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Tomasz Brzeziński - One of the best experts on this subject based on the ideXlab platform.
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Actions of Hopf Quasigroups
Communications in Algebra, 2012Co-Authors: Tomasz Brzeziński, Zhengming JiaoAbstract:Definitions of actions of Hopf Quasigroups are discussed in the context of Long dimodules and smash products. In particular, Long dimodules are defined for Hopf Quasigroups and coQuasigroups, and solutions to Militaru's 𝒟-equation are constructed. A necessary compatibility condition between action and multiplication of a Hopf quasigroup acting on its quasimodule Hopf quasigroup for a smash product construction is derived. This necessary condition takes the form of the associativity of the action up to an antipodal operation. In the case of a Hopf quasigroup with a bijective antipode it is simply the associativity of the action, a condition assumed a priori by Klim and Majid in their original construction of smash products of Hopf Quasigroups.
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Actions of Hopf Quasigroups
arXiv: Quantum Algebra, 2010Co-Authors: Tomasz Brzeziński, Zhengming JiaoAbstract:Definitions of actions of Hopf Quasigroups are discussed in the context of Long dimodules and smash products. In particular, Long dimodules are defined for Hopf Quasigroups and coQuasigroups, and solutions to Militaru's D-equation are constructed. A necessary compatibility condition between action and multiplication of a Hopf quasigroup acting on its quasimodule Hopf quasigroup for a smash product construction is derived.
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hopf modules and the fundamental theorem for hopf co Quasigroups
International Electronic Journal of Algebra, 2010Co-Authors: Tomasz BrzezińskiAbstract:The notion of a Hopf module over a Hopf (co)quasigroup is intro- duced and a version of the fundamental theorem for Hopf (co)Quasigroups is proven. Mathematics Subject Classiflcation (2010): 16T05, 16T15
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hopf modules and the fundamental theorem for hopf co Quasigroups
arXiv: Quantum Algebra, 2009Co-Authors: Tomasz BrzezińskiAbstract:The notion of a Hopf module over a Hopf (co)quasigroup is introduced and a version of the fundamental theorem for Hopf (co)Quasigroups is proven.
Shahn Majid - One of the best experts on this subject based on the ideXlab platform.
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Comment.Math.Univ.Carolin. 51,2 (2010) 287–304 287
2016Co-Authors: Bicrossproduct Hopf Quasigroups, Jennifer Klim, Shahn MajidAbstract:Abstract. We recall the notion of Hopf Quasigroups introduced previously by the authors. We construct a bicrossproduct Hopf quasigroup kM⊲◭k(G) from every group X with a finite subgroup G ⊂ X and IP quasigroup transversal M ⊂ X subject to certain conditions. We identify the octonions quasigroup GO as transversal in an order 128 group X with subgroup Z3 2 and hence obtain
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Bicrossproduct Hopf Quasigroups
2010Co-Authors: Jennifer Klim, Shahn MajidAbstract:We recall the notion of Hopf Quasigroups introduced previously by the authors. We construct a bicrossproduct Hopf quasigroup $kM {\triangleright\blacktriangleleft} k(G)$ from every group $X$ with a finite subgroup $G\subset X$ and IP quasigroup transversal $M\subset X$ subject to certain conditions. We identify the octonions quasigroup $G_{\mathbb O}$ as transversal in an order 128 group $X$ with subgroup $\mathbb Z_2^3$ and hence obtain a Hopf quasigroup $kG_{\mathbb O}{{>\blacktriangleleft}} k(\mathbb Z_2^3)$ as a particular case of our construction.
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Bicrossproduct Hopf Quasigroups
arXiv: Quantum Algebra, 2009Co-Authors: Jennifer Klim, Shahn MajidAbstract:We recall the notion of Hopf Quasigroups introduced previously. We construct a bicrossproduct Hopf quasigroup $kM\bicross k(G)$ from every group $X$ with a finite subgroup $G\subset X$ and IP quasigroup transversal $M\subset X$ subject to certain conditions. We identify the octonions quasigroup $G_O$ as transversal in an order 128 group $X$ with subgroup $Z_2^3$ and hence obtain a Hopf quasigroup $kG_O\lcocross k(Z_2^3)$ as a particular case of our construction.
Danilo Gligoroski - One of the best experts on this subject based on the ideXlab platform.
