ADCAIJ: Advances in Distributed Computing and Artificial Intelligence Journal
Regular Issue, Vol. 14 (2025), e32318
eISSN: 2255-2863
DOI: https://doi.org/10.14201/adcaij.32318

Bit-Independent Criteria Evaluation of Custom S-Boxes: Enhanced AES Encryption Security

Amira S. El Batouty^a, Taymour A. Hamdalla^b, Heba A. Fayed^c, and Moustafa H. Aly^d

^a Electronics Department, Alexandria Higher Institute of Engineering and Technology, Alexandria, Egypt

^b Physics Department, University of Tabuk, Tabuk, Saudi Arabia

^c Electronics and Communications Engineering Department, Arab Academy for Science, Technology, and Maritime Transport, Alexandria, Egypt

^d Electronics and Communications Engineering Department, Arab Academy for Science, Technology, and Maritime Transport, Alexandria, Egypt

✉ amira.elbatouty@aiet.eg.com, t-ahmed@ut.edu.sa, hebam@aast.edu, mosaly@aast.edu

ABSTRACT

KEYWORDS

1. Introduction

Cryptography plays a pivotal role in protecting data against unauthorized access, employing encryption techniques to secure data during transmission across various systems and to standardize cryptographic practices (Chari et al., 1999). Encryption ensures data integrity, enhances security, and protects data across different devices while preserving privacy. Commonly utilized encoding methods such as AES and DES play a crucial role in ensuring the security of information. To the best of our knowledge, no previous studies have specifically applied the Bit Independent Criterion (BIC) to evaluate customized S-Boxes integrated into the AES algorithm. This distinguishes our approach from prior research that focused primarily on nonlinearity and avalanche effect metrics.

Claude Shannon identified the fundamental concepts of diffusion and confusion as crucial for ensuring the security of any cipher system (Stallings, 2006). Diffusion conceals the relationship between ciphertext (CT) and plaintext (PT) by ensuring, each symbol in the CT is based on many symbols from the PT. Obscuration conceals the relationship between CT and the key, ensuring that a change in one bit of the key leads to changes in most or all bits of CT. The semi-field substitution boxes were investigated by Hussain et al. (2023) using permutation of symmetric group on a set of size 8 S₈. An effective procedure for generating S₈ semi-field substitution boxes with the same algebraic properties was established by Alharbi et al. (2023), where an innovative approach was proposed for constructing dynamic S-boxes using Gaussian distribution-based pseudo-random sequences. Cryptosystems are broadly categorized into two major classes: asymmetric key and symmetric key. Symmetric key encryption is additionally divided into two subdivisions, namely block ciphers and stream ciphers (Stallings, 2006). Among these, the S-Box stands out as one of the most critical components, serving as the sole nonlinear element that ensures the confusion characteristic in traditional block ciphers such as AES. The strength of this encryption algorithm depends on the meticulous design of a cryptographically robust S-Box (Shannon, 1998).

This paper focuses on the examination of the Bit Independent Criterion (BIC) for the evaluation of block cipher security, specifically by assessing the degree of confusion in elements resulting from the alteration of a single bit in the PT.

The following assumptions and limitations apply to our study.

• The evaluation is based on the standard AES block cipher structure, without incorporating hardware-specific optimizations or resistance to side-channel attacks.

• All S-Boxes are assumed to be invertible and have a fixed size of 16×16.

• Plaintexts (PTs) used during the BIC test are 128-bit blocks, where single-bit variations are introduced in a controlled manner.

• The encryption key size is fixed at 128 bits.

• The test environment is purely algorithmic and does not account for real-world conditions such as network latency, memory overhead, or concurrent processing scenarios.

The structure of this paper is presented as follows. Section 2 introduces a survey on different types of S-Boxes, while Section 3 explains the different kinds of security analysis. Section 4 delves into an in-depth study of the BIC test, while Section 5 analyzes the results of utilizing the BIC test with different types of S-Boxes. Finally, Section 6 concludes the paper.

