Number of Distinct Homomorphic Images in 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.

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

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.

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

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.

Equivalent Pairs of Words and Points of Connection

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.

Adjacency Matrices of PSL(2,5) and Resemblance of Its Coset Diagrams with Fullerene C60

The action of PSL(2,ℤ) on PL(F5n) yields PSL(2,5). In this paper, we use the adjacency matrices of the abstract group PSL(2,5), eventually of fullerene C60, to draw the coset diagram and investigate various properties of the diagram with reference to the intransitivity of the action and the number of orbits.