# N-Dimensional Binary Vector Spaces

Kenichi Arai; Hiroyuki Okazaki

Formalized Mathematics (2013)

- Volume: 21, Issue: 2, page 75-81
- ISSN: 1426-2630

## Access Full Article

top## Abstract

top## How to cite

topKenichi Arai, and Hiroyuki Okazaki. "N-Dimensional Binary Vector Spaces." Formalized Mathematics 21.2 (2013): 75-81. <http://eudml.org/doc/267159>.

@article{KenichiArai2013,

abstract = {The binary set \{0, 1\} together with modulo-2 addition and multiplication is called a binary field, which is denoted by F2. The binary field F2 is defined in [1]. A vector space over F2 is called a binary vector space. The set of all binary vectors of length n forms an n-dimensional vector space Vn over F2. Binary fields and n-dimensional binary vector spaces play an important role in practical computer science, for example, coding theory [15] and cryptology. In cryptology, binary fields and n-dimensional binary vector spaces are very important in proving the security of cryptographic systems [13]. In this article we define the n-dimensional binary vector space Vn. Moreover, we formalize some facts about the n-dimensional binary vector space Vn.},

author = {Kenichi Arai, Hiroyuki Okazaki},

journal = {Formalized Mathematics},

keywords = {formalization of binary vector space; binary vector space},

language = {eng},

number = {2},

pages = {75-81},

title = {N-Dimensional Binary Vector Spaces},

url = {http://eudml.org/doc/267159},

volume = {21},

year = {2013},

}

TY - JOUR

AU - Kenichi Arai

AU - Hiroyuki Okazaki

TI - N-Dimensional Binary Vector Spaces

JO - Formalized Mathematics

PY - 2013

VL - 21

IS - 2

SP - 75

EP - 81

AB - The binary set {0, 1} together with modulo-2 addition and multiplication is called a binary field, which is denoted by F2. The binary field F2 is defined in [1]. A vector space over F2 is called a binary vector space. The set of all binary vectors of length n forms an n-dimensional vector space Vn over F2. Binary fields and n-dimensional binary vector spaces play an important role in practical computer science, for example, coding theory [15] and cryptology. In cryptology, binary fields and n-dimensional binary vector spaces are very important in proving the security of cryptographic systems [13]. In this article we define the n-dimensional binary vector space Vn. Moreover, we formalize some facts about the n-dimensional binary vector space Vn.

LA - eng

KW - formalization of binary vector space; binary vector space

UR - http://eudml.org/doc/267159

ER -

## References

top- [1] Jesse Alama. The vector space of subsets of a set based on symmetric difference. FormalizedMathematics, 16(1):1-5, 2008. doi:10.2478/v10037-008-0001-7.[Crossref]
- [2] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.
- [3] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990. Zbl06213858
- [4] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
- [5] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
- [6] Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.
- [7] Czesław Bylinski. Finite sequences and tuples of elements of a non-empty sets. FormalizedMathematics, 1(3):529-536, 1990.
- [8] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.
- [9] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
- [10] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
- [11] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.
- [12] Eugeniusz Kusak, Wojciech Leonczuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.
- [13] X. Lai. Higher order derivatives and differential cryptoanalysis. Communications andCryptography, pages 227-233, 1994. Zbl0840.94017
- [14] Robert Milewski. Associated matrix of linear map. Formalized Mathematics, 5(3):339-345, 1996.
- [15] J.C. Moreira and P.G. Farrell. Essentials of Error-Control Coding. John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, 2006.
- [16] Hiroyuki Okazaki and Yasunari Shidama. Formalization of the data encryption standard. Formalized Mathematics, 20(2):125-146, 2012. doi:10.2478/v10037-012-0016-y.[Crossref] Zbl1288.94079
- [17] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990.
- [18] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.
- [19] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.
- [20] Wojciech A. Trybulec. Subspaces and cosets of subspaces in vector space. FormalizedMathematics, 1(5):865-870, 1990.
- [21] Wojciech A. Trybulec. Linear combinations in vector space. Formalized Mathematics, 1 (5):877-882, 1990.
- [22] Wojciech A. Trybulec. Basis of vector space. Formalized Mathematics, 1(5):883-885, 1990.
- [23] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
- [24] Edmund Woronowicz. Many argument relations. Formalized Mathematics, 1(4):733-737, 1990.
- [25] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.
- [26] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.
- [27] Mariusz Zynel. The Steinitz theorem and the dimension of a vector space. FormalizedMathematics, 5(3):423-428, 1996.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.