2. Types of S-Boxes

S-Boxes serve as the sole nonlinear component within block cipher algorithms, playing a pivotal role in bolstering the confusion function of the encryption process. An S-Box takes an input of n bits and produces an output of m bits, with m and n not necessarily being equal. Crucially, an S-Box must be designed to be invertible for decryption purposes. A defining characteristic of S-Boxes is the absence of any discernible relationship between columns or rows.

2.1. The Static S-Box

1. Begin the initialization of the S-Box by arranging the byte values row by row in an ascending order.

2. Link every byte within the S-Box to its corresponding multiplicative inverse in the finite field GF (2⁸) (Fahmy and Salama, 2005).

3. Finally, apply the Affine Transformation (AT) Equation to each bit of each byte in the S-Box as follows:

$$\left[\begin{array}{l}{\mathrm{b}}_0^{\prime }\\ {\mathrm{b}}_1^{\prime }\\ {\mathrm{b}}_2^{\prime }\\ {\mathrm{b}}_3^{\prime }\\ {\mathrm{b}}_4^{\prime }\\ {\mathrm{b}}_5^{\prime }\\ {\mathrm{b}}_6^{\prime }\\ {\mathrm{b}}_7^{\prime }\end{array}\right]=\left[\begin{array}{llllllll}1& 0& 0& 0& 1& 1& 1& 1\\ 1& 1& 0& 0& 0& 1& 1& 1\\ 1& 1& 1& 0& 0& 0& 1& 1\\ 1& 1& 1& 1& 0& 0& 0& 1\\ 1& 1& 1& 1& 1& 0& 0& 0\\ 0& 1& 1& 1& 1& 1& 0& 0\\ 0& 0& 1& 1& 1& 1& 1& 0\\ 0& 0& 0& 1& 1& 1& 1& 1\end{array}\right]\left[\begin{array}{l}{\mathrm{b}}_0\\ {\mathrm{b}}_1\\ {\mathrm{b}}_2\\ {\mathrm{b}}_3\\ {\mathrm{b}}_4\\ {\mathrm{b}}_5\\ {\mathrm{b}}_6\\ {\mathrm{b}}_6\end{array}\right]+\left[\begin{array}{l}1\\ 1\\ 0\\ 0\\ 0\\ 1\\ 1\\ 0\end{array}\right] (1)$$

Where each byte in the S-Box consists of 8 bits labelled (${\mathrm{b}}_7,{\mathrm{b}}_6,{\mathrm{b}}_5,{\mathrm{b}}_4,{\mathrm{b}}_3,{\mathrm{b}}_2,{\mathrm{b}}_1,{\mathrm{b}}_0$), and the new S-Box consists of 8 bits labelled (${\mathrm{b}}_7^{\prime },{\mathrm{b}}_6^{\prime },{\mathrm{b}}_5^{\prime },{\mathrm{b}}_4^{\prime },{\mathrm{b}}_3^{\prime },{\mathrm{b}}_2^{\prime },{\mathrm{b}}_1^{\prime },{\mathrm{b}}_0^{\prime }$).

2.2. The Dynamic S-Box

The dynamic S-Box is an S-Box influenced by the encryption key with its functionality tied to the Key Scheduling Algorithm (KSA) Felicisimo et al., 2015). It can be explained as follows.

1. Initialization involves two arrays, both with 256 elements. The first array, S[256] is initialized with values ranging from 0 to 255, while the second array, K[256] is loaded with the shared secret key.

2. The commonly established secret key is divided into bytes and systematically mapped into the K array, one byte at a time.

$$T[i]=k\text{[i mod Key length]} (2)$$

3. S is a vector encompassing the scheduled key, while i and j denote the iterations within the round range. The result comprises 256 distinct values, entirely contingent on the input key.

$$j=(j+s[i]+t[i])mod256 (3)$$

4. AT is applied to the generated values (Felicisimo et al., 2015). This transformation serves the purpose of preventing fixed points and configuring the S-Box to be invertible.

