Application of fuzzy soft sets to analyze the statistical strength of S-boxes

For the evaluation of the substitution boxes, the majority logic criterion is used to analyze the statistical strength of the existing substitution boxes. The main objective of this paper is to make a decision on the analysis and selection of the most appropriate S-box based on a fuzzy soft-aggregation operator. Instead of the usual practice in which a single parameter is considered, we are considering several parameters that will definitely give us a comprehensive analysis of the S-boxes.


Introduction
*For our imperative understanding, the material world is complex. Many difficulties in various fields such as applied sciences, social sciences, medical skills, computer sciences, and artificial intelligence are generally not precise. The researchers develop a number of tools of certainty that simplify the various uncertain aspects of the tangible world. Regrettably, these mathematical tools are complicated, and we cannot find the precise results. The usual methodologies used to share with these uncertainties are appropriate for certain environments. These may be ascribable to the uncertainties of ordinary environmental phenomena of human consciousness around the actual creation. For instance, vagueness in the boundary between provinces or between urban and rural areas or the precise increase in population in land or making decisions using database information. For that reason, the conventional set theory may not be appropriate to carry out such uncertain problems.
The concept of soft set theory was initiated by Molodtsov (1999). In his work, he gave many important results that are being used to resolve the uncertainty problems in different research areas such as game theory, operation research, probability theory, etc. Basically, soft set theory is considered to be a valuable mathematical tool to cope with the problems of uncertainty. As the soft set theory is based on a number of parameters, so intuitively, this theory is considered to be more comprehensive and effective as compared to other conventional theories. Nowadays, the soft set theory is taking so much attention from the researchers to make its applications in different fields.
In the current decade, the soft set theory is playing a vital role in decision making. For the optimal selection of the objects based on the reduction of parameters, Maji et al. (2002;2003) used the soft sets in decision making. Chen (2005) defined the soft sets parameterization reduction in another way and compared it with the rough set theory attribute reduction. Soft sets are defined as a class of special information systems by Pei and Miao (2005). The soft sets data analysis approach was investigated by Zou and Xiao (2008).  gave the idea of FP-soft sets and discussed their several characteristics, and proposed an algorithm for decision making.
If we look into the literature, we can find a number of proposed encryption methods. These methods seem to be capable, but their effectiveness is not yet installed, and these are preparing to become standards. The most usual methods utilized to study the statistical effectiveness of S-boxes are the differential approximation probability, strict avalanche criterion, correlation analysis, linear approximation probability, and so forth.
In this research article, the fuzzy soft aggregation operator is applied to key out the effectiveness point of S-boxes. The statistical features of S-boxes (average correlation, average entropy, average contrast, average homogeneity, average energy, and an average mean of absolute deviation) have been studied by counting the values of all the analyses of various S-boxes. An algorithm based on a fuzzy soft aggregation operator is applied to select the best Sbox among the different S-boxes.

Historical perspective of soft sets, fuzzy soft sets, and FS-aggregation operator
We recall some definitions from Molodtsov (1999), Abuelyman and Alsehibani (2008) Throughout this paper, is considered as the universal set, ( ) represents the power set. The set of parameters is denoted by while is considered as a subset of .
Definition (Molodtsov, 1999): If U denotes the universal set and the set of parameters is E, then consider a function F: A  P(U). We call ( , ) a soft set over U, where P(U) denotes a power set, A is a non-empty subset of parametrized set E.
Definition (Maji et al., 2001) (Maji et al., 2001): If (F, A) and (G, B) are soft subsets of each other, then these are said to be soft equal.
Definition (Maji et al., 2001): Definition (Acar et al., 2010): ( , ) is said to be a bi-intersection of (F, A) and (G, B) if it satisfies: Definition (Acar et al., 2010): The union of two soft sets (F, A) and (G, B) is a soft set (H, C) satisfying the following: For each ∈ , Definition (Acar et al., 2010): The support of (F, A) is given by Supp( , ) = { ∈ ∶ ( ) ≠ ∅}. Cagman et al. (2011) defined -aggregation operator on the fuzzy sets, which actually produces a single fuzzy set and is said to be an aggregate fuzzy set of the -set. In the following, we recollect some important definitions and algorithms from the work of .
Definition (Abuelyman and Alsehibani, 2008): Let be an fs-set. is given by a mapping : is said to be a fuzzy approximate function of and ( ) is said to be an -element of the -set for every ∈ . Consequently, an -set is represented as follows: The set of all -sets is represented by ( ).
where the membership function of is denoted by It is important to note that   is the cardinality of the universal set , and γ A ( ) is the scalar cardinality of fuzzy set ( ).
Moreover, the set of all cardinal sets of -sets is denoted by ( ). And obviously, ( ) ⊆ ( ).
Definition : If  ( ) and c  ( ), then the -aggregation operator is defined by, where, * ={ * ( )/ :  } is a fuzzy set over . * is said to be an aggregate fuzzy set of theset . The membership function * of is given as follows: * : →[0, 1], where   shows the cardinality of .
Theorem : If  ( ) and ⊆ . Assume that , and * are the matrices of , and * respectively. Then,

