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The optimal LLL algorithm is still polynomial in fixed dimension

2003
*
Theoretical Computer Science
*

ACM 32(1) (1985) 229) seem to show that

doi:10.1016/s0304-3975(02)00616-3
fatcat:w2txuny4vvee7m4zkgfbdi6t34
*the**algorithm*remains*polynomial**in*average. ... However, no bound better than a naive exponential order one*is*established for*the*worst-case complexity of*the**optimal**LLL**algorithm*, even for ÿxed small*dimension*(higher than 2). ... Acknowledgements I am indebted to Brigitte VallÃ ee for drawing my attention to*algorithmic*problems*in*lattice theory and for regular helpful discussions. ...##
###
Worst-Case Complexity of the Optimal LLL Algorithm
[chapter]

2000
*
Lecture Notes in Computer Science
*

However no bound better than a naive exponential order one

doi:10.1007/10719839_35
fatcat:uzqcj2goejeylmnimxabdepdei
*is*established for*the*worstcase complexity of*the**optimal**LLL**algorithm*, even for*fixed*small*dimension*(higher than ¾). ... Here we prove that, for any*fixed**dimension*Ò,*the*number of iterations of*the**LLL**algorithm**is*linear with respect to*the*size of*the*input. ... I am indebted to Brigitte Vallée for drawing my attention to*algorithmic*problems*in*lattice theory and for regular helpful discussions. I wish to thank her also for her help to improve this paper. ...##
###
Adaptive Precision Floating Point LLL
[chapter]

2013
*
Lecture Notes in Computer Science
*

*The*

*LLL*

*algorithm*

*is*one of

*the*most studied lattice basis reduction

*algorithms*

*in*

*the*literature. ...

*In*its classic setting,

*the*floating point precision

*is*a

*fixed*value, determined by

*the*

*dimension*of

*the*input basis at

*the*initiation of

*the*

*algorithm*. ...

*The*

*LLL*

*algorithm*, named after its inventors, Lenstra, Lenstra and Lovász [11] ,

*is*a

*polynomial*time lattice reduction

*algorithm*. ...

##
###
Integer Programming and Algorithmic Geometry of Numbers
[chapter]

2009
*
50 Years of Integer Programming 1958-2008
*

Lenstra's

doi:10.1007/978-3-540-68279-0_14
fatcat:c7iusb6esbgpnbjnmohiokwegy
*algorithm*runs*in**polynomial*time*in**the*input encoding length if*the**dimension**is**fixed*. ... It turns out that*the*parametric shortest vector problem can be solved*in*linear time when*the**dimension**is**fixed*with a cascaded*LLL*-*algorithm*. ...##
###
Formalizing the LLL Basis Reduction Algorithm and the LLL Factorization Algorithm in Isabelle/HOL

2020
*
Journal of automated reasoning
*

*The*

*LLL*basis reduction

*algorithm*was

*the*first

*polynomial*-time

*algorithm*to compute a reduced basis of a given lattice, and hence also a short vector

*in*

*the*lattice. ...

*The*

*algorithm*has applications

*in*number theory, computer algebra and cryptography.

*In*this paper, we provide an implementation of

*the*

*LLL*

*algorithm*. ... Sebastiaan

*is*now working at University of Twente,

*the*Netherlands, and supported by

*the*NWO VICI 639.023.710 Mercedes project. ...

##
###
Cryptanalysis of RSA: A Special Case of Boneh-Durfee's Attack
[article]

2020
*
IACR Cryptology ePrint Archive
*

*The*core objective

*is*to explore RSA

*polynomials*underlying algebraic structure so that we can improve

*the*performance of weak key attacks. ... Boneh-Durfee proposed (at Eurocrypt 1999) a

*polynomial*time attacks on RSA small decryption exponent which exploits lattices and sub-lattice structure to obtain an

*optimized*bounds d < N 0.284 and d < ... After that

*LLL*

*algorithm*process

*the*scanned lattice efficiently especially it

*is*useful for large

*dimension*matrices. Only pick specific vectors whose

*dimension*approximately remains same. ...

##
###
A Formalization of the LLL Basis Reduction Algorithm
[chapter]

2018
*
Lecture Notes in Computer Science
*

We additionally integrate one application of

doi:10.1007/978-3-319-94821-8_10
fatcat:fpazlnxfyrhvzltnl23t6u36e4
*LLL*, namely a verified factorization*algorithm*for univariate integer*polynomials*which runs*in**polynomial*time. ...*The**LLL*basis reduction*algorithm*was*the*first polynomialtime*algorithm*to compute a reduced basis of a given lattice, and hence also a short vector*in**the*lattice. ... This research was supported by*the*Austrian Science Fund (FWF) project Y757. Jose Divasón*is*partially funded by*the*Spanish projects MTM2014-54151-P and MTM2017-88804-P. ...##
###
The History of the LLL-Algorithm
[chapter]

2009
*
The LLL Algorithm
*

*Polynomial*Factorization Arjen Lenstra's connection with

*the*

*LLL*-

*algorithm*began while he

*still*was a student. ... If

*the*solution set K

*is*full-

*dimensional*, then vol(K) > 0 and one can prove that log(1/vol(K))

*is*bounded by a

*polynomial*

*in*

*the*

*dimension*n and

*the*length of

*the*rest of input for K. ...

