Brownian-Perturbed Hénon Map for Image Encryption: Application in Biomedical Images

1  Introduction

Cryptography is the practice and study of techniques for secure communication in the presence of adversaries. It is all about protecting and encrypting data so that if someone intercepts it during transmission, they are unable to read or understand the message [1]. The process of securing data in a certain way is called encryption [2]. Throughout history, encryption techniques have evolved from ancient methods like the Caesar cipher, key used method to today’s highly advanced systems [3]. As computational power and mathematical understanding have grown, old methods became easier to crack, making it essential to develop stronger and more sophisticated algorithms in order to stay one step ahead and keep communication safer.

A key theoretical leap was Kerckhoffs’s principle, which states that a cryptosystem should remain secure even if everything about the system except the secret key is publicly known [4], subsequently formulated by Claude Shannon as “The enemy knows the system” by which the approach to encryption can be publicly known, but not the key. From this, Shannon in 1945–1949 developed the basis of modern cryptography with his information theoretic analysis of secrecy systems [5]. Shannon defined the principles of confusion and diffusion, fundamental principles for block cipher design. Confusion hides relationships between key and cipher text, while diffusion disseminates the plain text structure throughout the output to prevent statistical attacks. Shannon also demonstrated that perfect secrecy, such that cipher text never provides any information on plain text, is attainable only with a key length equivalent to that of the message, as with the one time pad. In the contemporary age, cryptography divides into symmetric and asymmetric techniques, symmetric encryption employs one and the same key for decryption and encryption [2]. Block ciphers such as DES (Data Encryption Standard) and AES (Advanced Encryption Standard) employ substitution–permutation networks, where S-boxes introduce confusion and permutation layers provide diffusion. Asymmetric encryption is based on pairs of keys, one public and another private, to encrypt and decrypt communication [6]. Such schemes usually rely on difficult mathematical problems such as integer factorization or discrete logarithms. Elliptic Curve Cryptography (ECC) is a new, compact scheme for safe key exchange.

S-boxes play a pivotal role in symmetric key cryptography as the primary source of nonlinearity and confusion in block cipher algorithms [7]. A cryptographically secure S-box should exhibit certain properties [8], it must be a bijective mapping (permutation) to prevent cipher degeneracies, and it should have high Nonlinearity (NL) to resist linear cryptanalysis [9]. It should also satisfy the Strict Avalanche Criterion (SAC), a single-bit change in input flips about 50 percent of output bits, and the Bit Independence Criterion (BIC), output bit changes should be statistically independent [10]. Furthermore, a good S-box has low Differential Uniformity (DU) (minimizing the probability that a given input difference leads to a particular output difference), thereby resisting differential cryptanalysis [11]. Low maximum linear approximation probability (LAP) is desired for resistance to linear cryptanalysis, and high algebraic complexity (e.g., high algebraic degree) helps resist algebraic attacks [12]. Achieving all these properties simultaneously in an S-box is challenging, especially for larger S-box sizes (8×8), which motivates continued research into new design methods. Chaos theory has emerged as a promising approach for S-box construction due to the inherent randomness and sensitivity of chaotic maps [13]. Chaotic systems are deterministic but exhibit pseudo random behavior, a tiny perturbation in the initial state yields vastly different trajectories [14], akin to the avalanche effect in cryptography. These properties can be harnessed to generate S-boxes with good confusion properties. Indeed, numerous works have explored chaos-based S-box generation using maps like the logistic map, tent map, Lorenz system, etc., S-boxes with competitive cryptographic properties. Chaotic S-boxes offer flexibility and often satisfy bijectivity and NL requirements. However, using chaotic maps in digital cryptography also presents challenges [15]. Finite precision effects in digital implementations cause chaotic sequences to eventually repeat (entering periodic orbits) [16], potentially undermining unpredictability. Many 1D maps (e.g. logistic map) have limited chaotic ranges or smaller Lyapunov exponents, reducing entropy and making sequences more predictable or unevenly distributed [17]. Consequently, naive use of a single chaotic map may produce S-boxes with hidden weaknesses such as short cycle lengths, biases, or insufficient randomness. To overcome these limitations, researchers have combined multiple sources of chaos or introduced perturbations to enrich chaotic behavior [18]. Reference [19] proposed a discrete memristive conservative chaotic map for secure communication. Recently, several chaos based image encryption schemes have been proposed for biomedical applications, focusing on improved diffusion and resistance against statistical attacks [20–24].

