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An LLL-reduction algorithm with quasi-linear time complexity: extended abstract

Authors:

Andrew Novocin,

Damien Stehlé,

Gilles VillardAuthors Info & Claims

STOC '11: Proceedings of the forty-third annual ACM symposium on Theory of computing

Pages 403 - 412

https://doi.org/10.1145/1993636.1993691

Published: 06 June 2011 Publication History

Abstract

We devise an algorithm, L1, with the following specifications: It takes as input an arbitrary basis B=(b_i)_i ∈ Z^{d x d} of a Euclidean lattice L; It computes a basis of L which is reduced for a mild modification of the Lenstra-Lenstra-Lovász reduction; It terminates in time O(d^5+ε β + d^ω+1+ε β^1+ε) where β = log max |b_i| (for any ε>0 and ω is a valid exponent for matrix multiplication). This is the first LLL-reducing algorithm with a time complexity that is quasi-linear in β and polynomial in d.

The backbone structure of L1 is able to mimic the Knuth-Schönhage fast gcd algorithm thanks to a combination of cutting-edge ingredients. First the bit-size of our lattice bases can be decreased via truncations whose validity are backed by recent numerical stability results on the QR matrix factorization. Also we establish a new framework for analyzing unimodular transformation matrices which reduce shifts of reduced bases, this includes bit-size control and new perturbation tools. We illustrate the power of this framework by generating a family of reduction algorithms.

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An LLL-reduction algorithm with quasi-linear time complexity: extended abstract

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cover image ACM Conferences

STOC '11: Proceedings of the forty-third annual ACM symposium on Theory of computing

June 2011

840 pages

ISBN:9781450306911

DOI:10.1145/1993636

General Chair:
Lance Fortnow
Northwestern University
,
Program Chair:
Salil Vadhan
Harvard University

Copyright © 2011 ACM.

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Published: 06 June 2011

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June 6 - 8, 2011

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https://dl.acm.org/doi/10.1007/978-3-031-38548-3_1
Kauers MKoutschan CMaza MZhi L(2022)Guessing with Little DataProceedings of the 2022 International Symposium on Symbolic and Algebraic Computation10.1145/3476446.3535486(83-90)Online publication date: 4-Jul-2022
https://dl.acm.org/doi/10.1145/3476446.3535486
Xu JSarkar SHu L(2022)Revisiting orthogonal lattice attacks on approximate common divisor problemsTheoretical Computer Science10.1016/j.tcs.2022.02.005Online publication date: Feb-2022
https://doi.org/10.1016/j.tcs.2022.02.005
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