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Hardness of approximating the shortest vector problem in lattices

Subhash Khot
2005 Journal of the ACM  
We show that assuming NP ⊆ RP, there is no polynomial time algorithm that approximates the Shortest Vector Problem (SVP) in p norm within a constant factor.  ...  Under the stronger assumption NP ⊆ RTIME(2 poly(log n) ), we show that there is no polynomial-time algorithm with approximation ratio 2 (log n) 1/2− where n is the dimension of the lattice and > 0 is an  ...  Thanks to Oded Regev, Ravi Kumar, Venkatesan Guruswami and anonymous referees for their valuable comments on the earlier drafts of the paper. Thanks also to Miki Ajtai, D.  ... 
doi:10.1145/1089023.1089027 fatcat:oylx4lei5bgu7judmz6tyswfy4

Hardness of Approximating the Shortest Vector Problem in Lattices

S. Khot
45th Annual IEEE Symposium on Foundations of Computer Science  
We show that assuming NP ⊆ RP, there is no polynomial time algorithm that approximates the Shortest Vector Problem (SVP) in p norm within a constant factor.  ...  Under the stronger assumption NP ⊆ RTIME(2 poly(log n) ), we show that there is no polynomial-time algorithm with approximation ratio 2 (log n) 1/2− where n is the dimension of the lattice and > 0 is an  ...  Thanks to Oded Regev, Ravi Kumar, Venkatesan Guruswami and anonymous referees for their valuable comments on the earlier drafts of the paper. Thanks also to Miki Ajtai, D.  ... 
doi:10.1109/focs.2004.31 dblp:conf/focs/Khot04 fatcat:h2qy5wkx7bestea3gvjmxshri4

Shortest Vector Problem [chapter]

Daniele Micciancio, Shafi Goldwasser
2002 Complexity of Lattice Problems  
A g-approximation algorithm for SVP is an algorithm that on input a lattice L, outputs a nonzero lattice vector of length at most g times the length of the shortest vector in the lattice.  ...  The Shortest Vector Problem (SVP) is the most famous and widely studied computational problem on lattices.  ... 
doi:10.1007/978-1-4615-0897-7_4 fatcat:oxy3pnk4mvbi7kzobipwdd47ha

Shortest Vector Problem [chapter]

Daniele Micciancio
2016 Encyclopedia of Algorithms  
A g-approximation algorithm for SVP is an algorithm that on input a lattice L, outputs a nonzero lattice vector of length at most g times the length of the shortest vector in the lattice.  ...  The Shortest Vector Problem (SVP) is the most famous and widely studied computational problem on lattices.  ... 
doi:10.1007/978-1-4939-2864-4_374 fatcat:7qcbdmgki5e33px4vj4tdxsmue

Shortest Vector Problem [chapter]

Daniele Micciancio
2008 Encyclopedia of Algorithms  
A g-approximation algorithm for SVP is an algorithm that on input a lattice L, outputs a nonzero lattice vector of length at most g times the length of the shortest vector in the lattice.  ...  The Shortest Vector Problem (SVP) is the most famous and widely studied computational problem on lattices.  ... 
doi:10.1007/978-0-387-30162-4_374 fatcat:6dgzpwmppfcrlbs3kvpfmqhsoq

Shortest Vector Problem [chapter]

Carlisle Adams, Mark Stephens, Ernst M. Gabidulin, Dan Boneh, Mike Just, Claudio A. Ardagna, Ernesto Damiani, Caroline Fontaine, Paul England, Ali Bagherzandi, Bijit Hore, Sharad Mehrotra (+126 others)
2011 Encyclopedia of Cryptography and Security  
A g-approximation algorithm for SVP is an algorithm that on input a lattice L, outputs a nonzero lattice vector of length at most g times the length of the shortest vector in the lattice.  ...  The Shortest Vector Problem (SVP) is the most famous and widely studied computational problem on lattices.  ... 
doi:10.1007/978-1-4419-5906-5_434 fatcat:2dctv256nvbkzppflgfrja56ua

A relation of primal-dual lattices and the complexity of shortest lattice vector problem

Jin-Yi Cai
1998 Theoretical Computer Science  
In a forthcoming paper [8], Cai and Nerurkar also improve the NP-hardness result of Ajtai [2] to show that the problem of approximating the shortest vector length up to a factor of 1 +( l/n"), for any  ...  We give a simplified proof of a theorem of Lagarias, Lenstra and Schnorr [ 171 that the problem of approximating the length of the shortest lattice vector within a factor of Cn, for an appropriate constant  ...  Acknowledgements We thank the anonymous referees for helpful comments. We also thank Miki Ajtai, Oded Goldreich, Shafi Goldwasser and Ajay Nerurkar for valuable discussions and comments.  ... 
doi:10.1016/s0304-3975(98)00058-9 fatcat:vw4dtpvg4jfpfpmf3appzcnwxa

Approximating shortest lattice vectors is not harder than approximating closest lattice vectors

O. Goldreich, D. Micciancio, S. Safra, J.-P. Seifert
1999 Information Processing Letters  
We show that given oracle access to a subroutine which returns approximate closest vectors in a lattice, one may nd in polynomial-time approximate shortest vectors in a lattice.  ...  The result holds for any norm, and preserves the dimension of the lattice, i.e., the closest vector oracle is called on lattices of exactly the same dimension as the original shortest vector problem.  ...  In M] and DMS] the Shortest Vector Problem and the Minimum Distance Problem are proved NP-hard to approximate by reduction from the Closest Vector Problem and the Nearest Codeword Problem respectively.  ... 
doi:10.1016/s0020-0190(99)00083-6 fatcat:qaala2vudbdkhauj5o3ukjfds4

