Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama (2012)
Formalized Mathematics
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In this article, we formalize Z-module, that is a module over integer ring. Z-module is necassary for lattice problems, LLL (Lenstra-Lenstra-Lovász) base reduction algorithm and cryptographic systems with lattices [11].
Kamal, M.A., Elmnophy, O.A. (2005)
Acta Mathematica Universitatis Comenianae. New Series
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Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama (2012)
Formalized Mathematics
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In this article we formalize a quotient module of Z-module and a vector space constructed by the quotient module. We formally prove that for a Z-module V and a prime number p, a quotient module V/pV has the structure of a vector space over Fp. Z-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lov´asz) base reduction algorithm and cryptographic systems with lattices [14]. Some theorems in this article are described by translating theorems in [20] and [19] into theorems...
Jackson, Nicholas (2005)
Homology, Homotopy and Applications
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Nido V., Juan A., Mendoza I., Pablo, Villegas S., Luis M. (2009)
Revista Colombiana de Matemáticas
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Wisbauer, Robert, Yousif, Mohamed F., Zhou, Yiqiang (2002)
Beiträge zur Algebra und Geometrie
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Ivan Penkov, Vera Serganova (2012)
Annales de l’institut Fourier
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Let be a complex reductive Lie algebra and be any reductive in subalgebra. We call a -module bounded if the -multiplicities of are uniformly bounded. In this paper we initiate a general study of simple bounded -modules. We prove a strong necessary condition for a subalgebra to be bounded (Corollary 4.6), to admit an infinite-dimensional simple bounded -module, and then establish a sufficient condition for a subalgebra to be bounded (Theorem 5.1). As a result we are...
Ali, Majid M., Smith, David J. (2001)
Beiträge zur Algebra und Geometrie
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Wongwai, S. (2002)
Acta Mathematica Universitatis Comenianae. New Series
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Oneto R., Ángel V. (1996)
Divulgaciones Matemáticas
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Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama (2013)
Formalized Mathematics
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In this article, we formalize a free Z-module and its property. In particular, we formalize the vector space of rational field corresponding to a free Z-module and prove formally that submodules of a free Z-module are free. Z-module is necassary for lattice problems - LLL (Lenstra, Lenstra and Lov´asz) base reduction algorithm and cryptographic systems with lattice [20]. Some theorems in this article are described by translating theorems in [11] into theorems of Z-module, however their...