The substitution box, often known as an S-box, is a nonlinear component that is a part of several block ciphers. Its purpose is to protect cryptographic algorithms from a variety of cryptanalytic assaults. A Multi-Criteria Decision Making (MCDM) problem has a complex selection procedure because of having many options and criteria to choose from. Because of this, statistical methods are necessary to assess the performance score of each S-box and decide which option is the best one available based on this score. Using the Pythagorean Fuzzy-based Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method, the major objective of this investigation is to select the optimal S-box to be implemented from a pool of twelve key choices. With the help of the Pythagorean fuzzy set (PFS), the purpose of this article is to evaluate whether this nonlinear component is suitable for use in a variety of encryption applications. In this article, we have considered various characteristics of S-boxes, including nonlinearity, algebraic degree, strict avalanche criterion (SAC), absolute indicator, bit independent criterion (BIC), sum of square indicator, algebraic immunity, transparency order, robustness to differential cryptanalysis, composite algebraic immunity, signal to noise ratio-differential power attack (SNR-DPA), and confusion coefficient variance on some standard S-boxes that are Advanced Encryption Following this, the findings of the investigation are changed into Pythagorean fuzzy numbers in the shape of a matrix. This matrix is then subjected to an analysis using the TOPSIS method, which is dependent on the Pythagorean fuzzy set, to rank the most suitable S-box for use in encryption applications.

In this age of the 21st century where technology reaches its new heights, secure communication is a big challenge for researchers. Different communication channels are increasingly being utilized for online data transfer from one location to another which requires security and confidentiality that can be achieved by the use of cryptography. Cryptography is used to obscure the meaning of the data which results in the protection of information from unauthorized access. In cryptography, efficient algorithms are used for encryption purposes, which are characterized by private key and public key cryptographic algorithms. A modern block cipher is a part of private key cryptographic algorithms, which uses the similar key both for encoding and decoding purposes. S-box is an essential nonlinear part in many modern block cipher techniques, responsible for creating confusion during encryption. S-box with strong confusion ability is more sustainable in distorting the input information. Numerous techniques have been presented in writing for the structure of a reliable S-box [

Decision-making (DM) entails, choosing the best option from the set of feasible options. It is an essential part of human life. As humans make decisions almost every day to perform their daily tasks. The human-based decision involves uncertainty and vagueness in their preferences. L. A. Zadeh [

Multicriteria decision-making (MCDM) is a set of systems used in various decision-making applications. The goal of MCDM is to choose the most excellent option from a group of alternatives characterized by multiple, usually conflicting criteria. Researchers have introduced various MCDM methods in the past few decades to successfully tackle decision-making problems [

In this research, an MCDM approach was employed to select the desired S-box used in encryption applications. In this regard, we first investigate the results of cryptographic properties of some standard S-boxes [

In this article, a decision-making algorithm is utilized to select the suitable S-box. Our contributions are summarized as follows:

We first investigate into the results by investigating the cryptographic properties of some standard S-boxes.

Secondly, the TOPSIS procedure depending on the interval-valued Pythagorean fuzzy (IVPF) set is applied to analyze the above outcoming results to reach the final decision.

The remainder of the research is categorized as follows: segment 2 is devoted to the background; cryptographic analysis is presented in segment 3; IVPF-based TOPSIS structure is employed to choose the desired S-box in segment 4; last, the conclusion is discussed in segment 5.

A fuzzy set

An intuitionistic fuzzy set

Let

A Pythagorean fuzzy set (PFS)

For simplicity, Zhang and Xu called the pair (

Let

Let

For any

Let P = (

Let

For any two PFNs

Let

Clearly s (

But if we consider

For any two (PF) numbers

In this segment, we examine some cryptographic features of S-boxes used in this work. We also show the findings obtained using the extended TOPSIS structure recommended by Zhang et al. [

The least distance of any Boolean function f to the set of all affine functions

Nonlinearity determines the robustness of an S-box. The high value of NL is desired since it enhances resistance to cryptanalytic attack [

Strict Avalanche Criteria (SAC) is an examination test of an S-box in which each resultant bit is changed with the probability of 1/2. SAC has an optimum value of 0.5 [

Bit independent criteria (BIC) are applied to the input bits that remain unchanged. The correlation coefficient is employed to compute the BIC value. It is observed that if nonlinearity and SAC are satisfied then BIC is also satisfied [

The highest absolute value of

The algebraic degree is the highest quantity of confusion elements in the truth table. A high-value of algebraic degree is required to resist any cryptanalytic attack [

Algebraic immunity (AI) provides a challenge to an S-box against any cryptanalytic attack. A high value of AI is required to overcome the algebraic attacks in breaking an encryption system [

A low value of transparency order is required to resist any differential power analysis (DPA) attack [

Suppose _{1}, _{2}_{s}) be an _{j} (

The robustness of DPA counter to differential and linear cryptanalysis correlates positively with the signal-to-noise ratio (SNR) of the algorithm. Good quality cryptographic S-boxes are assessed because of high SNR. A high SNR value refers to strong signal strength concerning noise level [

The confusion coefficient variance is the resistance of an S-boxes against any cryptanalytic attack. A high value of confusion coefficient variance is required [