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Simulation of a quasigroup error-detecting linear code
2015 38th International Convention on Information and Communication Technology Electronics and Microelectronics (MIPRO), 2015Co-Authors: N. Ilievska, Danilo GligoroskiAbstract:We analyze an error-detecting code with a fixed length redundancy, based on Quasigroups. We obtain experimental results for the probability of undetected errors when Quasigroups of different order are used for coding and conclude how the probability of undetected errors depends on the order of the quasigroup used for coding and the parameter of the code.
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error detecting code using linear Quasigroups
International Conference on ICT Innovations, 2014Co-Authors: N. Ilievska, Danilo GligoroskiAbstract:In this paper we consider an error-detecting code based on linear Quasigroups of order 2 q defined in the following way: The input block a 0 a 1...a n − 1 is extended into a block a 0 a 1...a n − 1 d 0 d 1...d n − 1, where redundant characters d 0 d 1...d n − 1 are defined with d i = a i *a i + 1*a i + 2, where * is a linear quasigroup operation and the operations in the indexes are modulo n. We give a proof that the probability of undetected errors is independent from the distribution of the characters in the input message. We also calculate the probability of undetected errors, if Quasigroups of order 8 are used. We found a class of Quasigroups of order 8 that have smallest probability of undetected errors, i.e. the Quasigroups which are the best for coding. We explain how the probability of undetected errors can be made arbitrary small.
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Quasigroups as boolean functions their equation systems and grobner bases
Gröbner Bases Coding and Cryptography, 2009Co-Authors: Danilo Gligoroski, Vesna Dimitrova, Smile MarkovskiAbstract:In this short note we represent Quasigroups of order 2 n as vector valued Boolean functions f:{0,1}2n →{0,1} n . The representation of finite Quasigroups as vector valued Boolean functions allows us systems of quasigroup equations to be solved by using Grobner bases.
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classification of Quasigroups by random walk on torus
Journal of Applied Mathematics and Computing, 2005Co-Authors: Smile Markovski, Danilo Gligoroski, Jasen MarkovskiAbstract:Quasigroups are algebraic structures closely related to Latin squares which have many different applications. There are several classifications of Quasigroups based on their algebraic properties. In this paper we propose another classification based on the properties of strings obtained by specific quasigroup transformations. More precisely, in our research we identified some quasigroup transformations which can be applied to arbitrary strings to produce pseudo random sequences. We performed tests for randomness of the obtained pseudo-random sequences by random walks on torus. The randomness tests provided an empirical classification of quasi-groups.
Ajith Abraham - One of the best experts on this subject based on the ideXlab platform.
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genetic algorithms evolving Quasigroups with good pseudorandom properties
International Conference on Computational Science and Its Applications, 2010Co-Authors: Vaclav Snasel, Eliska Ochodkova, Jan Platos, Pavel Kromer, Jiři Dvorský, Ajith AbrahamAbstract:Quasigroups are a well-known combinatorial design equivalent to more familiar Latin squares. Because all possible elements of a quasigroup occur with equal probability, it makes it an interesting tool for the application in computer security and for production of pseudorandom sequences. Prior implementations of Quasigroups were based on look-up table of the quasigroup, on system of distinct representatives etc. Such representations are infeasible for large Quasigroups. In contrast, presented analytic quasigroup can be implemented easily. It allows the generation of pseudorandom sequences without storing large amount of data (look-up table). The concept of isotopy enables consideration of many Quasigroups and genetic algorithms allow efficient search for good ones.
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testing quasigroup identities using product of sequence
DATESO, 2010Co-Authors: Eliska Ochodkova, Vaclav Snasel, Jiri Dvorský, Ajith AbrahamAbstract:Non-associative Quasigroups are well known combinatorial designs with many dierent applications. Many cryptographic algorithms based on Quasigroups primitives have been published. There are several classifications of Quasigroups based on their algebraic properties. In this paper we propose a new classification of Quasigroups based upon strings (product elements) obtained by a product of a sequence. It is shown in this paper that the more various results of the product elements, the less associative quasigroup.
