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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

4 Issues per year


SCImago Journal Rank (SJR) 2016: 0.207
Source Normalized Impact per Paper (SNIP) 2016: 0.315

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Online
ISSN
1898-9934
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Volume 24, Issue 1

Issues

Lattice of ℤ-module

Yuichi Futa / Yasunari Shidama
Published Online: 2016-08-31 | DOI: https://doi.org/10.1515/forma-2016-0005

Summary

In this article, we formalize the definition of lattice of ℤ-module and its properties in the Mizar system [5].We formally prove that scalar products in lattices are bilinear forms over the field of real numbers ℝ. We also formalize the definitions of positive definite and integral lattices and their properties. Lattice of ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [14], and cryptographic systems with lattices [15] and coding theory [9].

MSC: 15A03; 15A63; 11E39; 03B35

Keywords: ℤ-lattice; Gram matrix; integral ℤ-lattice; positive definite ℤ-lattice

MML : identifier: ZMODLAT1; version: 8.1.04 5.36.1267

References

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About the article

Received: 2015-12-30

Published Online: 2016-08-31

Published in Print: 2016-03-01


Citation Information: Formalized Mathematics, Volume 24, Issue 1, Pages 49–68, ISSN (Online) 1898-9934, DOI: https://doi.org/10.1515/forma-2016-0005.

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© by Yuichi Futa. This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License. BY-SA 3.0

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[1]
Yuichi Futa and Yasunari Shidama
Formalized Mathematics, 2017, Volume 25, Number 2

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