2.3. Key and PT-Dependent S-Boxes in the RC4 Algorithm

Hosseinkhani and Haj et al. (2012) introduced S-Boxes dependent on both key and PT within the RC4 algorithm. In this context, the term “dependent S-Box” implies that the S-Box relies on the specific key and PT used in RC4 encryption, distinguishing it from the key and PT utilized in AES.

The stream cipher, RC4 encryption algorithm commonly employed in real-time communication scenarios (Stallings, 2006), relies on nonlinear data table modifications. It consists of two fundamental components: the Key Scheduling Algorithm (KSA) and the Pseudo-Random Generation Algorithm (PRGA). The dynamic S-Box section (Ejaz et al., 2021) delves into the KSA, while the PRGA is utilized to generate a single byte from the KSA. To visually illustrate the effectiveness of key and PT-dependent S-Boxes in the RC4 algorithm, please consult Figure 1.

2.4. S-Boxes Customized for PT and Key in the RC4

The utilization of PT and key dependent S-Boxes within the RC4 algorithm involves a profound integration of key components, specifically PRGA and KSA. This integration results in the creation of robust S-Boxes, subsequently subjected to an AT to generate entirely new 256-byte S-Boxes. This strategic approach is adopted to enhance security against various forms of attacks (Agrawal and Himani, 2013; Ahmad and Hwang, 2015; El Batouty et al., 2020).

2.5. S-Boxes in the RC4 Algorithm Customized for PT and Key Using the Henon Chaotic Map

The RC4 algorithm utilizes the Henon chaotic map to enhance time efficiency and create innovative S-Boxes (Agrawal and Himani, 2013). By integrating two-dimensional Henon chaotic mapping into PRGA, the algorithm takes advantage of the complex dynamic behavior inherent in RC4 and the beneficial characteristics of chaotic systems. This integration of chaotic maps adds extra layers of data security by improving both confusion and diffusion, all the while reducing the algorithm’s time consumption without compromising its security (El Batouty et al., 2020; Wen, 2014; El Batouty et al., 2019).

2.6. Key and PT-Dependent S-Boxes in the RC4 Algorithm Employing the Logistic Chaotic Map

The S-Boxes are derived through the integration of the Logistic Chaotic Map (LCM) into the RC4 algorithm to create key and PT-dependent S-Boxes. This process is executed within the PRGA phase, resulting in the development of a dynamic S-Box that is reliant on RC4 and enforced by the LCM (El Batouty et al., 2020).

In a prior study (Webster and Travares, 1998), the implementation of RC4 encryption involved the integration of the Henon chaotic map to improve time efficiency. The infusion of a chaotic signal characterized by a high degree of randomness into the RC4 algorithm amplifies both the confusion and diffusion effects within the encryption process (Agrawal and Himani, 2013). On the other hand, the LCM represents a simple yet distinctive algorithmic relationship that connects the output of one iteration to the input of the subsequent iteration, thereby further enhancing the cryptographic properties of the algorithm.

The mathematical equation for the LCM is provided as (Sosa, 2016):

$$x_{n+1}=r{x}_n(1-x_n) (4)$$

Where r is a real management parameter within the range of 0 to 4, x_n denotes the initial value within the interval [0, 1], and n represents the number of iterations.

Integrating the one-dimensional LCM into the RC4 algorithm leverages the complex dynamic behavior of RC4 and the beneficial characteristics of chaotic systems. The process begins with the initiation of the KSA following the input of the key and PT. Following this, the execution of the PRGA is carried out in conjunction with the LCM. The resulting output comprises 256 bytes, each possessing unique values. Subsequently, an Affine Transformation is applied to derive the suggested S-Box, which consists of 256 bytes, as illustrated in Figure 2.

