Sieve algorithms for some orthogonal integer lattices

Abstract

We propose in this work a Sieve algorithm that we called OrthogonalInteger sieve algorithm for some orthogonal integer lattices and particularly the case of integer lattices [Formula: see text], root lattices of type [Formula: see text] ([Formula: see text]) and of type [Formula: see text] ([Formula: see text]). In these cases, we use the famous [Formula: see text] algorithm to find the shortest vector of these lattices. Indeed, in general, a sieve algorithm builds a list of short random vectors which are not necessarily in the lattice, and try to produce short lattice vectors by taking linear combinations of the vectors in the list. But in our case, we built a list of short vectors in the lattice. From the first column of the [Formula: see text]-reduced basis of the considered basis, we have the list of at least [Formula: see text] and at most [Formula: see text] short vectors for the general case (where [Formula: see text] is the dimension of the lattice) of orthogonal integer lattices [Formula: see text] . For the lattices [Formula: see text], [Formula: see text] ([Formula: see text]) and [Formula: see text] ([Formula: see text]), we have, respectively, [Formula: see text], [Formula: see text] and [Formula: see text] short vectors. The proposed sieve algorithm for integer lattice [Formula: see text] runs in space [Formula: see text] and the OrthogonalInteger sieve algorithm performs [Formula: see text] arithmetic operations and is polynomial in space.