Classification of H-Circuits Having Different Length and Reduced Length of M-Circuits on Quadratic Irrational Numbers

The construction of circuits formed by reduced quadratic irrational numbers (RQINs) under the action of Mobius groups has attracted growing attention due to their deep algebraic structure and wide range of applications. Such orbits and circuits play a significant role in modern cryptographic systems, particularly in the design of robust substitution boxes (S-boxes), secure data encryption protocols, and image processing algorithms. The main objective of this novel study is to classify the types of H-circuits with different lengths contained in H-orbits ηH, where η is a RQIN and H is a Hecke group. For a specific η, the circuits of different lengths may contain η,η¯,−η and −η¯ either lie in one circuit or a different circuit of the same orbit. Also, we discuss the behavior of RQINs in the coset diagrams under the action of group M=u,f:u2=f6=1. Furthermore, the general form of reduced numbers in specific orbits under certain circumstances on prime p⌣ is investigated by applying the concept of congruence. Finally, special attention is given to the classification of M-circuits of length two in M-orbits ηM.