Faster Algorithms for Integer Lattice Basis Reduction

Fast LLL-type lattice reduction

We modify the concept of LLL-reduction of lattice bases in the sense of Lenstra, Lenstra, Lovász [LLL82] towards a faster reduction algorithm. We organize LLL-reduction in segments of the basis. Our SLLL-bases approximate the successive minima of the lattice in nearly the same way as LLL-bases. For integer lattices of dimension n given by a basis of length 2, SLLL-reduction runs in O(n) bit ope...

Adaptive precision LLL and Potential-LLL reductions with Interval arithmetic

Lattice reduction is fundamental in computational number theory and in computer science, especially in cryptography. The celebrated Lenstra–Lenstra–Lovász reduction algorithm (called LLL or L) has been improved in many ways through the past decades and remains one of the central tool for reducing lattice basis. In particular, its floating-point variants — where the long-integer arithmetic requi...

Generated the Integer Null Space and Conditions for Determination of an Integer Basis Using the ABS Algorithms

SIAG/OPT Views-and-News pdfsubject

Algebraic Methods for Integer Programming Introduction by the Guest Editor Karen Aardal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Second Generation Lift-And-Project Algorithms Daniel Bienstock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Short Rational Generating Functions and Their Applications to Integer Programming Kevin Woods and Ruriko Yos...

Basis Reduction Methods

where P is a polyhedron described by inequalities. The work of Lenstra in 1979 [1] answered one of the most challenging questions in the theory of integer programming by presenting an algorithm to solve (IP), whose running time is polynomial, when n, the dimension is fixed. This paper pioneered the use of lattice basis reduction in integer programming, and initiated an interest in polynomial re...