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evolving Quasigroups by genetic algorithms
DATESO, 2010Co-Authors: Vaclav Snasel, Jiri Dvorský, Eliska Ochodkova, Jan Platos, Pavel Kromer, Ajith AbrahamAbstract:Quasigroups are a well-known combinatorial design equivalent to more familiar Latin squares. Because all possible elements of a quasigroup occur with equal probability, it makes it an interesting tool for the application in computer security and for production of pseudorandom sequences. Most implementations of Quasigroups are based on look-up table of the quasigroup, on system of distinct representatives etc. Such representations are infeasible for large Quasigroups. An analytic quasigroup is a recent concept that allows usage of certain Quasigroups without the need of look-up table. The concept of isotopy enables consideration of many Quasigroups and genetic algorithms allow efficient search for good ones. In this paper we describe analytic quasigroup and genetic algorithms for its optimization.
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testing the properties of large Quasigroups
International Conference on Ultra Modern Telecommunications, 2009Co-Authors: Eliska Ochodkova, Vaclav Snasel, Jiri Dvorský, Ajith AbrahamAbstract:With the growing importance of data security a growing effort to find new approaches to the cryptographic algorithms designs appears. One of the trends is to research the use of other algebraic structures than the traditional, such as a quasigroup. Quasigroups are equivalent to the more familiar Latin squares. There are many characteristics that must Quasigroups have from the cryptography point of view. They have to be non-commutative, non-associative, non-idempotent and so on. If one want to work with Quasigroups of a large order, effective methods of testing their properties are necessary. In this paper we present several experiments on various types of guasigroups and their results.
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searching for Quasigroups for hash functions with genetic algorithms
Nature and Biologically Inspired Computing, 2009Co-Authors: Vaclav Snasel, Jiri Dvorský, Eliska Ochodkova, Ajith Abraham, Jan Platos, Pavel KromerAbstract:In this study we discuss a method for evolution of Quasigroups with desired properties based on genetic algorithms. Quasigroups are a well-known combinatorial design equivalent to the more familiar Latin squares. One of their most important properties is that all possible elements of certain quasigroup occur with equal probability. The Quasigroups are evolved within a framework of a simple hash function. Prior implementations of Quasigroups were based on look-up table of the quasigroup, which is infeasible for large Quasigroups. In contrast, analytic quasigroup can be implemented easily. It allows the evaluation of hash function without storing large amount of data (look-up table) and the concept of homotopy enables consideration of many quasigropus.
Jennifer Klim - One of the best experts on this subject based on the ideXlab platform.
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Comment.Math.Univ.Carolin. 51,2 (2010) 287–304 287
2016Co-Authors: Bicrossproduct Hopf Quasigroups, Jennifer Klim, Shahn MajidAbstract:Abstract. We recall the notion of Hopf Quasigroups introduced previously by the authors. We construct a bicrossproduct Hopf quasigroup kM⊲◭k(G) from every group X with a finite subgroup G ⊂ X and IP quasigroup transversal M ⊂ X subject to certain conditions. We identify the octonions quasigroup GO as transversal in an order 128 group X with subgroup Z3 2 and hence obtain
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Bicrossproduct Hopf Quasigroups
2010Co-Authors: Jennifer Klim, Shahn MajidAbstract:We recall the notion of Hopf Quasigroups introduced previously by the authors. We construct a bicrossproduct Hopf quasigroup $kM {\triangleright\blacktriangleleft} k(G)$ from every group $X$ with a finite subgroup $G\subset X$ and IP quasigroup transversal $M\subset X$ subject to certain conditions. We identify the octonions quasigroup $G_{\mathbb O}$ as transversal in an order 128 group $X$ with subgroup $\mathbb Z_2^3$ and hence obtain a Hopf quasigroup $kG_{\mathbb O}{{>\blacktriangleleft}} k(\mathbb Z_2^3)$ as a particular case of our construction.
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Bicrossproduct Hopf Quasigroups
arXiv: Quantum Algebra, 2009Co-Authors: Jennifer Klim, Shahn MajidAbstract:We recall the notion of Hopf Quasigroups introduced previously. We construct a bicrossproduct Hopf quasigroup $kM\bicross k(G)$ from every group $X$ with a finite subgroup $G\subset X$ and IP quasigroup transversal $M\subset X$ subject to certain conditions. We identify the octonions quasigroup $G_O$ as transversal in an order 128 group $X$ with subgroup $Z_2^3$ and hence obtain a Hopf quasigroup $kG_O\lcocross k(Z_2^3)$ as a particular case of our construction.
