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

2018
*
arXiv
*
pre-print

As

arXiv:1801.02358v2
fatcat:3yanvzdlg5adtiw3iu4aaaklha
*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*. ...##
###
Improved Algorithms for the Shortest Vector Problem and the Closest Vector Problem in the Infinity Norm

2018
*
International Symposium on Algorithms and Computation
*

As

doi:10.4230/lipics.isaac.2018.35
dblp:conf/isaac/AggarwalM18
fatcat:23h37gqehjemvhuufnnqqxjfi4
*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**the*∞*norm*. ... 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. ...##
###
Hardness of Approximate Nearest Neighbor Search under L-infinity
[article]

2020
*
arXiv
*
pre-print

This shows a conditional separation between ANN under

arXiv:2011.06135v1
fatcat:n4ywphahqnfqbf35gri33pj774
*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. ...##
###
Efficient lattice-based signature scheme

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. ...

##
###
Covering Cubes and the Closest Vector Problem
[article]

2010
*
arXiv
*
pre-print

We provide

arXiv:1012.2289v1
fatcat:n5ypnle5m5chtdnv2ut3cd2cx4
*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*. ...##
###
Parallel Improved Schnorr-Euchner Enumeration SE++ for the CVP and SVP

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. ...

##
###
Broadcast Attacks against Lattice-Based Cryptosystems
[chapter]

2009
*
Lecture Notes in Computer Science
*

These

doi:10.1007/978-3-642-01957-9_28
fatcat:ht7nrvkhlnb7zb4p3nlwjbe75u
*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). ...##
###
Estimating the Effectiveness of Lattice Attacks
[article]

2021
*
IACR Cryptology ePrint Archive
*

We also investigated

dblp:journals/iacr/AbeI21
fatcat:7kt5pc3qlvgk5nzngp7jfo3gcq
*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. ...##
###
Universal lattice decoding: a review and some recent results

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 ...

##
###
Maximum Distance Sub-Lattice Problem
[article]

2018
*
arXiv
*
pre-print

We prove that MDSP is isomorphic to a well-known

arXiv:1811.03019v1
fatcat:cdztcq67ivdxpedibjgfecfpyu
*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. ...##
###
Algorithms to Compute Minimum Cycle Basis in Directed Graphs

2007
*
Theory of Computing Systems
*

This paper presents anÕ(m 4 n)

doi:10.1007/s00224-006-1319-6
fatcat:4bzqoqwao5erpm7bfxdac3bwcm
*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 . . . . ...##
###
Adapting Density Attacks to Low-Weight Knapsacks
[chapter]

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

doi:10.1007/11593447_3
fatcat:us7noxwnofg6lpq3qgwjxmzk6m
*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. ...##
###
Parallel Improved Schnorr-Euchner Enumeration SE++ on Shared and Distributed Memory Systems, With and Without Extreme Pruning

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. ...

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

2011
*
Lecture Notes in Computer Science
*

We present

doi:10.1007/978-3-642-20901-7_10
fatcat:dxcj7djiybg2bjurx6rveejmsu
*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. ...##
###
Integer Programming and Algorithmic Geometry of Numbers
[chapter]

2009
*
50 Years of Integer Programming 1958-2008
*

I also want to thank Johannes Blömer

doi:10.1007/978-3-540-68279-0_14
fatcat:c7iusb6esbgpnbjnmohiokwegy
*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*. ...
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