In this work, we propose an enhanced chaos-based S-box generation method that synergistically combines the 2D Hénon map with Brownian motion. The Hénon map is a well known discrete time chaotic system with proven chaotic dynamics and a two dimensional state that provides more complexity than 1D maps. Brownian motion, on the other hand, is a stochastic process characterized by random, continuous movements, intuitively, the irregular zigzag movement of particles suspended in fluid (also known as a Wiener process in mathematical terms). By injecting a Brownian motion component into the iterative process of the Hénon map, we effectively introduce an external source of randomness that perturbs the chaotic trajectory in a controlled manner. This hybrid approach yields a composite chaotic system with improved unpredictability, reduced periodicity, and greater entropy compared to the use of the Hénon map alone. In essence, the Brownian perturbation continually perturbs the deterministic chaos, preventing it from settling into short cycles and ensuring a broader exploration of state space. We leverage this enhanced chaos to generate S-boxes that are highly random yet reproducible (with a secret key seed) and that fulfill the strict criteria for cryptographic substitution components. The Brownian perturbed Hénon (BPH) map is ideal for privacy safe encryption of biomedical and biofluid imaging in real time, without degrading diagnostic quality. The main contributions of the paper are:

1.   A hybrid Brownian perturbed Hénon chaotic map is proposed to mitigate finite-precision degradation in chaos based S-box generation.

2.   A deterministic, key dependent S-box construction framework is developed using rank order mapping, guaranteeing bijectivity and cryptographic soundness.

3.   The proposed hybrid map enhances robustness by combining deterministic chaos with controlled stochastic perturbations, resulting in increased unpredictability and improved statistical complexity compared with conventional chaotic maps.

4.   The controlled Brownian perturbations in the proposed map are specifically designed to suppress periodicity inherent in finite precision implementations while preserving determinism and key dependence.

The remainder of the paper is organized as follows. In Section 2, we review existing S-box design approaches and their limitations. Section 3 introduces the necessary preliminaries, including the Hénon chaotic map and discrete Brownian motion. Section 4 presents the proposed S-box generation method in detail. In Section 5, we evaluate the cryptographic properties of the proposed S-box and compare them with existing state-of-the-art designs. Section 6 demonstrates the effectiveness of the proposed S-box in image encryption through statistical and graphical analysis. Finally, Section 7 concludes the paper and outlines directions for future research.

2  Related Work

Over the past few decades, diverse approaches have been developed for constructing S-boxes, each with its own philosophy and trade-offs. We categorize them into a few broad categories, algebraic constructions, chaos-based methods, heuristic techniques and cipher derived S-boxes, and discuss how our work relates to and improves upon these.

Many classical ciphers use S-boxes designed with algebraic structures or human crafted criteria. The AES S-box, for example, is constructed by inverting elements in the finite field GF (28) and then applying an affine transformation [25]. This design gives the AES S-box excellent properties, it is bijective with an algebraic degree of 7, a DU of 4, and NL of 112 [26]. It is considered one of the best 8×8 S-boxes in terms of cryptographic strength. Similarly, the older DES cipher uses 4-bit S-boxes that were carefully constructed to satisfy known criteria [27]. Algebraic S-box design often involves ensuring that there are no simple mathematical relations, for example, AES is an affine equivalent of an almost perfect NL (APN) function, thus optimally resistant to differential attacks [28]. Chaos theory has inspired numerous S-box design methods, leveraging the complex dynamics of chaotic maps. Many works in this vein took simple 1D maps to produce pseudo random sequences that are then mapped to S-box permutations. For instance, an S-box pattern generation using an enhanced logistic map is proposed by El Gaabouri et al. (2024) [29], where the chaotic map’s output is used to shuffle values. Two dimensional chaotic maps and higher dimensional systems have also been used to achieve better randomness and avoid the limited chaos of 1D maps [30]. A notable challenge addressed in recent chaos-based designs is the problem of finite precision and short periodicity. When chaotic maps are implemented with finite bits, they can fall into short cycles. Some researchers have proposed perturbation or switching techniques to extend the effective period [31]. For example, Li et al. (2020) introduced an anterior perturbation to the logistic map to form an Enhanced Logistic Map with infinite chaotic length [32]. Similarly, composite systems that switch between multiple maps or use multiple chaotic variables have been explored [33]. Another vein of research treats S-box design as an optimization problem. Since metrics like NL, DU, SAC, etc., can be computed for a given S-box, one can attempt to search the space of all permutations (which is huge for 8-bit) for those with good properties. Direct brute force is infeasible, but heuristic search algorithms have been successful in finding S-boxes with excellent properties. Techniques include genetic algorithms, particle swarm optimization, hill-climbing, simulated annealing, and more recently, deep learning models (e.g., using generative adversarial networks to produce candidate S-boxes) [34]. For example, Wang et al. (2020) evolved S-boxes and achieved NL around 104–106 and DU 4–6. A 2023 study by Thakor et al. used a multi objective evolutionary approach to optimize boomerang and algebraic immunity along with standard criteria. A final category is the reuse or adaptation of S-boxes from existing cryptosystems. For lightweight cryptography, 4×4 S-boxes like that of the PRESENT cipher or GIFT cipher are often used as they are small and efficient. The PRESENT S-box, for instance, was designed to have no fixed points or opposite fixed points and to achieve DU 4 and relatively high NL ∼4 or 6. It has an affine algebraic representation which can be used in cryptanalysis, but overall is strong for its size. Researchers sometimes adapt such S-boxes to create new designs. While this approach benefits from known security of existing S-boxes, it lacks novelty and the flexibility of generating brand new S-boxes.

