Coset Diagram Research Articles

Number of Distinct Fragments in Coset Diagrams for P S L 2 , Z

Coset diagrams [1, 2] are used to demonstrate the graphical representation of the action of the extended modular group P G L 2 , Z over P L F q = F q ⋃ ∞ . In these sorts of graphs, a closed path of edges and triangles is known as a circuit, and a fragment is emerged by the connection of two or more circuits. The coset diagram evolves through the joining of these fragments. If one vertex of the circuit is fixed by a x ρ 1 a x − 1 ρ 2 a x ρ 3 ⋯ a x − 1 ρ k ∈ P S L 2 , Z , then this circuit is termed to be a length – k circuit, denoted by ρ 1 , ρ 2 , ρ 3 , ⋯ , ρ k . In this study, we consider two circuits of length − 6 as Ω 1 = α 1 , α 2 , α 3 , α 4 , α 5 , α 6 and Ω 2 = β 1 , β 2 , β 3 , β 4 , β 5 , β 6 with the vertical axis of symmetry that is α 2 = α 6 , α 3 = α 5 and β 2 = β 6 , β 3 = β 5 . It is supposed that Ω is a fragment formed by joining Ω 1 and Ω 2 at a certain point. The condition for existence of a fragment is given in [3] in the form of a polynomial in Z z . If we change the pair of vertices and connect them, then the resulting fragment and the fragment Ω may coincide. In this article, we find the total number of distinct fragments by joining all the vertices of Ω 1 with the vertices of Ω 2 provided the condition β 4 < β 3 < β 2 < β 1 < α 4 < α 3 < α 2 < α 1 .

Construction of nonlinear component of block cipher using coset graph

<abstract> <p>In recent times, the research community has shown interest in information security due to the increasing usage of internet-based mobile and web applications. This research presents a novel approach to constructing the nonlinear component or Substitution Box (S-box) of block ciphers by employing coset graphs over the Galois field. Cryptographic techniques are employed to enhance data security and address current security concerns and obstacles with ease. Nonlinear component is a keystone of cryptography that hides the association between plaintext and cipher-text. Cryptographic strength of nonlinear component is directly proportional to the data security provided by the cipher. This research aims to develop a novel approach for construction of dynamic S-boxes or nonlinear components by employing special linear group $ PSL(2, \mathbb{Z}) $ over the Galois Field $ GF\left({2}^{10}\right) $. The vertices of coset diagram belong to $ GF\left({2}^{10}\right) $ and can be expressed as powers of α, where α represents the root of an irreducible polynomial $ p\left(x\right) = {x}^{10}+{x}^{3}+1 $. We constructed several nonlinear components by using $ {GF}^{*}\left({2}^{10}\right) $. Furthermore, we have introduced an exceptionally effective algorithm for optimizing nonlinearity, which significantly enhances the cryptographic properties of the nonlinear component. This algorithm leverages advanced techniques to systematically search for and select optimal S-box designs that exhibit improved resistance against various cryptographic attacks.</p> </abstract>

Counting of Distinct Equivalence Classes of Circuits in PSL2, Z‐Space

Graham Higman was the first who studied the transitive actions of the extended modular group over PL(Fq) = Fq∪{∞} graphically and named it as coset diagram. In these sorts of graphs, a closed path of edges and triangles is known as a circuit. Coset diagrams evolve through the joining of these circuits. In a coset diagram, a circuit is termed as a length‐l circuit if its one vertex is fixed by , and it is denoted by (π1, π2, π3, …, πl). In this study, we shall formulate combinatorial sequences and find the number of distinct equivalence classes of a length‐6 circuit (π1, π2, π3, π4, π5, π6) for a fixed number of triangle Δ of class Π.

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.

A highly nonlinear substitution-box (S-box) design using action of modular group on a projective line over a finite field.

Cryptography is commonly used to secure communication and data transmission over insecure networks through the use of cryptosystems. A cryptosystem is a set of cryptographic algorithms offering security facilities for maintaining more cover-ups. A substitution-box (S-box) is the lone component in a cryptosystem that gives rise to a nonlinear mapping between inputs and outputs, thus providing confusion in data. An S-box that possesses high nonlinearity and low linear and differential probability is considered cryptographically secure. In this study, a new technique is presented to construct cryptographically strong 8×8 S-boxes by applying an adjacency matrix on the Galois field GF(28). The adjacency matrix is obtained corresponding to the coset diagram for the action of modular group on a projective line PL(F7) over a finite field F7. The strength of the proposed S-boxes is examined by common S-box tests, which validate their cryptographic strength. Moreover, we use the majority logic criterion to establish an image encryption application for the proposed S-boxes. The encryption results reveal the robustness and effectiveness of the proposed S-box design in image encryption applications.

