel method of reducing a vector under the l -norm and propose a digital signature scheme based on it.  ...  This article is an extended version of Plantard, Susilo and Win (2008) with a new and tighter proof of complexity.  ... 
doi:10.1504/ijact.2008.021085 fatcat:k36c4emernfzvamksv6qamkiee

Covering Cubes and the Closest Vector Problem [article]

Friedrich Eisenbrand, Nicolai Hähnle, Martin Niemeier
2010 arXiv   pre-print
We provide the currently fastest randomized (1+epsilon)-approximation algorithm for the closest vector problem in the infinity norm.  ...  Thereby, we obtain a method to boost any 2-approximation algorithm for closest-vector in the infinity norm to a (1+epsilon)-approximation algorithm that has the desired running time.  ...  The approximation algorithm We now present our (1 + ε)-approximation algorithm for the closest vector problem in the ℓ ∞ norm.  ... 
arXiv:1012.2289v1 fatcat:n5ypnle5m5chtdnv2ut3cd2cx4

Parallel Improved Schnorr-Euchner Enumeration SE++ for the CVP and SVP

Fabio Correia, Artur Mariano, Alberto Proenca, Christian Bischof, Erik Agrell
2016 2016 24th Euromicro International Conference on Parallel, Distributed, and Network-Based Processing (PDP)  
The Closest Vector Problem (CVP) and the Shortest Vector Problem (SVP) are prime problems in lattice-based cryptanalysis, since they underpin the security of many lattice-based cryptosystems.  ...  Despite the importance of these problems, there are only a few CVP-solvers publicly available, and their scalability was never studied.  ...  ACKNOWLEDGEMENTS We thankÖzgür Dagdelen and Michael Schneider, the authors of [10] , for providing us with their implementation.  ... 
doi:10.1109/pdp.2016.95 dblp:conf/pdp/CorreiaMPBA16 fatcat:jmljkbhgffhjzoec2vqipmuvba

Broadcast Attacks against Lattice-Based Cryptosystems [chapter]

Thomas Plantard, Willy Susilo
2009 Lecture Notes in Computer Science  
These problems are used in lattice based cryptography and to model attack on knapsack cryptosystems. In this work, we are able to present some attacks against both lattice and knapsack cryptosystems.  ...  Håstad's attack was demonstrated on the RSA algorithm, where low exponents are used.  ...  Find the closest vector v of c in L(B).  ... 
doi:10.1007/978-3-642-01957-9_28 fatcat:ht7nrvkhlnb7zb4p3nlwjbe75u

Estimating the Effectiveness of Lattice Attacks [article]

Kotaro Abe, Makoto Ikeda
2021 IACR Cryptology ePrint Archive  
We also investigated in detail the reasons for the failure of the attacks and proposed a model to estimate the feasibility of lattice attacks using the BKZ algorithm.  ...  Currently, the BKZ algorithm is frequently used as a lattice reduction algorithm for lattice attacks, and there are many reports on the conditions for successful attacks.  ...  Acknowledgement We would like to thank Editage (www.editage.com) for English language editing.  ... 
dblp:journals/iacr/AbeI21 fatcat:7kt5pc3qlvgk5nzngp7jfo3gcq

Universal lattice decoding: a review and some recent results

Wai Ho Mow
2004 2004 IEEE International Conference on Communications (IEEE Cat. No.04CH37577)  
The principle of universal lattice decoding can trace its roots back to the theory developed for solving the shortest/closest lattice vector problem.  ...  In addition, it will be shown that with some lattice preprocessing steps, impressive performance improvement and/or complexity reduction of some wellknown detectors (e.g.  ...  The principle of universal lattice decoding can trace its roots back to the theory and algorithms developed for solving the shortest/closest lattice vector problem for integer programming and cryptoanalysis  ... 
doi:10.1109/icc.2004.1313048 dblp:conf/icc/Mow04 fatcat:p2ijccxnobh6deyegeyawxyrue