Inapproximability Results for Computational Problems on Lattices [chapter]

Subhash Khot
2009 The LLL Algorithm  
In this article, we present a survey of known inapproximability results for computational problems on lattices, viz.  ...  The Shortest Vector Problem (SVP) The most studied computational problem on lattices is the Shortest Vector Problem (SVP), 1 where given a basis for an n-dimensional lattice, we seek the shortest non-zero  ...  vector in the lattice.  ... 
doi:10.1007/978-3-642-02295-1_14 dblp:series/isc/Khot10 fatcat:lsgk3bgpp5adzg7zctnydb3ihq

The Shortest Vector Problem in Lattices with Many Cycles [chapter]

Mårten Trolin
2001 Lecture Notes in Computer Science  
In this paper we investigate how the complexity of the shortest vector problem in a lattice Λ depends on the cycle structure of the additive group Z n /Λ.  ...  We give a proof that the shortest vector problem is NP-complete in the max-norm for n-dimensional lattices Λ where Z n /Λ has n − 1 cycles.  ...  Acknowledgements I would like to thank Johan Håstad for valueable feedback and ideas during the preparation of this paper.  ... 
doi:10.1007/3-540-44670-2_14 fatcat:kf6fd5lj5zfebhlqmcizax47ri

On Bounded Distance Decoding, Unique Shortest Vectors, and the Minimum Distance Problem [chapter]

Vadim Lyubashevsky, Daniele Micciancio
2009 Lecture Notes in Computer Science  
version of the Shortest Vector Problem).  ...  We prove the equivalence, up to a small polynomial approximation factor n/ log n, of the lattice problems uSVP (unique Shortest Vector Problem), BDD (Bounded Distance Decoding) and GapSVP (the decision  ...  Acknowledgements We thank the anonymous referees for very useful comments.  ... 
doi:10.1007/978-3-642-03356-8_34 fatcat:beqnj73tlzeihkvaa5y4pyhfpm

Explicit Hard Instances of the Shortest Vector Problem [chapter]

Johannes Buchmann, Richard Lindner, Markus Rückert
2008 Lecture Notes in Computer Science  
Building upon a famous result due to Ajtai, we propose a sequence of lattice bases with growing dimension, which can be expected to be hard instances of the shortest vector problem (SVP) and which can  ...  We use our sequence of lattice bases to create a challenge, which may be helpful in determining appropriate parameters for these schemes.  ...  Furthermore, we thank the PQCrypto 2008 program committee and the anonymous reviewers for their valuable comments.  ... 
doi:10.1007/978-3-540-88403-3_6 fatcat:4c2fiymsk5cwzpnq5tfs3g4dfm

On the complexity of computing short linearly independent vectors and short bases in a lattice

Johannes Blömer, Jean-Pierre Seifert
1999 Proceedings of the thirty-first annual ACM symposium on Theory of computing - STOC '99  
Under the assumption that problems in NP cannot be solved in DTIME(n p"'y'og(n)) we show that no polynomial time algorithm can approximate the length of a shortest set of linearly independent vectors (  ...  Motivated by Ajtai's worst-case to average-case reduction for lattice problems, we study the complexity of computing short linearly independent vectors (short basis) in a lattice.  ...  SVP(SHORTESTVECTOR PROBLEM) Find the length of a shortest non-xro vector in L within a factor of ncz.  ... 
doi:10.1145/301250.301441 dblp:conf/stoc/BlomerS99 fatcat:4ztzginc3vf4fp47hogty6wjsa

A Novel Lattice Reduction Algorithm

Dipayan Das, Vishal Saraswat
2018 Proceedings of the 15th International Joint Conference on e-Business and Telecommunications  
The cryptographic hardness of the lattice based constructions mainly lies on the difficulty of solving two problems, namely, shortest vector problem (SVP) and closest vector problem (CVP).  ...  The proposed algorithm is very simple -it calls the shortest vector oracle for n − 1 times and outputs an almost orthogonal lattice basis with running time O(n 3 ), n being the rank of the lattice. 496  ...  Part of the work was carried out while visiting the R.C.Bose Centre for Cryptology and Security, Indian Statistical Institute, Kolkata. We are thankful to Kajla Basu for her support.  ... 
doi:10.5220/0006862106620667 dblp:conf/icete/DasS18a fatcat:62qr7octyrdzncmhit5o5bgvuq

Improved hardness results for unique shortest vector problem [article]

Divesh Aggarwal, Chandan Dubey
2011 arXiv   pre-print
We give several improvements on the known hardness of the unique shortest vector problem. - We give a deterministic reduction from the shortest vector problem to the unique shortest vector problem.  ...  As a byproduct, we get deterministic NP-hardness for unique shortest vector problem in the ℓ_∞ norm. - We give a randomized reduction from SAT to uSVP_1+1/poly(n).  ...  Given a lattice L, the γ-approximate shortest vector problem (SVP γ ) is the problem of finding a non-zero lattice vector of length at most γλ 1 (L).  ... 
arXiv:1112.1564v1 fatcat:qawzkurarrdwjdyytjxdhn6izm
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