Symmetry operations of layers periodic in two dimensions restrict the geometry the lattice according to the five two-dimensional Bravais types of lattices. In order-disorder (OD) structures, the operations relating equivalent layers generally leave invariant only a sublattice of the layers. The thus resulting restrictions can be expressed in terms of linear relations of the a, b and a · b scala...

In this paper we introduce a new parallel variant of the LLL lattice basis reduction algorithm. Lattice theory and in particular lattice basis reduction continues to play an integral role in cryptography. Not only does it provide effective cryptanalysis tools but it is also believed to bring about new cryptographic primitives that exhibit strong security even in the presence of quantum computer...

In 1982, Arjen Lenstra, Hendrik Lenstra Jr. and László Lovász introduced an efficiently computable notion of reduction of basis of a Euclidean lattice that is now commonly referred to as LLL-reduction. The precise definition involves the R-factor of the QR factorization of the basis matrix. In order to circumvent the use of rational/exact arithmetic with large bit-sizes, it is tempting to consi...