Optimum criterion for lightweight nonlinear confusion component with multi-criteria decision making

1 Introduction

In the current world, everything has become so fast forward and technology-dependent. If anything has pros it also has its cons as well. Therefore, with all the technology now we have the main goal to achieve is to secure our communication. For this purpose, many cryptographic techniques have been used so far to secure our communication. Security and confidentiality are the new challenges of this world. To ensure privacy algebraic-based cryptosystems are constructed which are resilient to differential and linear cryptanalysis. Substitution-boxes (S-boxes) play a vital role in secure data encryption. Many techniques have been proposed in literature for secure S-box construction [1–3]. In this era, constructing the S-box with new mathematical structures to create confusion has become the target for every oncoming researcher in the field of information technology [4, 5].

Different types of mathematical structures are utilized for the construction of nonlinear confusion component. In addition to that, we have introduced another technique which is known as TOPSIS, to choose the best among all. Left almost semigroups are the algebraic structures that hold closure law and left inverted binary operation, abbreviated as LA-semigroup. After the 1970 s, the evolution of LA semigroup occurred after the generalization of groups and semigroups. After the significant number of research results, it has become a distinct branch within itself. Kazim and Naseeruddin familiarized the (LA semigroup) Left almost semigroup [6]. Further, the exploration of fuzzy semigroup was done by Kuroki and Modern [7, 8]. The applications of fuzzy algebra are so vast, that is why they are being applied in different fields. Some significant properties of inverse LA semigroup were deliberated by Younas et al. [9]. The generalization of quasi-prime ideals of the LA semigroup was introduced by Yiarayong [10]. The perception of ordered (LA-semigroups, LA*-semigroups, LA groups) and left (resp. right) simple ordered LA-semigroups are introduced. The relationship between ordered (LA-semigroups, LA*-semigroup) ideals and ordered (LA-semigroup) ideals (ordered LA-groups, left simple, right simple) are proved, in particular, that: 1. If an ordered LA*-semigroup (S, . , ⩽) has the left identity e, then Sa = aS = S ∀ aɛS. 2. If and only if it contains no proper quasi-ideal, an ordered LA* -semigroup (S, . , ⩽) with left identity e is a left simple and a right simple [11]. In a non-associative algebraic structure named as an ordered LA-Γ-semigroup, the concept of (m, n)-ideals is investigated by Basar [12]. Younas et al endeavor to establish a framework for defining an inverse LA- semigroup using generators and relations [13].

In this work, we have utilized inverse LA- semigroup, Moufang loops, and C-loops for the construction of the nonlinear confusion component. This article presents the structure of ten different substitution boxes with strong cryptographic properties due to their design dependence on loops. Moreover, we have applied a method for the selection of the best S-box among all the proposed ones known as the technique for order of preference by similarity to ideal solution (TOPSIS).

The problems of TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) occur and are extensively employed in numerous fields, such as medical sciences, social sciences, management sciences, and economics. TOPSIS problems are also known as mentioned as multi-criteria decision analysis (MCDA), multi-criteria decision making (MCDM), or multi-attribute decision-making (MADM) [14–16]. In the previous eras, the MCDM methods have become an influential part of operations research [17]. Mingwei et al. used this technique to take care of the site determination issue for vehicle sharing stations as selection of a suitable site for building a vehicle sharing station is a major test [18].

In various real-world circumstances, the problems of decision-making are exposed to certain restraints, purposes, and effects that are not precisely identified. Fuzzy set within MCDM was firstly introduced by Bellman and Zadeh [24] and afterward many researchers have been fixated by decision making in fuzzy environments. The combination of fuzzy set and multi-criteria decision making (MCDM) has headed to a different decision theory identified as fuzzy multi-criteria decision making (FMCDM). In FMCDM we can decide by uncertainty in incomplete information and knowledge. The decision-makers must rank or select all possible alternatives corresponding to the weights of the criteria.

