Displaying similar documents to “Multigraded modules.”

Matrix of ℤ-module1

Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama (2015)

Formalized Mathematics

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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...