Maximum Distance Sub-Lattice Problem [article]

Shashank K Mehta, Mahesh Sreekumar Rajasree, Rajendra Kumar
2018 arXiv   pre-print
We prove that MDSP is isomorphic to a well-known problem called closest vector problem (CVP). We give an exact and a heuristic algorithm for MDSP.  ...  In this paper, we define a problem on lattices called the Maximum Distance Sub-lattice Problem (MDSP). The decision version of this problem is shown to be in NP.  ...  Acknowledgements We would like to thank Naman Verma for helping us develop a C-library which contains algorithms related to this paper.  ... 
arXiv:1811.03019v1 fatcat:cdztcq67ivdxpedibjgfecfpyu

Algorithms to Compute Minimum Cycle Basis in Directed Graphs

Telikepalli Kavitha, Kurt Mehlhorn
2007 Theory of Computing Systems  
This paper presents anÕ(m 4 n) algorithm, which is the first polynomial time algorithm for computing a minimum cycle basis in G. We then improve it to anÕ(m 4 ) algorithm.  ...  In this problem a {−1, 0, 1} incidence vector is associated with each cycle and the vector space over Q generated by these vectors is the cycle space of G.  ...  The cost of updating (similarly, normalizing) the vector T j (for i + 1 ≤ j ≤ d) in iteration i depends on how large are the infinity norms of T j and T i . . . .  ... 
doi:10.1007/s00224-006-1319-6 fatcat:4bzqoqwao5erpm7bfxdac3bwcm

Adapting Density Attacks to Low-Weight Knapsacks [chapter]

Phong Q. Nguyễn, Jacques Stern
2005 Lecture Notes in Computer Science  
This approach is actually a bit misleading: we show that low-weight knapsacks do not prevent efficient reductions to lattice problems like the shortest vector problem, they even make reductions more likely  ...  On the one hand, it is well-known that this problem is NP-hard, and accordingly it is considered to be hard in the worst case.  ...  quantity is the norm of the expected shortest vector, while for CVP, it is the distance between the target vector and the lattice.  ... 
doi:10.1007/11593447_3 fatcat:us7noxwnofg6lpq3qgwjxmzk6m

Parallel Improved Schnorr-Euchner Enumeration SE++ on Shared and Distributed Memory Systems, With and Without Extreme Pruning

Fábio Correia, Artur Mariano, Alberto Proença, Christian H. Bischof, Erik Agrell
2016 Journal of Wireless Mobile Networks, Ubiquitous Computing, and Dependable Applications  
In this context, two lattice problems are especially relevant: the Shortest Vector Problem (SVP) and the Closest Vector Problem (CVP).  ...  The security of lattice-based cryptography relies on the hardness of problems based on lattices, such as the Shortest Vector Problem (SVP) and the Closest Vector Problem (CVP).  ...  Acknowledgments We thankÖzgür Dagdelen and Michael Schneider, the authors of [14] , for providing us with their implementation.  ... 
doi:10.22667/jowua.2016.12.31.001 dblp:journals/jowua/CorreiaMPBA16 fatcat:qq6irsu7rzffrcc6iet4umpqka

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

Guillaume Hanrot, Xavier Pujol, Damien Stehlé
2011 Lecture Notes in Computer Science  
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  ...  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

Integer Programming and Algorithmic Geometry of Numbers [chapter]

Friedrich Eisenbrand
2009 50 Years of Integer Programming 1958-2008  
I also want to thank Johannes Blömer for several discussions on shortest and closest vectors.  ...  Acknowledgments I am grateful to Damien Stehlé for numerous comments and suggestions which helped me a lot to improve this manuscript.  ...  In the same paper he proved that the closest vector problem is NP-hard for any ℓ p norm.  ... 
doi:10.1007/978-3-540-68279-0_14 fatcat:c7iusb6esbgpnbjnmohiokwegy
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