The TOPSIS technique [_{1b}, _{2b}, _{3b}, _{4b}, _{5b}, _{6b}} be six standard S-boxes which are to be evaluated. These S-boxes are evaluated with the aid of the following criterion:

Nonlinearity

BIC Nonlinearity

SAC

BIC SAC

Sum of square indicator

Transparency order

Composite algebraic immunity

Absolute Indicator

Robustness to differential cryptanalysis

Algebraic degree

Signal to noise ratio (SNR) (DPA)

Algebraic immunity

Confusion coefficient variance,

and is represented by the set

Let us consider equal weights for criteria that are W =

AES | APA | Gray | Prime | Skipjack | Xyi | |
---|---|---|---|---|---|---|

Nonlinearity | (0.8, 0.3) | (0.8, 0.3) | (0.8, 0.3) | (0.6, 0.5) | (0.8, 0.4) | (0.5, 0.6) |

BIC-Nonlinearity | (0.8, 0.3) | (0.8, 0.3) | (0.8, 0.3) | (0.8, 0.4) | (0.7, 0.4) | (0.8, 0.4) |

SAC | (0.8, 0.3) | (0.7, 0.4) | (0.8, 0.3) | (0.8, 0.4) | (0.7, 0.4) | (0.8, 0.5) |

BIC-SAC | (0.8, 0.3) | (0.7, 0.4) | (0.8, 0.4) | (0.8, 0.4) | (0.7, 0.4) | (0.8, 0.4) |

Absolute indicator | (0.7, 0.4) | (0.7, 0.3) | (0.7, 0.3) | (0.3, 0.8) | (0.5, 0.4) | (0.5, 0.4) |

Sum of square indicator | (0.7, 0.3) | (0.7, 0.3) | (0.7, 0.3) | (0.4, 0.7) | (0.5, 0.5) | (0.4, 0.6) |

Algebraic degree | (0.7, 0.4) | (0.7, 0.4) | (0.7, 0.4) | (0.7, 0.4) | (0.7, 0.4) | (0.7, 0.4) |

Algebraic immunity | (0.7, 0.4) | (0.7, 0.4) | (0.7, 0.4) | (0.7, 0.4) | (0.7, 0.4) | (0.7, 0.4) |

Transparency order | (0.7, 0.4) | (0.7, 0.4) | (0.7, 0.4) | (0.6, 0.4) | (0.6, 0.4) | (0.7, 0.5) |

Composite Algebraic immunity | (0.7, 0.4) | (0.7, 0.4) | (0.7, 0.4) | (0.7, 0.4) | (0.7, 0.4) | (0.7, 0.4) |

Robustness to differential cryptanalysis | (0.6, 0.5) | (0.6, 0.5) | (0.6, 0.5) | (0.5, 0.5) | (0.7, 0.3) | (0.7, 0.3) |

SNR(DPA) | (0.7, 0.3) | (0.6, 0.4) | (0.7, 0.3) | (0.8, 0.4) | (0.6, 0.4) | (0.8,0.3) |

Confusion coefficient variance | (0.6, 0.5) | (0.7, 0.4) | (0.7, 0.5) | (0.5, 0.5) | (0.8, 0.3) | (0.7, 0.4) |

In

Analysis | Analysis | ||
---|---|---|---|

(0.8, 0.3) | (0.5, 0.6) | ||

(0.8, 0.3) | (0.6, 0.5) | ||

(0.8, 0.3) | (0.7, 0.4) | ||

(0.8, 0.3) | (0.7, 0.4) | ||

(0.7, 0.3) | (0.6, 0.5) | ||

(0.7, 0.3) | (0.6, 0.5) | ||

(0.7, 0.4) | (0.7, 0.4) | ||

(0.7, 0.4) | (0.7, 0.4) | ||

(0.7, 0.4) | (0.6, 0.5) | ||

(0.7, 0.4) | (0.7, 0.4) | ||

(0.8, 0.4) | (0.6, 0.5) | ||

(0.5, 0.6) | (0.6, 0.4) | ||

(0.8, 0.4) | (0.6, 0.5) |

In

S-box | S-box | ||
---|---|---|---|

0.0508 | 0.1877 | ||

0.0685 | 0.1700 | ||

0.0415 | 0.2031 | ||

0.1992 | 0.0531 | ||

0.1092 | 0.1638 | ||

0.1154 | 0.1823 |

The distance between Gray S-box and PFPIS is minimum in

Now to determine the performance score of all S-boxes, we have to calculate the relative closeness. Closeness

_{i} |
−0.298 | −0.811 | 0 | −4.5349 | −1.8228 | −1.8801 |

Rank | 2 | 3 | 1 | 6 | 4 | 5 |

The S-box with the highest rank is considered the best S-box, and

The primary intention of this study is to figure out which S-box is better for encryption applications. To do this, a decision-making algorithm namely the extended TOPSIS technique based on PFS is applied to a matrix containing data in the form of a PF number as displayed in

This research was funded by Princess Nourah bint Abdulrahman University Researchers Supporting Project Number (PNURSP2022R87), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.