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Hardness of approximating the shortest vector problem in lattices

2005
*
Journal of the ACM
*

We show that assuming NP ⊆ RP, there is no polynomial time algorithm that

doi:10.1145/1089023.1089027
fatcat:oylx4lei5bgu7judmz6tyswfy4
*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. ...##
###
Hardness of Approximating the Shortest Vector Problem in Lattices

*
45th Annual IEEE Symposium on Foundations of Computer Science
*

We show that assuming NP ⊆ RP, there is no polynomial time algorithm that

doi:10.1109/focs.2004.31
dblp:conf/focs/Khot04
fatcat:h2qy5wkx7bestea3gvjmxshri4
*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. ...##
###
Shortest Vector Problem
[chapter]

2002
*
Complexity of Lattice Problems
*

A g-

doi:10.1007/978-1-4615-0897-7_4
fatcat:oxy3pnk4mvbi7kzobipwdd47ha
*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*. ...##
###
Shortest Vector Problem
[chapter]

2016
*
Encyclopedia of Algorithms
*

A g-

doi:10.1007/978-1-4939-2864-4_374
fatcat:7qcbdmgki5e33px4vj4tdxsmue
*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*. ...##
###
Shortest Vector Problem
[chapter]

2008
*
Encyclopedia of Algorithms
*

A g-

doi:10.1007/978-0-387-30162-4_374
fatcat:6dgzpwmppfcrlbs3kvpfmqhsoq
*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*. ...##
###
Shortest Vector Problem
[chapter]

2011
*
Encyclopedia of Cryptography and Security
*

*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*. ...

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

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. ...

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

1999
*
Information Processing Letters
*

We show that given oracle access to a subroutine which returns

doi:10.1016/s0020-0190(99)00083-6
fatcat:qaala2vudbdkhauj5o3ukjfds4
*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. ...##
###
Inapproximability Results for Computational Problems on Lattices
[chapter]

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*. ...

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

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. ...

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

2009
*
Lecture Notes in Computer Science
*

version

doi:10.1007/978-3-642-03356-8_34
fatcat:beqnj73tlzeihkvaa5y4pyhfpm
*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. ...##
###
Explicit Hard Instances of the Shortest Vector Problem
[chapter]

2008
*
Lecture Notes in Computer Science
*

Building upon a famous result due to Ajtai, we propose a sequence

doi:10.1007/978-3-540-88403-3_6
fatcat:4c2fiymsk5cwzpnq5tfs3g4dfm
*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. ...##
###
On the complexity of computing short linearly independent vectors and short bases in a lattice

1999
*
Proceedings of the thirty-first annual ACM symposium on Theory of computing - STOC '99
*

Under

doi:10.1145/301250.301441
dblp:conf/stoc/BlomerS99
fatcat:4ztzginc3vf4fp47hogty6wjsa
*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. ...##
###
A Novel Lattice Reduction Algorithm

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. ...

##
###
Improved hardness results for unique shortest vector problem
[article]

2011
*
arXiv
*
pre-print

We give several improvements on

arXiv:1112.1564v1
fatcat:qawzkurarrdwjdyytjxdhn6izm
*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). ...
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