##
###
Page 2614 of Mathematical Reviews Vol. , Issue 2001D
[page]

2001
*
Mathematical Reviews
*

Classical results concern

*the**optimal*order of n(e,d), for*fixed**dimension*d, as a function of ¢. Usually these results include an unknown constant Cy, which depends on d. ...*The*number n(e,d)*is**the*minimal number of function values needed for a worst case error €*in**the**dimension*d for*the*class Fy. ...##
###
Techniques for Solving Shortest Vector Problem

2021
*
International Journal of Advanced Computer Science and Applications
*

More precisely, this paper presents four

doi:10.14569/ijacsa.2021.0120598
fatcat:x2hfrvhtyjewnniblcyiu7deru
*algorithms*:*the*Lenstra-Lenstra-Lovasz (*LLL*)*algorithm*,*the*Block Korkine-Zolotarev (BKZ)*algorithm*, a Metropolis*algorithm*, and a convex relaxation of SVP. ... This problem has a great many applications such as*optimization*, communication theory, cryptography, etc. ... ACKNOWLEDGMENT*The*authors would like to thank*the*reviewers for their valuable comments. ...##
###
Approximate common divisors via lattices
[article]

2012
*
arXiv
*
pre-print

*The*multivariate approximate common divisor problem

*is*

*the*number-theoretic analogue of multivariate

*polynomial*reconstruction, and we develop a corresponding lattice-based

*algorithm*for

*the*latter problem ... While these results do not challenge

*the*suggested parameters, a 2^(n^epsilon) approximation

*algorithm*with epsilon<2/3 for lattice basis reduction

*in*n

*dimensions*could be used to break these parameters ...

*The*

*algorithm*runs

*in*

*polynomial*time for

*fixed*m. ...

##
###
Fast Lattice Point Enumeration with Minimal Overhead
[chapter]

2014
*
Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms
*

*The*last technique

*is*used to obtain a new SVP enumeration procedure withÕ(n n/2e ) running time, matching (even

*in*

*the*constant

*in*

*the*exponent)

*the*

*optimal*worst-case analysis (Hanrot and Stehlé, CRYPTO ... Enumeration

*algorithms*are

*the*best currently known methods to solve lattice problems, both

*in*theory (within

*the*class of

*polynomial*space

*algorithms*), and

*in*practice (where they are routinely used to ...

*polynomial*

*in*

*the*

*dimension*n. ...

##
###
New Results for Partial Key Exposure on RSA with Exponent Blinding

2015
*
Proceedings of the 12th International Conference on Security and Cryptography
*

This type of attacks

doi:10.5220/0005571701360147
dblp:conf/secrypt/CimatoMS15
fatcat:z4xlwabdrvadblonvhaln6izvy
*is*of particular interest*in**the*context of side-channel attacks. ... Additionally, we apply partial key exposure attacks to CRT-RSA when exponent blinding*is*used, a case not yet analyzed*in*literature. ... ACKNOWLEDGEMENTS This work was partly supported by*the*Italian MIUR project SecurityHorizons (c.n. 2010XSEMLC). ...##
###
Decoding by Sampling: A Randomized Lattice Algorithm for Bounded Distance Decoding

2011
*
IEEE Transactions on Information Theory
*

Of particular interest

doi:10.1109/tit.2011.2162180
fatcat:tju7blaa65g4nlxrn77tcltrpa
*is*that a*fixed*gain*in**the*decoding radius compared to Babai's decoding can be achieved at*polynomial*complexity. ...*The*technical contribution of this paper*is*two-fold: we analyze and*optimize**the*decoding radius of sampling decoding resulting*in*better error performance than Klein's original*algorithm*, and propose ...*The*third author gratefully acknowledges*the*Department of Computing of Macquarie University and*the*Department of Mathematics and Statistics of*the*University of Sydney, where part of this work was undergone ...##
###
Analyzing Blockwise Lattice Algorithms Using Dynamical Systems
[chapter]

2011
*
Lecture Notes in Computer Science
*

(or SVP) subroutine, then BKZ returns a basis whose first vector has norm ≤ 2ν n−1 2(β−1) + 3 2 β · (det L) 1 n , where ν β ≤ β

doi:10.1007/978-3-642-22792-9_25
fatcat:wmrqaut75bgwljwomdfvqyuiou
*is**the*maximum of Hermite's constants*in**dimensions*≤ β. ... Among them,*the*BKZ*algorithm*introduced by Schnorr and Euchner [FCT'91] seems to achieve*the*best time/quality compromise*in*practice. ... Nguyen for explaining to us their bound on*the*number of tours of*the*original BKZ*algorithm*. We also thank C.-P. Schnorr for helpful discussions. ...
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