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 under the action of group , where is square free integer and . We discuss the type and reduced cardinality of the orbit . By using the notion of congruence, we give the general form of reduced numbers (RNs) in particular orbits under certain conditions on prime . Further, we classify that for a reduced number whether lying in orbit or not. AMS Mathematics subject classification (2010): 05C25, 20G401.
1. Introduction and Preliminaries
Groups are very helpful algebraic structures, carrying other algebraic structures on them. In abstract algebra, almost all typical structures are illustration of groups. The significance of groups was derived from their action on special structures or spaces. Cryptography is the technique of converting secret knowledge into information and a type of data pretending to reach its terminus without leaking data safely. Modern cryptography is classified into several branches. Although there are two main research fields such that symmetric and public key cryptography, the public and private keys are used in public key cryptography. The same keys are used at both ends to encrypt and decrypt data/information in symmetric key cryptography. It is well known that the substitution box is a standout in symmetric key cryptography. Shahzad et al. investigated the efficient technique for the construction of an -box by using action of a . For constructing an box, the vertices of the coset diagram are considered in a special way. In this way, the generated box is highly safe and also closely meeting the optimal values of the standard -box. In [1–6], the construction of substitution boxes based on coset graphs under the action of modular group has been discussed. In this piece of work, we investigate the structure of coset graphs under the action of modular group This work will be more helpful for construction of strong substitution boxes.
The -circuits of the set upon which the groups act are the equivalence classes of group action. Group can be written in the form of relations and generators as .
Assume that is nonsquare integer, then . In , Cayley was the first who introduced the technique of analysis of the groups through graphs. To investigate the action of infinite groups generated by finite elements on the infinite field by the coset diagram was first introduced by Higman in . A number is said to be ambiguous number (AN) if and have opposite signs. If is not ambiguous, then it is either totally positive or negative. The real quadratic irrational (RQI) numbers of the form , where and is nonzero integer, make the set represented as . A RQI number is known as RQIN if and . In this paper, we will denote the reduced number by If there are reduced numbers, then they are denoted by . For , in the orbit of , the count of RQINs is called the reduced length (RL), which is denoted by . These numbers are very less in and play a significant part in the circuit of an orbit. A circuit made of vertices of a square and edges existing in -orbits of , under the Mobius group in coset diagram. If is the type of a circuit, then it makes an element of group of This fixes some element exists in this circuit.
In [7, 8], Mushtaq and Aslam presented that there are only finite number of ambiguous numbers (ANS); in the coset diagram for the orbit of the ambiguous numbers form unique closed path. A cost diagram is introduced in [7, 8] to investigate the action of an infinite group on the projective line over real quadratic field (RQF). Malik and Zafar [9] investigated the properties of RQI numbers under the action of Zafar and Malik [10, 11] investigated the type and ambiguous lengths of the orbit of . Farkhanda and Qamar discussed the real quadratic irrational and action of . Razaq et al. [12, 13] investigated the circuits of length 4 in PSL, group theoretic construction of highly nonlinear substitution box, and its applications in image encryption. Ali and Malik [14, 15] discussed the classification of PSL-circuits and investigated the RQIN and types of -circuits with length four and six. Chen et al. [16] investigate reduced numbers which play an important role in the study of modular group action on the -subset. For more studies of group action on various field, we recommend reading of [17, 18]. The application of group theory and group action is obvious to encryption, physics, and mechanics to construct models and their structures [5, 19–21]. Mateen et al. [22–27] investigated the structure of power digraphs associated with the congruence , the partitioning of a set into two or more disjoint subsets of equal sums, and the symmetry of complete graphs and, moreover, investigated the importance of power digraphs in computer science. Alolaiyan et al. [28] discussed the homomorphic copies in coset graphs for the modular group.
