Mobius Group Generated by Two Elements of Order 2, 4, and Reduced Quadratic Irrational Numbers

Abstract

The construction of circuits for the evolution of orbits and reduced quadratic irrational numbers under the action of Mobius groups have many applications like in construction of substitution box (s-box), strong-substitution box (s.s-box), image processing, data encryption, in interest for security experts, and other fields of sciences. In this paper, we investigate the behavior of reduced quadratic irrational numbers (RQINs) in the coset diagrams of the set $Q^{\prime \prime }\left(\sqrt{m}\right)=\left\{\eta /s:\eta \in Q^{\ast }\left(\sqrt{m}\right),s=1,2\right\}$ under the action of group $H=<x^{\prime },y^{\prime }:{x^{\prime }}^2={y^{\prime }}^4=1>$ , where $m$ is square free integer and $Q^{\ast }\left(\sqrt{m}\right)=\left\{\left(a^{\prime }+\sqrt{m}\right)/c^{\prime },\left(a^{\prime },\left({a^{\prime }}^2-m\right)/c^{\prime }{c}^{\prime }\right)=1,c^{\prime }\ne 0\right\}$ . We discuss the type and reduced cardinality of the orbit $Q^{\prime \prime }\left(\sqrt{p}\right)$ . By using the notion of congruence, we give the general form of reduced numbers (RNs) in particular orbits under certain conditions on prime $p$ . Further, we classify that for a reduced number $r$ whether $-r,\overline{r},-\overline{r}$ lying in orbit or not. AMS Mathematics subject classification (2010): 05C25, 20G401.