Technical Report CS0676

TR#:CS0676
Class:CS
Title: FACTORIZATION PROPERTIES OF LATTICES OVER THE INTEGERS
Authors: A. Paz and M. Lempel
PDF - RevisedCS0676.revised.pdf
Abstract: Let $A$ be a nonsingular $n \times n$ matrix over the integers $L = L(A)$ denoting the lattice whose elements are combinations with integer coefficients of the rows of $A$. $L$ is cyclic if it can be defined in the modular form $L = \{x = (x_{i}) : \sum a_{i} x_{i} \equiv 0\ (\mbox{mod\ d}) \}$ where the $a_{i}$'s and $d$ are integers and $0 \leq a_{i} < d$. Let $L, L_{1}, L_{2} (B)$ be lattices over the integers $L = L_{1} L_{2}$ is a factorization of $L$ if every element of $L$ is a combination of the rows of $B$ such that the vector of combination coefficients is in $L_{1}$, and $B$ is a nonsingular $n \times n$ matrix. The following results are proved in this paper: Every lattice can be expressed as a product of cyclic factors in polynomial time; Every cyclic lattice can be factored into `simple' (term explained in the text) factors in polynomial time; Every simple lattice can be factored into `prime' factors in polynomial time if a prime factorization of the determinant of its basis is given. In addition, we provide polynomial algorithms for the following problems: Transform a cyclic lattice given by a basis into a modular form and vice versa; find a basis of finite modular lattice, given in modular form.
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