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A modification of the LLL reduction algorithm

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M. PohstAuthors Info & Claims

Journal of Symbolic Computation, Volume 4, Issue 1

Pages 123 - 127

https://doi.org/10.1016/S0747-7171(87)80061-5

Published: 01 August 1987 Publication History

Abstract

The reduction algorithm of Lenstra et al. (1982) is modified in a way that the input vectors can be linearly dependent. The output consists of a basis of the lattice generated by the input vectors as well as non-trivial linear combinations of O by the input vectors if those are linearly dependent.

References

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Computation of independent units in number fields by Dirichlet's method. Math. Comp.

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Polynomial time algorithms for finding integer relations among real numbers. Lecture Notes in Computer Science.

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Factoring polynomials with rational coefficients. Math. Ann. v26. 513-534.

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Cryptoanalytic attacks on the multiplicative knapsack cryptosystem and on Shamir's fast signature system. IEEE Trans. Inform. Theory. vIT-30. 594-600.

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Li JNguyen P(2025)A Complete Analysis of the BKZ Lattice Reduction AlgorithmJournal of Cryptology10.1007/s00145-024-09527-038:1Online publication date: 1-Jan-2025
https://dl.acm.org/doi/10.1007/s00145-024-09527-0
Yasuda MNakamura SYamaguchi J(2020)Analysis of DeepBKZ reduction for finding short lattice vectorsDesigns, Codes and Cryptography10.1007/s10623-020-00765-488:10(2077-2100)Online publication date: 1-Oct-2020
https://dl.acm.org/doi/10.1007/s10623-020-00765-4
Li JNguyen PDavenport JWang DKauers MBradford R(2019)Computing a Lattice Basis RevisitedProceedings of the 2019 International Symposium on Symbolic and Algebraic Computation10.1145/3326229.3326265(275-282)Online publication date: 8-Jul-2019
https://dl.acm.org/doi/10.1145/3326229.3326265
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Index Terms

A modification of the LLL reduction algorithm
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
2. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Number-theoretic computations

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Abstract
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. It approximates an NP-hard problem where the approximation quality solely ...

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cover image Journal of Symbolic Computation

Journal of Symbolic Computation Volume 4, Issue 1

August 1987

129 pages

ISSN:0747-7171

Editor:
Bruno Buchberger
Johannes Kepler Univ., Linz, Austria

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Publisher

Academic Press, Inc.

United States

Publication History

Published: 01 August 1987

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Li JNguyen P(2025)A Complete Analysis of the BKZ Lattice Reduction AlgorithmJournal of Cryptology10.1007/s00145-024-09527-038:1Online publication date: 1-Jan-2025
https://dl.acm.org/doi/10.1007/s00145-024-09527-0
Yasuda MNakamura SYamaguchi J(2020)Analysis of DeepBKZ reduction for finding short lattice vectorsDesigns, Codes and Cryptography10.1007/s10623-020-00765-488:10(2077-2100)Online publication date: 1-Oct-2020
https://dl.acm.org/doi/10.1007/s10623-020-00765-4
Li JNguyen PDavenport JWang DKauers MBradford R(2019)Computing a Lattice Basis RevisitedProceedings of the 2019 International Symposium on Symbolic and Algebraic Computation10.1145/3326229.3326265(275-282)Online publication date: 8-Jul-2019
https://dl.acm.org/doi/10.1145/3326229.3326265
Nagashima HShirayanagi K(2018)Effect of the interval-symbol method with correct zero rewriting on the δ-LLL algorithmACM Communications in Computer Algebra10.1145/3282678.328267952:2(24-31)Online publication date: 1-Oct-2018
https://dl.acm.org/doi/10.1145/3282678.3282679
Chen JStehlé DVillard GKauers MOvchinnikov ASchost É(2018)Computing an LLL-reduced Basis of the Orthogonal LaticeProceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation10.1145/3208976.3209013(127-133)Online publication date: 11-Jul-2018
https://dl.acm.org/doi/10.1145/3208976.3209013
Tagliasacchi MVisentini-Scarzanella MDragotti PTubaro S(2016)Identification of Transform Coding ChainsIEEE Transactions on Image Processing10.1109/TIP.2015.250798525:3(1109-1123)Online publication date: 1-Mar-2016
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Micciancio DWalter M(2016)Practical, Predictable Lattice Basis ReductionProceedings, Part I, of the 35th Annual International Conference on Advances in Cryptology --- EUROCRYPT 2016 - Volume 966510.1007/978-3-662-49890-3_31(820-849)Online publication date: 8-May-2016
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Chen JStehlé DVillard GMonagan MCooperman GGiesbrecht M(2013)A new view on HJLS and PSLQProceedings of the 38th International Symposium on Symbolic and Algebraic Computation10.1145/2465506.2465936(149-156)Online publication date: 26-Jun-2013
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Storjohann AJohnson JPark HKaltofen E(2009)Integer matrix rank certificationProceedings of the 2009 international symposium on Symbolic and algebraic computation10.1145/1576702.1576748(333-340)Online publication date: 28-Jul-2009
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Previtali A(2006)Irreducible constituents of monomial representationsJournal of Symbolic Computation10.1016/j.jsc.2006.09.00241:12(1345-1359)Online publication date: 1-Dec-2006
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Segment LLL-Reduction of Lattice Bases

Fast LLL-type lattice reduction

Formalizing the LLL Basis Reduction Algorithm and the LLL Factorization Algorithm in Isabelle/HOL

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