In this study, we have applied TOPSIS for the selection of substitution boxes proposed by using loops. Also, we have presented a brief comparison of proposed substitution-boxes properties with already existing techniques [19–30]. These criteria rank substitution boxes according to the weight assigned to each characteristic of substitution-box and ideal best and worst values from all possible events.

The remaining manuscript is arranged as follows: Section 2 presents some basic concepts and proposed technique for construction of nonlinear confusion component; some standard cryptographic benchmarks are applied to proposed s-boxes are performed in section 3; In section 4 we have presented comparative analysis; TOPSIS technique is applied for best s-box selection in section 5, finally, work is concluded in the last section.

2 Basic concepts and proposed methodology for confusion component

In this section, we discussed some basic definitions and our proposed methodology for the construction of the nonlinear confusion component of the block cipher.

Definition 2.1. An operation which is performed on the elements of set S to produce output which also lies in set S. Elements of set S can be written as a pair set (a, b). Elements of Cartesian productS × S are writable as (a, b), so binary operation of set S can also be described as: f : S × S → S

Definition 2.2. A non-empty set S = (S, .) with a binary operation that holds closure law is known as groupoid.

Definition 2.3. Semigroup can be defined as a groupoid that also satisfies associative law.

Definition 2.4. Let (S, .) be a groupoid. We call (S, .) an LA-semigroup if ∀ a , b , c ∈ S : ( a . b ) . c = ( c . b ) . a .

We write (a . b) as(ab).

Definition 2.5. The LA-semigroup (S, .) is called inverse if ∀x ∈ S there exists a unique x′ ∈ S such that xx′x = x and x′xx′ = x′ .

To authenticate the cryptographic strength of the S-boxes we have generated (shown in Tables 1–7) different analysis techniques are used, and their results are discussed in this section.

In this section, we have presented a brief comparison of our proposed substitution box with already existing schemes. Table 11 presents the comparative analysis of nonlinearity, SAC, and BIC nonlinearity, and contrast of BIC-SAC, LP, and DP is listed in Table 12. From the presented contrast we can observe that our proposed scheme produces substitution-boxes that exhibit cryptographic properties that are much better than the already existing schemes.

Multi-criteria for decision-making (MCDM) are criteria for the selection of best evaluation from a given set. This criterion defines how the decision-maker’s evaluations of the considered alternatives hinge on the fact of the best selection. The probabilities of the considered alternatives are assumed to be known. The choices are typically associated based on the probable values and the inconsistencies of their assessments. Therefore, we consider the problem of selection of the best substitution box which is demonstrated by fuzzy sets defined on the universal set on which the probability distribution is specified, and the assessments of the choices are uttered by fuzzy numbers. Here in this section, we are using MCDM by using fuzzy sets for the best S-box selection from the ten S-boxes constructed by loops defined in the previous segment.

Now consider the probability space be (S, μ, P) where S denotes a non-empty universal set of all possible elementary events, μ denotes the set of considered random events, and P : μ→ { 0, 1 } represents the probability measures. Table 13 shows the decision matrix with {S₁, S₂, S₃, … , S₁₀ } as all possible alternatives of the decision-maker. X₁, X₂, X₃, X₄, X₅, and X₆ are the states of the problem which represents nonlinearity, SAC, BIC-SAC, BIC nonlinearity, DP, and LP ( See Fig. 1). Therefore, the sets S and μ can be written as:

μ = { X 1 , X 2 , X 3 , X 4 , X 5 , X 6 } , S = { S 1 , S 2 , S 3 , … , S 10 } .

Table 13 defines the average cryptographic properties of constructed substitution boxes.

Now, let us determine the normalized vector of each state in the probability space (S, μ, P) by using: U i , m = X i , m ∑ i , m ( X i , m ) 2 , where 1 ⩽ i ⩽ 10 denotes all possible alternatives, 1 ⩽ m ⩽ 6 shows all possible states of the event. The set of normalized decision vectors is denoted by the matrix U_i,m and defined by the vectors as U i , m = { U 1 i , U 2 i , U 3 i , U 4 i , U 5 i , U 6 i } .

