A computational algebraic geometry approach to analyze pseudo-random sequences based on Latin squares

Latin squares are used as scramblers on symmetric-key algorithms that generate pseudo-random sequences of the same length. The robustness and effectiveness of these algorithms are respectively based on the extremely large key space and the appropriate choice of the Latin square under consideration....

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Bibliographic Details
Published in:Advances in computational mathematics Vol. 45; no. 4; pp. 1769 - 1792
Main Authors: Falcón, Raúl M., Álvarez, Víctor, Gudiel, Félix
Format: Journal Article
Language:English
Published: New York Springer US 01.08.2019
Springer Nature B.V
Subjects:
Algebra
Algorithms
Arrays
Computational mathematics
Computational Mathematics and Numerical Analysis
Computational Science and Engineering
Invariants
Isomorphism
Latin square design
Mathematical and Computational Biology
Mathematical Modeling and Industrial Mathematics
Mathematics
Mathematics and Statistics
Pseudorandom sequences
Visualization
20N05
Affine algebraic set
14G50
Partial Latin square
Isomorphism
Image pattern
05B15
Symmetric-key algorithm
ISSN:1019-7168, 1572-9044
Online Access:Get full text
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Summary:Latin squares are used as scramblers on symmetric-key algorithms that generate pseudo-random sequences of the same length. The robustness and effectiveness of these algorithms are respectively based on the extremely large key space and the appropriate choice of the Latin square under consideration. It is also known the importance that isomorphism classes of Latin squares have to design an effective algorithm. In order to delve into this last aspect, we improve in this paper the efficiency of the known methods on computational algebraic geometry to enumerate and classify partial Latin squares. Particularly, we introduce the notion of affine algebraic set of a partial Latin square L = ( l i j ) of order n over a field K as the set of zeros of the binomial ideal 〈 x i x j − x l ij : ( i , j ) is a non-empty cell in L 〉 ⊆ K [ x 1 , … , x n ] . Since isomorphic partial Latin squares give rise to isomorphic affine algebraic sets, every isomorphism invariant of the latter constitutes an isomorphism invariant of the former. In particular, we deal computationally with the problem of deciding whether two given partial Latin squares have either the same or isomorphic affine algebraic sets. To this end, we introduce a new pair of equivalence relations among partial Latin squares: being partial transpose and being partial isotopic.
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ISSN:1019-7168
1572-9044
DOI:10.1007/s10444-018-9654-0