The distinctive feature setting apart this S-Box from the Dynamic S-Box lies in the use of the modified PRGA with the LCM for S-Box generation. The incorporation of the one-dimensional LCM into the RC4 algorithm aims to harness the complex dynamic behavior of RC4 and the advantageous properties inherent in chaotic systems. The process involves several distinct steps. Initially, the KSA is initialized on the basis of the provided input key and PT. Subsequently, the PRGA is executed with the integration of LCM. The resulting output comprises 256 distinct byte values. Following this stage, an Affine Transformation is applied to produce the proposed S-Box, which consists of 256 bytes, as illustrated in Figure 2. It is noteworthy that the S-Box generated in this approach differs from the Dynamic S-Box due to the utilization of a modified PRGA with LCM during the S-Box generation process. This innovative methodology enhances the cryptographic properties of the RC4 algorithm, combining the strengths of chaotic dynamics and traditional cryptographic techniques.

3. Security Analysis

This section discusses three key security analysis metrics that are essential in evaluating the strength of S-Boxes in cryptographic systems. These are: nonlinearity, avalanche effect, and execution time performance efficiency (ETPE). These criteria assess the confusion, diffusion, and efficiency properties, respectively. The PT sequence “ 01, 11,…., FF” is chosen due to its structured format, which makes it suitable for tracking changes caused by single-bit modifications during testing. The overall time complexity for evaluating the BIC criterion for an S-Box is o(n²), where n is the number of PTs (typically 32 for each S-Box variant). The function o(.) gives the minimum number of calculation steps, leading to a minimum processing time. Each input triggers a full AES encryption, and the correlation coefficient computation involves linear scans over output vectors. Given five different keys, the total computational complexity of the full evaluation framework becomes o(k×s×n²), where k=3 keys, s=5 S-Boxes.

3.1. Nonlinearity

Nonlinearity is a key objective for S-box design, aiming to introduce a nonlinear transformation from the PT to the encrypted data. In this context, nonlinearity refers to achieving a probability of 0.5 for the number of bits inverted from input to output, indicating an equal likelihood for fifty percent of all inputs and altered bits (Sosa, 2016; Hamdi, et al. 2015). While it is uncommon to precisely attain this probability, the goal is to approach or reach as close to 0.5 as feasible. The findings indicate that utilizing Henon chaotic maps in the S-box design enhances security, as evidenced by improved nonlinearity test results.

The primary objective of the S-box lies in effecting a nonlinear transformation from the original PT to its encrypted counterpart. Nonlinearity, in this context, denotes a scenario where there is a 50 % probability of bits being inverted from the input to the output, implying that half of all input bits undergo a change (Sosa, 2016; Hamdi, et al. 2015). While achieving an exact 0.5 probability which is rare in practical applications, the goal is to approximate this probability as closely as possible. In reality, achieving precisely half probability is an infrequent outcome, prompting a focus on attaining a probability as near to 0.5 as practicable. Our investigation results indicate that employing Henon chaotic maps in the construction of the S-Box leads to heightened security, particularly evident in nonlinearity tests. These findings suggest that the use of Henon chaotic maps in the S-Box yields superior nonlinearity characteristics when compared to alternative approaches.

3.2. Avalanche Effect

The avalanche effect characterizes an intriguing aspect of cryptanalytic algorithms (Eshmawi and Mahmoud, 2020). An S-Box is considered to exhibit the avalanche effect when a change occurs in one input bit of the PT, resulting in alterations in each output bit of the CT. This transformation occurs under the condition that at least half of the output bits will be flipped (El Batouty et al., 2020). Consider a new PT as follows: “01, 11, 22, 33, 44, 55, 66, 77, 88, 99, AA, BB, CC, DD, EE, FF”.

