Orthogonal matrix and its application in Bloom’s threshold scheme

Applying the Gram–Schmidt process (also called Gram–Schmidt orthogonalization) to a matrix M ∈ G L ( n , R ) , set of n × n invertible matrices over the field of real numbers, with the usual inner product gives easily an orthogonal matrix. However, the orthogonality in the vector space F q k , where...

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Veröffentlicht in:Applicable algebra in engineering, communication and computing Jg. 30; H. 2; S. 147 - 160
Hauptverfasser: Mameri, Ahmed, Aissani, Amar
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Berlin/Heidelberg Springer Berlin Heidelberg 12.03.2019
Springer Nature B.V
Schlagworte:
Algorithms
Artificial Intelligence
Computer Hardware
Computer Science
Fields (mathematics)
Mathematical analysis
Matrix algebra
Matrix methods
Original Paper
Orthogonality
Real numbers
Symbolic and Algebraic Manipulation
Theory of Computation
MSC 94A62
MSC 65F25
MSC 94A60
Matrix
Orthogonalization
MSC 15A33
Secret sharing
Cryptography
MSC 15A09
ISSN:0938-1279, 1432-0622
Online-Zugang:Volltext
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Zusammenfassung:Applying the Gram–Schmidt process (also called Gram–Schmidt orthogonalization) to a matrix M ∈ G L ( n , R ) , set of n × n invertible matrices over the field of real numbers, with the usual inner product gives easily an orthogonal matrix. However, the orthogonality in the vector space F q k , where F q is a binary finite field, is quite tricky as there are non-zero vectors which are orthogonal to themselves. For this reason the computational variants of Gram–Schmidt orthogonalization can fail. This paper presents an algorithm for constructing random orthogonal matrices over binary finite fields. The approach is inspired from the Gram–Schmidt procedure. Since the inverse of orthogonal matrix is easy to compute, the orthogonal matrices are used to construct a proactive variant of Bloom’s threshold secret sharing scheme.
Bibliographie:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:0938-1279
1432-0622
DOI:10.1007/s00200-018-0365-z