scholarly journals Geometrical Representation of Automata over Some Abelian Groups

2014 ◽  
Vol 8 (1) ◽  
Author(s):  
Y.S. Gan ◽  
W.H. Fong ◽  
N.H. Sarmin ◽  
S. Turaev
Keyword(s):  
Abelian Groups ◽  
Finite Automata ◽  
Cayley Table ◽  
Geometrical Representation

One of the classic models of automata is finite automata, which determine whether a string belongs to a particular language or not. The string accepted by automata is said to be recognized by that automata. Another type of automata, so-called Watson-Crick automata, with two reading heads that work on double-stranded tapes using the complimentary relation. Finite automata over groups extend the possibilities of finite automata and allow studying the properties of groups using finite automata. In this paper, we consider finite automata over some Abelian groups ℤn and ℤn × ℤn. The relation of Cayley table to finite automata diagram is introduced in the paper. Some properties of groups ℤn and ℤn × ℤn in terms of automata are also presented in this paper.

scholarly journals Finite automata presentable abelian groups

2009 ◽  
Vol 161 (3) ◽  
pp. 458-467 ◽  
Author(s):  
André Nies ◽  
Pavel Semukhin
Keyword(s):  
Abelian Groups ◽  
Finite Automata

scholarly journals Completing Partial Transversals of Cayley Tables of Abelian Groups

10.37236/9386 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Jaromy Kuhl ◽  
Donald McGinn ◽  
Michael William Schroeder
Keyword(s):  
General Result ◽  
Abelian Groups ◽  
Cayley Table ◽  
Odd Order ◽  
Cayley Tables

In 2003 Grüttmüller proved that if $n\geqslant 3$ is odd, then a partial transversal of the Cayley table of $\mathbb{Z}_n$ with length $2$ is completable to a transversal. Additionally, he conjectured that a partial transversal of the Cayley table of $\mathbb{Z}_n$ with length $k$ is completable to a transversal if and only if $n$ is odd and either $n \in \{k, k + 1\}$ or $n \geqslant 3k - 1$. Cavenagh, Hämäläinen, and Nelson (in 2009) showed the conjecture is true when $k = 3$ and $n$ is prime. In this paper, we prove Grüttmüller’s conjecture for $k = 2$ and $k = 3$ by establishing a more general result for Cayley tables of Abelian groups of odd order.

scholarly journals Permutation Groups in Automata Diagrams

2013 ◽  
Vol 9 (1) ◽  
Author(s):  
Gan Yee Siang ◽  
Fong Wan Heng ◽  
Nor Haniza Sarmin ◽  
Sherzod Turaev
Keyword(s):  
Group Theory ◽  
Permutation Group ◽  
Finite Automata ◽  
Permutation Groups ◽  
Cayley Table ◽  
Classical Models

Automata act as classical models for recognition devices. From the previous researches, the classical models of automata have been used to scan strings and to determine the types of languages a string belongs to. In the study of automata and group theory, it has been found that the properties of a group can be recognized by the automata using the automata diagrams. There are two types of automata used to study the properties of a group, namely modified finite automata and modified Watson-Crick finite automata. Thus, in this paper, automata diagrams are constructed to recognize permutation groups using the data given by the Cayley table. Thus, the properties of permutation group are analyzed using the automaton diagram that has been constructed. Moreover, some theorems for the properties of permutation group in term of automata are also given in this paper.

scholarly journals Infinite Latin Squares: Neighbor Balance and Orthogonality

10.37236/8020 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Anthony B. Evans ◽  
Gage N. Martin ◽  
Kaethe Minden ◽  
M. A. Ollis
Keyword(s):  
Abelian Group ◽  
Infinite Order ◽  
Abelian Groups ◽  
Latin Squares ◽  
Infinite Group ◽  
Cayley Table ◽  
Orthogonal Latin Squares ◽  
Mutually Orthogonal Latin Squares ◽  
Complete Mapping ◽  
Infinite Abelian Group

Regarding neighbor balance, we consider natural generalizations of $D$-complete Latin squares and Vatican squares from the finite to the infinite. We show that if $G$ is an infinite abelian group with $|G|$-many square elements, then it is possible to permute the rows and columns of the Cayley table to create an infinite Vatican square. We also construct a Vatican square of any given infinite order that is not obtainable by permuting the rows and columns of a Cayley table.  Regarding orthogonality, we show that every infinite group $G$ has a set of $|G|$ mutually orthogonal orthomorphisms and hence there is a set of $|G|$ mutually orthogonal Latin squares based on $G$. We show that an infinite group $G$ with $|G|$-many square elements has a strong complete mapping; and, with some possible exceptions, infinite abelian groups have a strong complete mapping.

A novel cryptosystem based on abstract automata and Latin cubes

2015 ◽  
Vol 52 (2) ◽  
pp. 221-232
Author(s):  
Pál Dömösi ◽  
Géza Horváth
Keyword(s):  
Block Cipher ◽  
Finite Automata ◽  
Theoretical Background ◽  
Point Of View ◽  
Theoretical Point ◽  
Transition Matrices ◽  
Encryption And Decryption ◽  
Secret Keys ◽  
Automata Network ◽  
Fully Functioning

In this paper we introduce a novel block cipher based on the composition of abstract finite automata and Latin cubes. For information encryption and decryption the apparatus uses the same secret keys, which consist of key-automata based on composition of abstract finite automata such that the transition matrices of the component automata form Latin cubes. The aim of the paper is to show the essence of our algorithms not only for specialists working in compositions of abstract automata but also for all researchers interested in cryptosystems. Therefore, automata theoretical background of our results is not emphasized. The introduced cryptosystem is important also from a theoretical point of view, because it is the first fully functioning block cipher based on automata network.

Parallel Combining Different Approaches to Multi-pattern Matching for Fpga-based Security Systems

2017 ◽  
Vol 5 (1) ◽  
pp. 8-15
Author(s):  
Sergii Hilgurt ◽  
Keyword(s):  
Information Security ◽  
Pattern Matching ◽  
Intrusion Detection System ◽  
Detection System ◽  
Finite Automata ◽  
Network Intrusion Detection ◽  
Security Systems ◽  
Analytical Technique ◽  
Network Intrusion ◽  
Pros And Cons

The multi-pattern matching is a fundamental technique found in applications like a network intrusion detection system, anti-virus, anti-worms and other signature- based information security tools. Due to rising traffic rates, increasing number and sophistication of attacks and the collapse of Moore’s law, traditional software solutions can no longer keep up. Therefore, hardware approaches are frequently being used by developers to accelerate pattern matching. Reconfigurable FPGA-based devices, providing the flexibility of software and the near-ASIC performance, have become increasingly popular for this purpose. Hence, increasing the efficiency of reconfigurable information security tools is a scientific issue now. Many different approaches to constructing hardware matching circuits on FPGAs are known. The most widely used of them are based on discrete comparators, hash-functions and finite automata. Each approach possesses its own pros and cons. None of them still became the leading one. In this paper, a method to combine several different approaches to enforce their advantages has been developed. An analytical technique to quickly advance estimate the resource costs of each matching scheme without need to compile FPGA project has been proposed. It allows to apply optimization procedures to near-optimally split the set of pattern between different approaches in acceptable time.

Complexity and Universality of Iterated Finite Automata

2009 ◽  
Vol 18 (1) ◽  
pp. 145-158
Author(s):  
Jiang Zhang ◽  
Keyword(s):  
Finite Automata
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