The avalanche effect, a captivating characteristic of cryptanalytic algorithms (Eshmawi and Mahmoud, 2020), describes a phenomenon where an S-Box exhibits this effect when altering a single input bit in the PT results in every output bit in the CT undergoing a change, ensuring that at least half of the output bits are flipped (Ejaz et al., 2021). In the context of our study, we observe this effect with a new PT sequence:

$$\begin{array}{c}“01,11,22,33,44,55,66,77,88,99,\mathrm{AA},\mathrm{BB},\mathrm{CC},\mathrm{DD},\mathrm{EE},\mathrm{FF}”.\\ \text{Avalanche effect}=\left(\mathrm{NC}/\mathrm{NT}\right)\times 100\end{array} (5)$$

Here, NC represents the count of changed bits in the CT and TN signifies the total number of bits in the CT. The Avalanche effect tests demonstrate that the S-Box utilizing LCM is more secure and exhibits a robust S-Box.

3.3. Execution Time Performance Efficiency (ETPE)

Speed serves as a gauge for assessing the relative performance enhancement during algorithm execution. The percentage Execution Time Performance Efficiency (ETPE) is defined as the following equation (Eshmawi and Mahmoud, 2020):

$$\text{ETPE}\mathrm{\%}=(\frac{{\mathrm{T}}_{\text{cld}}}{{\mathrm{T}}_{\text{now}}}-1)\times 100 (6)$$

Where T_old is the average execution time of S-Box without modification and T_new is the average execution time of S-Box with modification.

Having reviewed the foundational security metrics of S-Boxes, we now proceed to a more advanced and focused evaluation method: the Bit Independent Criterion (BIC). This criterion offers a more granular analysis of how single-bit changes in the input affect the independence of output bits.

4. Bit Independent Criteria

In the context of BIC testing correlation coefficients are used to assess the degree of dependency between two output bits after a single-bit change in the input. Lower values (closer to 0) suggest higher independence between the bits, which reflects stronger diffusion properties and better cryptographic performance.

The Bit-Independent Criterion (BIC), initially introduced by “Webstar and Tavares” (Webster and Travares, 1998), represents a desirable cryptographic property. It serves to examine the nonlinear transformation behavior by modifying input bits, thereby inducing changes at the output bit level. This suggests that, for a vector set generated by the complement of a single PT bit, the entire avalanche variable must exhibit pair independence (José et al., 2020).

Webster and Tavares introduced a property known as the Bit Independence Criterion (BIC) for S-Boxes (Agrawal and Himani, 2013; Webster and Travares, 1998). A function f: {0, 1}ⁿ → {0, 1}ⁿ; satisfies the BIC if for all i, j, k ϵ{1,2,……,n}, with j ≠ k, inverting input bit i leads to independent changes in output bits j and k. To quantify the bit independence concept, it is necessary to calculate the correlation coefficient between the j^th and k^th components of the output difference string, referred to as the avalanche vector, a^ei. A bit independence parameter corresponding to the effect of the i^th input bit change on the j^th and k^th bits of a^ei is defined as:

$$\mathrm{BIC}({\mathrm{a}}_{\mathrm{j}};{\mathrm{a}}_{\mathrm{k}})=\underset{1\le i\le n}{max}\left|\mathrm{corr}(a_j^{ei},a_{jk}^{ei})\right| (7)$$

Overall, the BIC parameter for the s-Box function f is obtained as:

$$\begin{array}{c}\mathrm{BIC}\left(\mathrm{f}\right)=\underset{1\le j,k\le n}{max}\mathrm{BIC}(\mathrm{aj};\mathrm{ak})\\ \mathrm{j}\ne \mathrm{k}\end{array} (8)$$

To assess the adherence of f to BIC, one can examine the correlation coefficient, which is captured by BIC(f). This parameter varies between 0 and 1, with an optimal value of 0, while in the worst-case scenario, it reaches 1, which demonstrates how close f is to satisfying the BIC.

Figure 3 illustrates the flowchart detailing the BIC test. In an ideal scenario, the output of this test is expected to be precisely 0, signifying optimal performance. However, in the worst-case scenario, the output can reach a maximum value of 1.

