Equivalent Pairs of Words and Points of Connection

The Scientific World JOURNAL ◽

10.1155/2014/505496 ◽

2014 ◽

Vol 2014 ◽

pp. 1-8 ◽

Cited By ~ 1

Author(s):

Qaiser Mushtaq ◽

Abdul Razaq

Keyword(s):

Coset Diagrams

Higman has defined coset diagrams forPGL(2,Z). The coset diagrams are composed of fragments, and the fragments are further composed of two or more circuits. A condition for the existence of a certain fragmentγin a coset diagram is a polynomialfinZ[z], obtained by choosing a pair of wordsF[wi,wj]such that bothwiandwjfix a vertexvinγ. Two pairs of words are equivalent if and only if they have the same polynomial. In this paper, we find distinct pairs of words that are equivalent. We also show there are certain fragments, which have the same orientations as those of their mirror images.

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Construction of New S-Box Using Action of Quotient of the Modular Group for Multimedia Security

Security and Communication Networks ◽

10.1155/2019/2847801 ◽

2019 ◽

Vol 2019 ◽

pp. 1-13 ◽

Cited By ~ 5

Author(s):

Imran Shahzad ◽

Qaiser Mushtaq ◽

Abdul Razaq

Keyword(s):

Finite Field ◽

Modular Group ◽

Fibonacci Sequence ◽

Projective Line ◽

Multimedia Security ◽

New Technique ◽

Substitution Box ◽

Nonlinear Component ◽

Coset Diagrams ◽

A New Technique

Substitution box (S-box) is a vital nonlinear component for the security of cryptographic schemes. In this paper, a new technique which involves coset diagrams for the action of a quotient of the modular group on the projective line over the finite field is proposed for construction of an S-box. It is constructed by selecting vertices of the coset diagram in a special manner. A useful transformation involving Fibonacci sequence is also used in selecting the vertices of the coset diagram. Finally, all the analyses to examine the security strength are performed. The outcomes of the analyses are encouraging and show that the generated S-box is highly secure.

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A family of conformally asymmetric Riemann surfaces

Glasgow Mathematical Journal ◽

10.1017/s0017089500032109 ◽

1997 ◽

Vol 39 (2) ◽

pp. 221-225 ◽

Cited By ~ 2

Author(s):

Brent Everitt

Keyword(s):

Automorphism Group ◽

Riemann Surfaces ◽

Coset Diagrams ◽

Free Subgroups ◽

Torsion Free

AbstractWe give explicit examples of asymmetric Riemann surfaces (that is, Riemann surfaces with trivial conformal automorphism group) for all genera g ≥ 3. The technique uses Schreier coset diagrams to construct torsion-free subgroups in groups of signature (0; 2,3,r) for certain values of r.

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Coset Diagram for the Action of Picard Group on $\mathbb{Q}(i,\sqrt{3})$

Algebra Colloquium ◽

10.1142/s1005386716000055 ◽

2016 ◽

Vol 23 (01) ◽

pp. 33-44 ◽

Cited By ~ 1

Author(s):

Qaiser Mushtaq ◽

Saima Anis

Keyword(s):

Positive Integer ◽

Finite Number ◽

Picard Group ◽

Closed Path ◽

Irrational Numbers ◽

Unique Pattern ◽

Graphical Interpretation ◽

Quadratic Irrational ◽

Coset Diagrams ◽

Real Quadratic

In this paper coset diagrams, propounded by Higman, are used to investigate the behavior of elements as words in orbits of the action of the Picard group Γ=PSL(2,ℤ[i]) on [Formula: see text]. Graphical interpretation of amalgamation of the components of Γ is also given. Some elements [Formula: see text] of [Formula: see text] and their conjugates [Formula: see text] over ℚ(i) have different signs in the orbits of the biquadratic field [Formula: see text] when acted upon by Γ. Such real quadratic irrational numbers are called ambiguous numbers. It is shown that ambiguous numbers in these coset diagrams form a unique pattern. It is proved that there are a finite number of ambiguous numbers in an orbit Γα, and they form a closed path which is the only closed path in the orbit Γα. We also devise a procedure to obtain ambiguous numbers of the form [Formula: see text], where b is a positive integer.

