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

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 <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <msup> <mrow> <mi>Q</mi> </mrow> <mrow> <mo>′</mo> <mo>′</mo> </mrow> </msup> <mfenced open="(" close=")"> <mrow> <msqrt> <mrow> <mi>m</mi> </mrow> </msqrt> </mrow> </mfenced> <mo>=</mo> <mfenced open="{" close="}"> <mrow> <mi>η</mi> <mo>/</mo> <mi>s</mi> <mo>:</mo> <mi>η</mi> <mo>∈</mo> <msup> <mrow> <mi>Q</mi> </mrow> <mrow> <mo>∗</mo> </mrow> </msup> <mfenced open="(" close=")"> <mrow> <msqrt> <mrow> <mi>m</mi> </mrow> </msqrt> </mrow> </mfenced> <mo>,</mo> <mi>s</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </mfenced> </math> under the action of group <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <mi>H</mi> <mo>=</mo> <mo>&lt;</mo> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mo>′</mo> </mrow> </msup> <mo>,</mo> <msup> <mrow> <mi>y</mi> </mrow> <mrow> <mo>′</mo> </mrow> </msup> <mo>:</mo> <msup> <mrow> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mo>′</mo> </mrow> </msup> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <msup> <mrow> <mi>y</mi> </mrow> <mrow> <mo>′</mo> </mrow> </msup> </mrow> <mrow> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>&gt;</mo> </math> , where <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3"> <mi>m</mi> </math> is square free integer and <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4"> <msup> <mrow> <mi>Q</mi> </mrow> <mrow> <mo>∗</mo> </mrow> </msup> <mfenced open="(" close=")"> <mrow> <msqrt> <mrow> <mi>m</mi> </mrow> </msqrt> </mrow> </mfenced> <mo>=</mo> <mfenced open="{" close="}"> <mrow> <mfenced open="(" close=")"> <mrow> <msup> <mrow> <mi>a</mi> </mrow> <mrow> <mo>′</mo> </mrow> </msup> <mo>+</mo> <msqrt> <mrow> <mi>m</mi> </mrow> </msqrt> </mrow> </mfenced> <mo>/</mo> <msup> <mrow> <mi>c</mi> </mrow> <mrow> <mo>′</mo> </mrow> </msup> <mo>,</mo> <mfenced open="(" close=")"> <mrow> <msup> <mrow> <mi>a</mi> </mrow> <mrow> <mo>′</mo> </mrow> </msup> <mo>,</mo> <mfenced open="(" close=")"> <mrow> <msup> <mrow> <msup> <mrow> <mi>a</mi> </mrow> <mrow> <mo>′</mo> </mrow> </msup> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>m</mi> </mrow> </mfenced> <mo>/</mo> <msup> <mrow> <mi>c</mi> </mrow> <mrow> <mo>′</mo> </mrow> </msup> <msup> <mrow> <mi>c</mi> </mrow> <mrow> <mo>′</mo> </mrow> </msup> </mrow> </mfenced> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msup> <mrow> <mi>c</mi> </mrow> <mrow> <mo>′</mo> </mrow> </msup> <mo>≠</mo> <mn>0</mn> </mrow> </mfenced> </math> . We discuss the type and reduced cardinality of the orbit <math xmlns="http://www.w3.org/1998/Math/MathML" id="M5"> <msup> <mrow> <mi>Q</mi> </mrow> <mrow> <mo>′</mo> <mo>′</mo> </mrow> </msup> <mfenced open="(" close=")"> <mrow> <msqrt> <mrow> <mi>p</mi> </mrow> </msqrt> </mrow> </mfenced> </math> . By using the notion of congruence, we give the general form of reduced numbers (RNs) in particular orbits under certain conditions on prime <math xmlns="http://www.w3.org/1998/Math/MathML" id="M6"> <mi>p</mi> </math> . Further, we classify that for a reduced number <math xmlns="http://www.w3.org/1998/Math/MathML" id="M7"> <mi>r</mi> </math> whether <math xmlns="http://www.w3.org/1998/Math/MathML" id="M8"> <mo>−</mo> <mi>r</mi> <mo>,</mo> <mover accent="true"> <mi>r</mi> <mo stretchy="true">¯</mo> </mover> <mo>,</mo> <mo>−</mo> <mover accent="true"> <mi>r</mi> <mo stretchy="true">¯</mo> </mover> </math> lying in orbit or not. AMS Mathematics subject classification (2010): 05C25, 20G401.