5. Results and Discussion

The experimentation involved varying inputs with the alteration of only one bit, and the ensuing step involved assessing the correlation between the outputs when this singular bit was inverted. The analysis encompassed the application of the BIC test to evaluate the cryptographic properties. Specifically, for each of the five presented S-Boxes, 32 unique PTs were considered, where the sole point of distinction lay in the inversion of a single bit. The BIC test was systematically applied to all 32 possible PTs, and the outcomes of this evaluation are summarized in the subsequent tables, in Appendix, for different secret keys “0123456789ABCDEF”, “FEDCBA9876543210”, and “F9E8D7C6B5A4012”. Tables 1, 2, and 3 present measuring the average of BIC test for different secret keys and Figures 4, 5 and 6 represent the average of BIC test for different secret keys graphically. This study assumes a standard AES block cipher structure and does not incorporate hardware-specific factors or side-channel attacks.

Procedure for BIC

Tables 1-3 and Figures 4-6 are quite extensive, a general analysis is now offered and comments on several key aspects.

1. Correlation values:

ᴏ The correlation values in the tables represent the relationship between the outputs when changing only one bit in the input.

2. Variation across bit position:

ᴏ One noticeable trend is the variation in correlation values across different bit positions. Some bit positions lead to higher correlations, indicating that flipping those bits has a more predictable impact on the CT.

Comparing the results, the key and PT dependent S-Box victimization RC4 algorithm implemented using Henon chaotic map gives the least correlation between the outputs. That means, the utilization of the Henon chaotic map in implementing the key and PT-dependent S-Box in the RC4 algorithm results in enhanced security compared to the key and PT-dependent S-Box in the standard RC4 algorithm.

6. Conclusion

The BIC serves as a valuable tool for scrutinizing the nonlinear transformation dynamics induced by alterations in input bits, influencing the corresponding output bits. In an ideal scenario, BIC registers a value of 0, indicating optimal performance, while its worst-case manifestation reaches a value of 1. A comparative analysis of the results reveals noteworthy findings for both the standard S-Box and the dynamic S-Box, where the BIC consistently hovers around zero or even below, denoting an absence of correlation between output variations and the inversion of a single input bit.

Contrastingly, the key and PT-dependent S-Box, implemented within the RC4 algorithm, presents a distinct behavior. When the input is set to “00012233445566778899aabbccddeeff”, the correlation manifests in its worst-case scenario. However, intriguingly, using any other input yields no discernible correlation between output variations. Further exploration into the application of the key and PT-dependent S-Box within the RC4 algorithm, implemented through the Henon chaotic map, reveals the least correlation between output changes. This observation implies that the implementation of the key and PT-dependent S-Box using the RC4 algorithm and Henon chaotic map is more secure compared to its counterpart utilizing the standard RC4 algorithm.

The correlation outcomes exhibit variations depending on the AES secret key employed. Specifically utilizing the key “0123456789ABCDEF”, the S-box generated by Henon Chaotic map demonstrates a minimal BIC value of -0.009373, surpassing the performance of AES with the standard S-Box. Similarly, when the key is set to “FEDCBA9876543210”, the S-Box generated by logistic chaotic maps yields a minimal BIC value of -0.01473. Furthermore, with the key “F9E8D7C6B5A40123”, the S-Box generated by RC4 attains a minimal BIC value of -0.00339.

To summarize, the BIC test results for the AES encryption algorithm, generated by different chaotic maps or RC4, consistently outperform the standard S-Box when considering various secret keys. These findings emphasize the potential advantages of incorporating diverse chaotic maps or the RC4 algorithm in the generation of S-Boxes for enhanced security within the AES encryption algorithm.

Future research could extend the BIC evaluation framework to real time encryption scenarios and IoT-constrained environments. This includes exploring adaptive or AI-driven dynamic S-Box generation techniques, applying the methodology to other cryptographic primitives such as lightweight ciphers or stream ciphers, and evaluating hardware implementations and side-channel resistance in future studies. These extensions aim to broaden the applicability and robustness of the proposed approach.

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Appendix