Dual Lattice of ℤ-module Lattice-Reference-Cited by-同舟云学术

Dual Lattice of ℤ-module Lattice

Published:2017-07-01 Issue:2 Volume:25 Page:157-169
ISSN:1898-9934
Container-title:Formalized Mathematics
language:en
Short-container-title:

Author:

Futa Yuichi¹,Shidama Yasunari²

Affiliation:

1. Tokyo University of Technology , Tokyo , Japan

2. Shinshu University , Nagano , Japan

Abstract

Summary In this article, we formalize in Mizar [5] the definition of dual lattice and their properties. We formally prove that a set of all dual vectors in a rational lattice has the construction of a lattice. We show that a dual basis can be calculated by elements of an inverse of the Gram Matrix. We also formalize a summation of inner products and their properties. Lattice of ℤ-module is necessary for lattice problems, LLL(Lenstra, Lenstra and Lovász) base reduction algorithm and cryptographic systems with lattice [20], [10] and [19].

Publisher

Walter de Gruyter GmbH

Subject

Applied Mathematics,Computational Mathematics

Reference27 articles.

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5. [5] Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi: 10.1007/978-3-319-20615-817.10.1007/978-3-319-20615-817