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Submodule of free Z-module

100%

Futa Y., Okazaki H., Shidama Y.

Formalized Mathematics

2013

tom 21

nr 4

273-282

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 proofs are different.

Rank of Submodule, Linear Transformations and Linearly Independent Subsets of Z-module

70%

Nakasho K., Futa Y., Okazaki H., Shidama Y.

Formalized Mathematics

2014

tom 22

nr 3

189-198

In this article, we formalize some basic facts of Z-module. In the first section, we discuss the rank of submodule of Z-module and its properties. Especially, we formally prove that the rank of any Z-module is equal to or more than that of its submodules, and vice versa, and that there exists a submodule with any given rank that satisfies the above condition. In the next section, we mention basic facts of linear transformations between two Z-modules. In this section, we define homomorphism between two Z-modules and deal with kernel and image of homomorphism. In the last section, we formally prove some basic facts about linearly independent subsets and linear combinations. These formalizations are based on [9](p.191-242), [23](p.117-172) and [2](p.17-35).

Torsion Z-module and Torsion-free Z-module

70%

Futa Y., Okazaki H., Nakasho K., Shidama Y.

Formalized Mathematics

2014

tom 22

nr 4

277-289

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].

Ograniczanie wyników

3 Formalized Mathematics

3 Futa Y.

3 Okazaki H.

3 Shidama Y.

2 Nakasho K.

2 2014

1 2013

Wyniki wyszukiwania

Submodule of free Z-module

Rank of Submodule, Linear Transformations and Linearly Independent Subsets of Z-module

Torsion Z-module and Torsion-free Z-module