Matrix of Z - module 1 Yuichi Futa Japan Advanced Institut of Science and Technology Ishikawa , Japan Hiroyuki

In this article, we formalize a matrix of Z-module and its properties. Specially, we formalize a matrix of a linear transformation of Z-module, a bilinear form and a matrix of the bilinear form (Gramian matrix). We formally prove that for a finite-rank free Z-module V , determinant of its Gramian matrix is constant regardless of selection of its basis. Z-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm and cryptographic systems with lattices [? ]. Some theorems in this article are described by translating theorems in [22], [24] and [18] into theorems of Z-module.