The major contributions of this paper are given below. (1)This paper presents a graphical study of the action of a Mobius group on the real quadratic field (RQF)(2)We discuss the classification of -circuits and find the numbers that play vital role in the structure of -circuits(3)We investigate the RQINs and the types of -circuits with different length(4)We give the number of reduced numbers and their general form in different orbits for different values of under a certain condition on by using the concept of congruences
Theorem 1. [29]. If is symmetric continued fraction (CF) and , then
Theorem 2. [9]. The set is unchanged under the action of .
Theorem 3. [10]. Let . Then, splits into four -subsets. In particular, , , , and are at least four -orbits of .
Theorem 4. [9]. Let . Then, splits into three -subsets. In particular, , , and are at least three -orbits of .
Lemma 5. Every RQIN in is ambiguous number.
Theorem 6. [29]. If is symmetric continued fraction and if , then
Theorem 7. [9]. The set is unchanged under the action of .
Theorem 8. [10]. Let . Then, splits into four -subsets. In particular, , , , and are at least four -orbits of .
Theorem 9. [9]. Let . Then, splits into three -subsets. In particular, , , and are at least three -orbits of .
Lemma 10. Every RQIN in is an ambiguous number.
Lemma 11. [14]. is an ambiguous number if and only if and or and
Remark 12. [9]. Let and . Then, (1) = = (2) = = .(3) =
Remark 13. It should be noted here that for a reduced number , we have , and
2. Properties of Reduced Quadratic Irrational Numbers in
This section is devoted to study the behavior of reduced numbers.
Lemma 14. If is an RQIN, then is an ambiguous number but not RQIN.
Proof. Let be a reduced quadratic irrational number such that , where . Then, by using the Mobius transformation , we have , where and . Since by using Remark 12, is not RQIN.
Theorem 15. . Let or such that Then, the circuit of a reduced number has the type Moreover, , , and each exists on the turning points of the circuit and not reduced.
Proof. , , , where is the number of squares inside the circuit. which shows that one circuit is lying between the inside and outside boundary of the circuit. , where is the number of squares inside the circuit. which implies that one of the squares is lying between the inside and outside boundary of the circuit.
Theorem 16. For or such that and = be a reduced number, then , , and map onto the nonreduced number under the action of .
Proof. Let and . By using linear fractional transformation and Table 1, where , where By using Remark 12, it is not a reduced number. Similarly, is not a reduced number. and ; where . By Remark 12, hence is not a RQIN.
Theorem 17. Let be a RQIN moved to under a Mobius transformation . Then,
Proof. Suppose be RQIN under Mobius transformation moved to half of their conjugate, i.e., by using Table 1, as . By using Theorem 1, and have symmetric periodic part, since, in the form of continued fraction, every RQIN has unique description. In similar fashion, and with symmetric periodic parts are identical. By Lemma 5 and are not identical. Hence, we conclude that
Lemma 18. Let which moves to half of their conjugate under the linear fractional transformation . Then, has at least distinct circuits Example 1 reflects Lemma 18.
Example 1. Suppose and be reduced quadratic irrational transformed to half of their conjugate under the transformation and is the type of . be reduced quadratic irrational transformed to half of their conjugate under the transformation and is the type of . It is easy to see that and are not equivalent, so that as shown in figures below.
Figures 1 and 2 reflect Lemma 18.
3. Reduced Length of the -Circuits of
The circuit generates an element of the form of and fixes some vertex of a square on the closed orbit, and thus, the reduced length of closed orbit is the count of RNs in this closed circuit.
Example 2. The circuit of the type represents that the circuit generates an element of and fixes the vertex . Suppose , , , , , , , , , . Equations (1), (2), (3), and (4) follow that are only reduced numbers in the orbit. Thus, the reduced length of this orbit is 4.
Now, we investigate the reduced cardinalities of -orbits.