On circuit structure of PSL (2, Z)-orbits of length eight

Background/Objectives: Modular group or PSL(2,Z) is a well-known group of the non-singular square matrices of order two by two with unit determinant. An Action of this group on real quadratic fields is represented by coset diagrams consisting of closed paths known as circuits. For a particular value of n , one or more than one circuits combine to form an orbit. The length of a circuit identifies all the circuits of an orbit. The main objective of this study is to explore all the circuits of length eight and their corresponding orbits. Methods: Some circuit equivalent properties, the group theoretical approach, along with the statistical methodology, is adopted to classify the orbits containing circuits of length eight. We also use already discovered results related to circuits of lengths two, four, and six to formulate the basis for our new results of length eight. Findings: We have discovered all the equivalence classes for the circuits of length eight, which are twenty-one in number. For a particular reduced number α, circuits of length eight can have all four; α,−α, its algebraic conjugateα¯ and − α¯ , either in one circuit or (α) G = (−α) G with (−α) G = (−α¯) G or (α) G = (α¯) G with (−α) G = (−α) G or (α) G = (−α) G with (−α) G = (α) G , depending upon the equivalent class of the circuit. Moreover, we have introduced reduced positions and G-midway and discovered that for any reduced numbers αi starting from α1 , reduced positions have a recurring pattern α2,−α3,−α4,α5, and so on. Applications: PSL(2, Z) orbits are entirely classified and drawn, along with the cyclically equivalent circuits of length eight. Keywords: Modular group; coset diagram; reduced numbers; G-midway; reduced positions

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 diagrams of the modular group and continued fractions

The coset diagram for each orbit under the action of the modular group on Q(n)⁎=Q(n)∪{∞} contains a circuit Ci. For any α∈Q(n), the path leading to the circuit Ci and the circuit itself are obtained through continued fractions in this paper. We show that the structure of the continued fractions of a reduced quadratic irrational element is weaved with the structure or type of the circuit. The three types of circuits of the action of V4 on Q(n)⁎ are also interconnected with the structure of continued fractions. The action of the modular group on Q(5)⁎ is chosen specifically because a circuit of it is related to the ratio of the Fibonacci numbers being the solution to the continued fractions of the golden ratio.

On contraction of vertices of the circuits in coset diagrams for $${\varvec{PSL}}\varvec{(2}, \pmb {\mathbb {Z}} \varvec{)}$$ PSL ( 2 , Z )

Coset diagrams for the action of $$ PSL (2,\mathbb {Z})$$ on real quadratic irrational numbers are infinite graphs. These graphs are composed of circuits. When modular group acts on projective line over the finite field $$ F_{q}$$ , denoted by $$ PL ( F_{q}) $$ , vertices of the circuits in these infinite graphs are contracted and ultimately a finite coset diagram emerges. Thus the coset diagrams for $$ PL ( F_{q}) $$ is composed of homomorphic images of the circuits in infinite coset diagrams. In this paper, we consider a circuit in which one vertex is fixed by $$( xy) ^{m_{1}}( xy^{-1}) ^{m_{2}}$$ , that is, $$( m_{1},m_{2}) $$ . Let $$\alpha $$ be the homomorphic image of $$( m_{1},m_{2}) $$ obtained by contracting a pair of vertices v, u of $$ ( m_{1},m_{2}) $$ . If we change the pair of vertices and contract them, it is not necessary that we get a homomorphic image different from $$ \alpha $$ . In this paper, we answer the question: how many distinct homomorphic images are obtained, if we contract all the pairs of vertices of $$( m_{1},m_{2}) ?$$ We also mention those pairs of vertices, which are ‘important’. There is no need to contract the pairs, which are not mentioned as ‘important’. Because, if we contract those, we obtain a homomorphic image, which we have already obtained by contracting ‘important’ pairs.

Some studies on algebraic integers in Q(i, √ 3) by using coset diagram

Some studies on algebraic integers in Q(i, √ 3) by using coset diagram

$$PSL\left( 2,7\right) $$ P S L 2 , 7 and carbon allotrope D168 Schwarzite

We investigate actions of the modular group $$PSL(2, \mathbb {Z} )$$ on the projective line over finite fields $$PL(\mathbf {F}_{7^{n}})$$ and find interesting relation between the coset diagram of orbits and the carbon allotrope with negative curvature D168 Shewarzite. We also highlighted some topological aspects of these diagrams.

Actions of [formula omitted] on projective line

Each conjugacy class of actions of the triangle group Δ(3,n,k) over the projective line PL(Fq) can be represented by a coset diagram D(θ,q), where θ∈Fq and q is a prime number. In this paper, we have considered conjugacy classes which arise from the actions of Δ(3,n,k)=〈r,s:r3=sn=(rs)k=1〉 over PL(Fq), where Fq is finite field. The points of PL(Fq) are the elements of Fq together with the additional point ∞.

