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2014 | 22 | 4 | 277-289
Tytuł artykułu

Torsion Z-module and Torsion-free Z-module

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In this article, we formalize a torsion Z-module and a torsionfree Z-module. Especially, we prove formally that finitely generated torsion-free Z-modules are finite rank free. We also formalize properties related to rank of finite rank free Z-modules. The notion of Z-module is necessary for solving lattice problems, LLL (Lenstra, Lenstra, and Lov´asz) base reduction algorithm [20], cryptographic systems with lattice [21], and coding theory [11].
Wydawca
Rocznik
Tom
22
Numer
4
Strony
277-289
Opis fizyczny
Daty
wydano
2014-12-01
otrzymano
2014-11-29
online
2014-12-31
Twórcy
autor
  • Japan Advanced Institute of Science and Technology Ishikawa, Japan
  • Shinshu University Nagano, Japan
  • Shinshu University Nagano, Japan
  • Shinshu University Nagano, Japan
Bibliografia
  • [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.
  • [2] Grzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543-547, 1990.
  • [3] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.
  • [4] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
  • [5] Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.
  • [6] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.
  • [7] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
  • [8] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
  • [9] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
  • [10] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.
  • [11] Wolfgang Ebeling. Lattices and Codes. Advanced Lectures in Mathematics. Springer Fachmedien Wiesbaden, 2013.
  • [12] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Z-modules. Formalized Mathematics, 20(1):47-59, 2012. doi:10.2478/v10037-012-0007-z.[Crossref]
  • [13] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Quotient module of Z-module. Formalized Mathematics, 20(3):205-214, 2012. doi:10.2478/v10037-012-0024-y.[Crossref]
  • [14] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Free Z-module. Formalized Mathematics, 20(4):275-280, 2012. doi:10.2478/v10037-012-0033-x.[Crossref]
  • [15] Yuichi Futa, Hiroyuki Okazaki, Daichi Mizushima, and Yasunari Shidama. Gaussian integers. Formalized Mathematics, 21(2):115-125, 2013. doi:10.2478/forma-2013-0013.[Crossref]
  • [16] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Submodule of free Z-module. Formalized Mathematics, 21(4):273-282, 2013. doi:10.2478/forma-2013-0029.[Crossref]
  • [17] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.
  • [18] Eugeniusz Kusak, Wojciech Leonczuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.
  • [19] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829-832, 1990.
  • [20] A. K. Lenstra, H. W. Lenstra Jr., and L. Lov´asz. Factoring polynomials with rational coefficients. Mathematische Annalen, 261(4), 1982.
  • [21] Daniele Micciancio and Shafi Goldwasser. Complexity of lattice problems: a cryptographic perspective. The International Series in Engineering and Computer Science, 2002.
  • [22] Kazuhisa Nakasho, Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Rank of submodule, linear transformations and linearly independent subsets of Z-module. Formalized Mathematics, 22(3):189-198, 2014. doi:10.2478/forma-2014-0021.[Crossref]
  • [23] Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263-264, 1990.
  • [24] Christoph Schwarzweller. The binomial theorem for algebraic structures. Formalized Mathematics, 9(3):559-564, 2001.
  • [25] Christoph Schwarzweller. The ring of integers, Euclidean rings and modulo integers. Formalized Mathematics, 8(1):29-34, 1999.
  • [26] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1): 115-122, 1990.
  • [27] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.
  • [28] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.
  • [29] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.
  • [30] Wojciech A. Trybulec. Operations on subspaces in vector space. Formalized Mathematics, 1(5):871-876, 1990.
  • [31] Wojciech A. Trybulec. Basis of vector space. Formalized Mathematics, 1(5):883-885, 1990.
  • [32] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
  • [33] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.
  • [34] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_forma-2014-0028
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