Theorem 19. Let or such that and then the circuit of the reduced number has the type , and
Proof. In order to prove that it is enough to find in such a manner . The proof was followed by the following four steps: , , , and Thus, we obtain . Hence, the circuit of the reduced number has the type . Now, we have to prove that . Let be reduced number. Now, by using Theorem 15 and Theorem 16, the numbers ,, and are on the turning point of the circuit and are not reduced numbers; furthermore, when we will apply linear fractional transformation on ,, and , then in result, we get no reduced number. So, is only reduced number in . Hence, .
Example 3. Take a prime number such that and . It is observed from the coset diagram given below that the reduced number is fixed by the word this shows that type of the circuit is , and it can be seen from the coset diagram given below; is only reduced number in , and hence,
Example 4. Take a prime number such that and . It is observed from the coset diagram given below that the reduced number is fixed by the word ; this shows that type of of the circuit is , and it can be seen from the coset diagram given below that is only reduced number in , and hence,
Lemma 20. For or such that , then the orbits of reduced numbers and have the type and .
Proof. To show that it is enough to discover in such a manner , where using Remark 12(1) and (3), we obtain . Now, and . In Table 1, and Finally, . Hence, be the type of circuit of reduce number . Similarly, the type of is same as first one and from equations (1) and (2); hence, .
Example 5. Take a prime number such that and . It is observed from the coset diagram given below that the reduced number is fixed by the word ; this shows that type of the circuit is , and it can be seen from the coset diagram given below; and are only reduced number in , and hence, .
Lemma 21. If and , then and circuits have the type Moreover and
Proof. Similar proof as of Lemma 20.
Remark 22. (i) It is not necessary that every circuit contains reduced number. As we can see in the figure given below, the circuit of contains no reduced number.
Figure 6 reflects Remark 22(i).
3.1. Detection of Reduced Numbers
In the orbits of and of , where or such that , then (i) and are only reduced numbers in the circuit of (ii) and are only reduced numbers in the circuit of
Remark 23. For such that . (i)If a reduced number , then its negative conjugate (ii)If a reduced number then, its negative conjugate
Lemma 24. If and then, the circuit of the reduced number has the type , and hence,
Proof. To illustrate this, it is sufficient to find such that , where by Remark 12, then ; now, by using Table 1. and , again by (1.1), we have , and , by Table 1, and ; finally, , and . Thus, and from equations (i), (ii), (iii), and (iv), we get which are only 4 reduced numbers in the circuit of .
Hence, 4.
Example 6. Take a prime number such that . It is observed from the coset diagram given below that the reduced number is fixed by the word ; this shows that type of of the circuit is , and also, as can be seen from the coset diagram given below, , , , and are only reduced number in and hence .
Lemma 25. If and then, the circuit of the reduced number has the type , and hence, .
Proof. Similar proof as of Lemma 24.
Example 7. Take such that . It is observed from the coset diagram given below that the reduced number is fixed by the word ; this shows that type of the circuit is , and it can be seen from the coset diagram given below, and and are only reduced numbers in , and hence, .
3.2. Detection of Reduced Numbers
In the circuits of and of where and , then, (i), , , and are only reduced numbers in the circuit of (ii) and are only reduced numbers in the circuit of
Remark 26. For such that . (1)If then (2)If , then (3).
4. Conclusion
The idea of types of -circuits in -orbits of RQF by Mobius group, which is given in this paper, is new and original. We have presented type of -circuits with different length in -orbits , where is RQIN and be Mobius group. We have investigated properties of RQINs and classified -orbits of different length. Furthermore, we proposed reduced length and general form of reduced numbers in different orbits. This work can be extended for the Mobius group and as well as examined the -circuits in -orbits and the -circuits in -orbits. Moreover, the reduced length and general form of reduced numbers for different orbits can be discussed.
Data Availability
No real data were used to support this study. The data used in this study are hypothetical, and anyone can use them by citing this article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors would like to thank the Deanship of Scientific Research of King Abdulaziz University, Jeddah, Saudi Arabia, for technical and financial support.