A Novel Construction of Substitution Box Involving Coset Diagram and a Bijective Map

The substitution box is a basic tool to convert the plaintext into an enciphered format. In this paper, we use coset diagram for the action of PSL(2,Z) on projective line over the finite field GF29 to construct proposed S-box. The vertices of the cost diagram are elements of GF29 which can be represented by powers of α, where α is the root of irreducible polynomial px=x9+x4+1 over Z2. Let GF⁎29 denote the elements of GF29 which are of the form of even powers of α. In the first step, we construct a 16×16 matrix with the elements of GF⁎29 in a specific order, determined by the coset diagram. Next, we consider h:GF⁎29⟶GF28 defined by hα2n=ωn to destroy the structure of GF28. In the last step, we apply a bijective map g on each element of the matrix to evolve proposed S-box. The ability of the proposed S-box is examined by different available algebraic and statistical analyses. The results are then compared with the familiar S-boxes. We get encouraging statistics of the proposed box after comparison.

Homomorphic Images of Circuits in $$\hbox {PSL}(2,{\mathbb {Z}})$$ PSL ( 2 , Z ) -Space

Each conjugacy class of actions of \(PGL\left( {2,{\mathbb {Z}}} \right) \) on the projective line over a finite field \(F_q \) denoted by \(PL\left( {F_q } \right) \), can be represented by a coset diagram \(D\left( {\theta ,q} \right) \), where \(\theta \in F_q \) and q is a prime power. The coset diagrams are composed of fragments, and the fragments are further composed of two or more circuits at a certain common point. Professor Graham Higman raised a question: for what values of q and \(\theta \), can a fragment \(\gamma \) be found in \(D\left( {\theta ,q} \right) ?\) Mushtaq in 1983 found that the condition for the existence of a fragment in \(D\left( {\theta ,q} \right) \) is a polynomial f in \({\mathbb {Z}}\left[ z \right] \). In this paper, we answer the question: how many polynomials are obtained from the fragments, composed by joining the circuits \(\left( {n_1 ,n_2 } \right) \) and \(\left( {m_1 ,m_2 } \right) \), where \(n_2 <n_1 <m_2 <m_1\), at all points of connection.

JOINING OF CIRCUITS IN PSL(2, ℤ)-SPACE

The coset diagrams are composed of fragments, and the fragments are further composed of circuits at a certain common point. A condition for the existence of a certain fragment <TEX>${\gamma}$</TEX> of a coset diagram in a coset diagram is a polynomial f in <TEX>${\mathbb{Z}}$</TEX>[z]. In this paper, we answer the question: how many polynomials are obtained from the fragments, evolved by joining the circuits (n, n) and (m, m), where n < m, at all points.

Equivalent pairs of words and points of connection.

Higman has defined coset diagrams for PGL(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 fragment γ in a coset diagram is a polynomial f in ℤ[𝔃], obtained by choosing a pair of words F[w i, w j] such that both w i and w j fix a vertex v in γ. 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.

A GRAPHICAL TECHNIQUE TO FIND THE PERMUTATION REPRESENTATION OF THE TRIANGLE GROUPS △(2, 3, k)

In this paper we have developed a graphical technique by which a suitably created fragment of a coset diagram for the action of PSL(2, ℤ) on projective lines over Galois fields can be used to obtain a complete coset diagram depicting finite homomorphic images of groups Δ(2, 3, k) = 〈x, y : x2 = y3 = (xy)k = 1〉.

Parametrization of Actions of the Automorphism Subgroup of a Cubic Tree on PL(F p )

The group is important because it is one of the seven finitely presented isomorphism types of subgroups of the full automorphism group (Aut(Γ3)) of a cubic tree Γ3. These seven groups act arc-transitively on the arcs of Γ3 with a finite vertex stabilizer. In this article, we have parametrized the actions of on the projective line over the finite field, PL(F p ), where p is the Pythagorean prime and has thus shown that there is only one coset diagram depicting the sole conjugacy class of these actions.

Pell Numbers, Pell–Lucas Numbers and Modular Group

We show that the matrix A(g), representing the element g = ((xy)2(xy2)2)m (m ≥ 1) of the modular group PSL(2,Z) = 〈x,y : x2 = y3 = 1〉, where [Formula: see text] and [Formula: see text], is a 2 × 2 symmetric matrix whose entries are Pell numbers and whose trace is a Pell–Lucas number. If g fixes elements of [Formula: see text], where d is a square-free positive number, on the circuit of the coset diagram, then d = 2 and there are only four pairs of ambiguous numbers on the circuit.