Therefore, after finding the normalized vector of each state of all possible alternatives, the results are listed in Table 14.

In solving decision-making problems, the considered parameters may have different importance due to diverse human perception that forces us to give different weight to each of them. Here, in this case, we have considered equal weight for each state because the selection of the best S-box is made by treating all the states equally important. The probability of the sum of each state for one alternative must be one. The weight matrix is defined as: γ m i = { γ 1 i , γ 2 i , γ 3 i , γ 4 i , γ 5 i , γ 6 i } .

Weight for each state and alternative is defined in Table 15.

The significance of each state is defined by multiplying weight against each event with a normalized decision matrix which is defined as: δ m i = { U 1 i × γ 1 i , U 2 i × γ 2 i , U 3 i × γ 3 i , U 4 i × γ 4 i , U 5 i × γ 5 i , U 6 i × γ 6 i } , δ m i = { δ 1 i , δ 2 i , δ 3 i , δ 4 i , δ 5 i , δ 6 i } .

Normalized weight matrix resultants are listed in Table 16.

The positive and negative matrices are constructed from a normalized weight matrix by taking the ideal best and ideal worst value of each state of the event. Ideal best values are stored in the positive matrix and listed in Table 17 and ideal worst values are stored in the negative matrix and depicted in Table 18. Ideal best values for the possible approximations are determined as follows:

IdealBest = { ( NL ) , ( SAC ) , ( BIC SAC ) , ( BIC NL ) , ( DP ) , min ( LP ) } , I m + = { ( δ 1 i ) , ( δ 2 i ) , ( δ 3 i ) , ( δ 4 i ) , ( δ 5 i ) , min ( δ 6 i ) } .

Similarly, ideal worst values are calculated by the opposite selection of ideal worst states such as: IdealWorst = { ( NL ) , ( SAC ) , ( BIC SAC ) , ( BIC NL ) , ( DP ) , max ( LP ) } , I m - = { ( δ 1 i ) , ( δ 2 i ) , ( δ 3 i ) , ( δ 4 i ) , ( δ 5 i ) , max ( δ 6 i ) } .

After the selection of deal best and worst values of the state, the positive and negative ideal separations are performed by the following procedure: S m + = ∑ m = 1 6 ( δ m i - I m + ) 2 , ∑ m = 1 6 ( δ m i - I m - ) 2 .

And relative closeness to t ideal solution is determined by, C m = S m - S m + + S m -

The ranking is performed on the rule (C _m ) = Rank. Positive, negative separation, relative closeness, and ranking is listed in Table 19. From the depicted table we can see that S₈ is ranked as one which shows that S₈ is the best substitution box selected from MCDM.

For better visual observation we have plotted a bar chart for relative closeness and its respective rank for all possible alternatives against each event. Figure 2 displays the bar chart of ranks against relative closeness and we can examine that S-box 8 is ranked as one with all superlative cryptographic properties.

OPEN IN VIEWER

Fig. 1

Bar chart representation of average cryptographic properties of proposed s-boxes.

OPEN IN VIEWER

Fig. 2

Bar chart representation of relative closeness and raking values for all possible alternatives.

In this work, we have utilized different types of loops for the construction of different nonlinear confusion components. We have employed LA inverse loop, C-loop, and Moufang loops for the basic structure of S-boxes. The suggested nonlinear confusion components are passed through some cryptographic benchmarks which depict that offered nonlinear components satisfy all the properties of strong structure. Moreover, we have presented a brief comparison of the properties of substitution boxes generated by our anticipated scheme with already existing S-boxes. The comparison elucidates that our offered S-boxes exhibit strong properties relative to the existing ones. we have applied TOPSIS for the selection of the best substitution-box among all the proposed structures. The raking with TOPSIS is applied to each substitution box concerning the achievement of cryptographic properties.