A lattice reduction algorithm, named after discoverers Lenstra, Lenstra, and Lovasz (1982), that produces a lattice basis of "short"
vectors. It was noticed by Lenstra et al. (1982) that the algorithm could
be used to obtain factors of univariate polynomials, which amounts to the determination
of integer relations. However, this application
of the algorithm, which later came to be one of its primary applications, was not
stressed in the original paper.
For a complexity analysis of the LLL algorithm, see Storjohann (1996).
More recently, other algorithms such as PSLQ, which can be significant faster than LLL, have been developed for finding integer
relations. PSLQ achieves its performance because of clever techniques that allow
machine arithmetic to be used at many intermediate steps, whereas LLL must use moderate
precision (although generally not as much as the HJLS
algorithm).
Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. "Integer Relation Detection."
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Mathematics in Action. Wellesley, MA: A K Peters, pp. 29-31, 2007.Borwein,
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by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A
K Peters, pp. 51-52, 2003.Borwein, J. M. and Corless, R. M.
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899-909, 1999.Borwein, J. M. and Lisonek, P. "Applications
of Integer Relation Algorithms." Disc. Math.217, 65-82, 2000.Centre
for Experimental & Constructive Mathematics. "Integer Relations." http://www.cecm.sfu.ca/projects/IntegerRelations/.Cohen,
H. A
Course in Computational Algebraic Number Theory. New York: Springer-Verlag,
1993.Lenstra, A. K.; Lenstra, H. W.; and Lovasz, L. "Factoring
Polynomials with Rational Coefficients." Math. Ann.261, 515-534,
1982.Matthews, K. "Keith Matthews' LLL Page." http://www.numbertheory.org/lll.html.Mignotte,
M. Mathematics
for Computer Algebra. New York: Springer-Verlag, 1991.Storjohann,
A. "Faster Algorithms for Integer Lattice Basis Reduction." Technical Report
249. Zurich, Switzerland: Department Informatik, ETH. July 30, 1996.Zimmerman,
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