E × M Γ A * = M Γ A × M c Γ A T
where   denotes the cardinality of and c denotes the transpose of c .
Algorithm:  proposed a decision-making algorithm which is given in the following: Step1. Construct an -set over , U Step2. Find the cardinal set of , Step3. Find the aggregate fuzzy set * of , Step4. Obtain the best alternative from this set that holds the largest membership grade by maxμ * (u).

Statistical analysis of S-boxes
It is of the essence to be conversant with the meaning and relationship among the outcomes of several types of analyses. In Tran et al. (2008), the authors employed statistical analysis to define the suitability of an S-box to image encryption application. In reality, the process begins with the correlation analysis. This analysis, under some conditions, does not furnish enough information in deciding the effectiveness of encryption. Hussain et al. (2012) projected a generalized majority logic criterion, where the authors considered several images. The diversity in image contents made this algorithm more appealing to a wider range of data samples. The generalized majority logic criterion seems to be an alluring option due to its applications and suitability of multiple types of images in the selection of the optimal S-box. Fig. 1 and Fig. 2 give the complete picture of the work of Hussain et al. (2012).

Consider multiple plain images
Encrypt plain images using S-box transformations Analyze the texture of the cipher images w,r,t plain images End Fig. 1: Generalized majority logic criterion   If we conclude the work of Hussain et al. (2012), it is evident that S8 AES S-box has been considered as the best S-box as compared with other S-boxes. This conclusion can be observed in Table 3 obtained from different statistical analyses.
If we look into the current developments in the field of cryptography, many researchers gave different methods to analyze and choose the optimal S-box. On the other hand, it is worth mentioning that the soft set theory has a remarkable contribution to decision-making problems. Here in our discussion, we intend to choose the best S-box by applying an algorithm based on a fuzzy soft set aggregation operator.  S-box satisfied majority of these conditions is better ( = 1, 2, 3, 4, 5, 6, 7, 8). To evaluate the S-boxes, let the set of parameters is = { 1 , 2 , 3 , 4 , 5 , 6 }, where = 1, 2, 3, 4, 5, 6 and stands for average entropy, average contrast, average correlation, average energy, average homogeneity, and average mean absolute deviation, respectively. After a detailed discussion, we have chosen a subset of parameters = = { 1 , 2 , 3 , 4 , 5 , 6 }. Now, we are in a position to make a decision to analyze S-boxes by using the following steps: Step1. Foremost, we will build an -set over .  Step4. From step3, it is obvious that the largest membership grade is maxμ * ( ) = 0.055. This means that the alternative 6 ( 8 AES) can be considered as the best S-box among all the other Sboxes.

Conclusion
In this research work, we tried to analyze the quality and strength of the S-boxes by using an algorithm based on a fuzzy soft aggregation operator to make a decision. This method is effective and appropriate for the selection of the best S-boxes among several S-boxes. In the luminosity of our findings, we may conclude that our study is going to be a good addition to the list of efficient methods to pick the best S-box among various S-box.

Conflict of interest
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.