Integer lattices have numerous important applications, but some of them may have been overlooked because of the common assumption that a lattice basis is part of the problem instance. This paper gives an application that requires finding a basis for a lattice defined in terms of linear constraints. We show how to find such a basis efficiently.

Let L be a k-dimensional lattice in IR m with basis B = (b 1 ; : : : ; b k). Let A = (a1; : : : ; ak) be a rational approximation to B. Assume that A has rank k and a lattice basis reduction algorithm applied to the columns of A yields a transformation T = (t 1 ; : : : ; t k) 2 GL(k; Z Z) such that At i sii(L(A)) where L(A) is the lattice generated by the columns of A, i(L(A)) is the i-th succe...

When analyzing lattice-based cryptosystems, we often need to solve the Shortest Vector Problem (SVP) in some lattice associated to the system under scrutiny. The go-to algorithms in practice to solve SVP are enumeration algorithms, which usually consist of a preprocessing step, followed by an exhaustive search. Obviously, the two steps offer a trade-off and should be balanced in their running t...

Lattice reduction (LR) aided multiple-inputmultiple-out (MIMO) linear detection can achieve the maximum receive diversity of the maximum likelihood detection (MLD). By emloying the most commonly used Lenstra, Lenstra, and L. Lov ́asz (LLL) algorithm, an equivalent channel matrix which is shorter and nearly orthogonal is obtained. And thus the noise enhancement is greatly reduced by employing the...

There are very close connections between the arithmetic of integer lattices, algebraic properties of the associated ideals, and the geometry and the combinatorics of corresponding polyhedra. In this paper we investigate the generating sets (\Grr obner bases") of integer lattices that correspond to the Grr obner bases of the associated binomial ideals. Extending results by Sturmfels & Thomas, we...

A lattice metric singularity occurs when unit cells defining two (or more) lattices yield the identical set of unique calculated d-spacings. The minerals Mawsonite and Chatkalite are of especial interest as both are characterized by tetragonal unit cells that correspond to the second member of a quaternary lattice metric singularity. This singularity includes lattices that are Cubic I, Tetragon...

Lattice reduction algorithms have numerous applications in number theory, algebra, as well as in cryptanalysis. The most famous algorithm for lattice reduction is the LLL algorithm. In polynomial time it computes a reduced basis with provable output quality. One early improvement of the LLL algorithm was LLL with deep insertions (DeepLLL). The output of this version of LLL has higher quality in...

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...

In this paper, we investigate a variant of the BKZ algorithm, called progressive BKZ, which performs BKZ reductions by starting with a small blocksize and gradually switching to larger blocks as the process continues. We discuss techniques to accelerate the speed of the progressive BKZ algorithm by optimizing the following parameters: blocksize, searching radius and probability for pruning of t...