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On a result of Petersson concerning the modular group

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500015201 ◽

1981 ◽

Vol 87 (3-4) ◽

pp. 263-270 ◽

Cited By ~ 3

Author(s):

W. W. Stothers

Keyword(s):

Class Number ◽

Modular Group ◽

Asymptotic Estimate ◽

Coset Diagrams

SynopsisLet N(u, g, h) denote the number of subgroups of index u, genus g and parabolic class number h in the classical modular group. In 1974, Petersson proved that, for g = 0 and h = 1, 2, N(u, g, h) grows exponentially with u. We obtain a formula valid for all g and h. From this we derive an asymptotic estimate, showing that Petersson's result gives the correct type of growth. The proofs use coset diagrams.

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Number of Distinct Homomorphic Images in Coset Diagrams

Journal of Mathematics ◽

10.1155/2021/6669459 ◽

2021 ◽

Vol 2021 ◽

pp. 1-39

Author(s):

Muhammad Aamir ◽

Muhammad Awais Yousaf ◽

Abdul Razaq

Keyword(s):

Coding Theory ◽

Vertical Axis ◽

Closed Path ◽

Finite Graphs ◽

Nonlinearity Factor ◽

Axis Of Symmetry ◽

Coset Diagrams

The representation of the action of PGL 2 , Z on F t ∪ ∞ in a graphical format is labeled as coset diagram. These finite graphs are acquired by the contraction of the circuits in infinite coset diagrams. A circuit in a coset diagram is a closed path of edges and triangles. If one vertex of the circuit is fixed by p q Δ 1 p q − 1 Δ 2 p q Δ 3 … p q − 1 Δ m ∈ PSL 2 , Z , then this circuit is titled to be a length- m circuit, denoted by Δ 1 , Δ 2 , Δ 3 , … , Δ m . In this manuscript, we consider a circuit Δ of length 6 as Δ 1 , Δ 2 , Δ 3 , Δ 4 , Δ 5 , Δ 6 with vertical axis of symmetry, that is, Δ 2 = Δ 6 , Δ 3 = Δ 5 . Let Γ 1 and Γ 2 be the homomorphic images of Δ acquired by contracting the vertices a , u and b , v , respectively, then it is not necessary that Γ 1 and Γ 2 are different. In this study, we will find the total number of distinct homomorphic images of Δ by contracting its all pairs of vertices with the condition Δ 1 > Δ 2 > Δ 3 > Δ 4 . The homomorphic images are obtained in this way having versatile applications in coding theory and cryptography. One can attain maximum nonlinearity factor using this in the encryption process.

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Modular group acting on real quadratic fields

Bulletin of the Australian Mathematical Society ◽

10.1017/s000497270002685x ◽

1988 ◽

Vol 37 (2) ◽

pp. 303-309 ◽

Cited By ~ 16

Author(s):

Q. Mushtaq

Keyword(s):

Positive Integer ◽

Finite Number ◽

Modular Group ◽

Quadratic Fields ◽

Closed Path ◽

Irrational Numbers ◽

Real Quadratic Fields ◽

Quadratic Irrational ◽

Coset Diagrams ◽

Real Quadratic

Coset diagrams for the orbit of the modular group G = 〈x, y: x2 = y3 = 1〉 acting on real quadratic fields give some interesting information. By using these coset diagrams, we show that for a fixed value of n, a non-square positive integer, there are only a finite number of real quadratic irrational numbers of the form , where θ and its algebraic conjugate have different signs, and that part of the coset diagram containing such numbers forms a single circuit (closed path) and it is the only circuit in the orbit of θ.

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