Lattice of ℤ-module

Yuichi Futa; Yasunari Shidama

Formalized Mathematics (2016)

  • Volume: 24, Issue: 1, page 49-68
  • ISSN: 1426-2630

Abstract

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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].

How to cite

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Yuichi Futa, and Yasunari Shidama. "Lattice of ℤ-module." Formalized Mathematics 24.1 (2016): 49-68. <http://eudml.org/doc/286765>.

@article{YuichiFuta2016,
abstract = {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].},
author = {Yuichi Futa, Yasunari Shidama},
journal = {Formalized Mathematics},
keywords = {ℤ-lattice; Gram matrix; integral ℤ-lattice; positive definite ℤ-lattice; -lattice; integral -lattice; positive definite -lattice},
language = {eng},
number = {1},
pages = {49-68},
title = {Lattice of ℤ-module},
url = {http://eudml.org/doc/286765},
volume = {24},
year = {2016},
}

TY - JOUR
AU - Yuichi Futa
AU - Yasunari Shidama
TI - Lattice of ℤ-module
JO - Formalized Mathematics
PY - 2016
VL - 24
IS - 1
SP - 49
EP - 68
AB - 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].
LA - eng
KW - ℤ-lattice; Gram matrix; integral ℤ-lattice; positive definite ℤ-lattice; -lattice; integral -lattice; positive definite -lattice
UR - http://eudml.org/doc/286765
ER -

References

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  13. [13] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Matrix of ℤ-module. Formalized Mathematics, 23(1):29-49, 2015. doi:10.2478/forma-2015-0003. Zbl1317.11037
  14. [14] A. K. Lenstra, H. W. Lenstra Jr., and L. Lovász. Factoring polynomials with rational coefficients. Mathematische Annalen, 261(4), 1982. 
  15. [15] Daniele Micciancio and Shafi Goldwasser. Complexity of lattice problems: A cryptographic perspective. The International Series in Engineering and Computer Science, 2002. Zbl1140.94010
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