A Novel Method for Designing Substitution Boxes Based on Mobius Group

It is as of now realized that the use of web in the present period is expanding very quickly, numerous users are sharing open and private data over web/internet. They need to ensure security of information. Encryption is one of the most significant viewpoint which is helpful to verify secret data. In encryption, cryptography assume a significant role. There are many algorithms accessible to shield information from unapproved access. In this work, we will use the strategy for cryptography to build up an algorithm which is increasingly viable for correspondence and information trade. In this study, we use a procedure to assemble new substitution boxes (S-boxes) parallel to coset diagram for the action of Mobius group [1] on projective line over . Afterward, the analyses of constructed S-boxes are performed on the basis of algebraic and statistical analysis.

Semigroups of Isometries of the Hyperbolic Plane

Motivated by a problem on the dynamics of compositions of plane hyperbolic isometries, we prove several fundamental results on semigroups of isometries, thought of as real Möbius transformations. We define a semigroup $S$ of Möbius transformations to be semidiscrete if the identity map is not an accumulation point of $S$. We say that $S$ is inverse free if it does not contain the identity element. One of our main results states that if $S$ is a semigroup generated by some finite collection $\mathcal{F}$ of Möbius transformations, then $S$ is semidiscrete and inverse free if and only if every sequence of the form $F_n=f_1\dotsb f_n$, where $f_n\in \mathcal{F}$, converges pointwise on the upper half-plane to a point on the ideal boundary, where convergence is with respect to the chordal metric on the extended complex plane. We fully classify all two-generator semidiscrete semigroups and include a version of Jørgensen’s inequality for semigroups. We also prove theorems that have familiar counterparts in the theory of Fuchsian groups. For instance, we prove that every semigroup is one of four standard types: elementary, semidiscrete, dense in the Möbius group, or composed of transformations that fix some nontrivial subinterval of the extended real line. As a consequence of this theorem, we prove that, with certain minor exceptions, a finitely generated semigroup $S$ is semidiscrete if and only if every two-generator semigroup contained in $S$ is semidiscrete. After this we examine the relationship between the size of the “group part” of a semigroup and the intersection of its forward and backward limit sets. In particular, we prove that if $S$ is a finitely generated nonelementary semigroup, then $S$ is a group if and only if its two limit sets are equal. We finish by applying some of our methods to address an open question of Yoccoz.

The construction of the mKdV cyclic symmetric N-soliton solution by the Bäcklund transformation

We study group theoretical structures of the mKdV equation. The Schwarzian-type mKdV equation has the global Möbius group symmetry. The Miura transformation makes a connection between the mKdV equation and the KdV equation. We find the special local Möbius transformation on the mKdV one-soliton solution which can be regarded as the commutative KdV Bäcklund transformation and can generate the mKdV cyclic symmetric N-soliton solution. In this algebraic construction to obtain multi-soliton solutions, we could observe the addition formula.

Novel Representation of the General Fuchsian and Heun Equations and their Solutions

In the present article we introduce and study a novel type of solutions to the general Heun's equation. Our approach is based on the symmetric form of the Heun's differential equation yielded by development of the Papperitz-Klein symmetric form of the Fuchsian equations with an arbitrary number N≥4 of regular singular points. We derive the symmetry group of these equations which turns to be a proper extension of the Mobius group. We also introduce and study new series solutions of the proposed in the present paper symmetric form of the general Heun's differential equation (N=4) which treats simultaneously and on an equal footing all singular points.