The literature shows a trade-off between structured approaches (algebraic, manual design) that give provable optimality in some criteria, and automated or chaotic approaches that can generate many alternative S-boxes but require careful analysis to ensure no weaknesses.

There are various ways to counter the shortcomings in chaotic systems. Chaos anti control methods are aimed at producing and improving chaotic phenomena by the application of control inputs, and they were applied in the study of chaotic encryption algorithms [35]. Stochastic perturbation approaches add random or noising perturbations to modify trajectories in the system, which helps affecting the escape rates and fractal properties found in chaotic systems [36]. Hybrid perturbation delay methods involve periodically disturbing system parameters to account for the digital precision effect and to achieve ergodicity [37]. Even these methods can be used for increasing the complexity of a system, however, they involve either increased structural complexity or parameter adjustments. Brownian motion, on the other hand, is a well known stochastic process for diffusion, which can be coupled with a pre-existing chaotic map for controlled randomness, efficient trajectory diffusion within phase space, and compensation for finite precision without incurring too much computational expense. The use of Brownian motion in chaos based cryptographic systems has proven useful for improving pseudo random sequence statistical properties and encrypting performance [38].

Our work combines two different sources, Hénon and Brownian, to form a composite chaotic system. A closely related prior effort is the work by Harmouch and El Kouch (2018) on using Brownian motion to generate 8×8 S-boxes [39]. They argued that Brownian motion’s random nature increases the complexity for attackers. However, Brownian motion alone is not deterministic or key driven in the same way chaotic maps are, and using it directly can complicate reproducibility of the S-box. We improve on this by using Brownian motion in a controlled manner, as a perturbation to a deterministic chaotic map. Thereby ensuring the S-box generation is both pseudo-random and deterministically reproducible given the secret key (which seeds both the chaotic map and the pseudo Brownian sequence). The Hénon map provides structure and reproducibility, while Brownian motion injects entropy and irregularity. Addition of controlled Brownian perturbation in the Hénon map can counteract the effect of finite precision problems in digital chaotic systems, making it more unpredictable and less cyclic in the produced sequences. When compared to traditional chaotic mappings, this unique property of being more irregular adds to other cryptographic desirable characteristics of higher nonlinearity in produced S-boxes [40].

3  Preliminaries

This section introduces the basic concepts and mathematical tools used to construct cryptographically robust S-boxes. We cover the Hénon chaotic map, discrete Brownian motion, and rank order mapping for generating bijective permutations.

3.1 Hénon Chaotic Map

The Hénon map is a two dimensional discrete time dynamical system introduced by Michel Hénon [41]. It exhibits chaotic behavior for certain parameter values and is defined as:

{xn+1=1−axn2+yn,yn+1=bxn,n=0,1,2,…(1)

where (x0,y0)∈R2 is the initial condition and (a,b)∈R2 are parameters. For the canonical values a=1.4 and b=0.3, the map exhibits sensitive dependence on initial conditions, generating the famous Hénon attractor, a fractal strange attractor. The Jacobian matrix is:

J=[−2ax1b0](2)

which shows that the map is area preserving. An area preserving chaotic map is a conservative dynamical system whose Jacobian determinant equals unity, ensuring that phase-space area remains invariant under iteration. This property is fundamental to its chaotic dynamics. The Hénon attractor is presented in Fig. 1a and Orbit diagram for the Hénon map, keeping b = 0.3 and varying a is shown in Fig. 1b. The plot shows the classic period doubling route to chaos.

Figure 1: Hénon map results: (a) Attractor and (b) Orbit diagram.

3.2 Discrete Brownian Motion

Brownian motion is a continuous time stochastic process with independent Gaussian increments. In digital implementations, it is commonly approximated in discrete time as a cumulative sum of zero mean random increments, referred to as discrete Brownian motion [42].

Xn=ΔX1+ΔX2+⋯+ΔXn(3)

where ΔXi∼𝒩(0,σ2) are i.i.d. Gaussian variables. In a cryptographic setting, a key dependent pseudo random generator produces these increments to ensure deterministic reproducibility. As shown in Fig. 2, the simulated discrete Brownian motion path exhibits the characteristic zigzag behavior, illustrating how the particle undergoes random, small scale displacements at each time step [43].

Figure 2: Multiple independent realizations of discrete Brownian motion.

3.3 Rank Order Mapping: Generating Permutations

Rank-order mapping is a mathematical technique to transform a real-valued sequence into a unique permutation of integers [44]. This is particularly useful when a continuous sequence, such as one generated by a chaotic map or a discrete Brownian motion needs to be converted into a discrete bijective sequence. Let N=2n denote the length of the sequence. Given a real-valued sequence

H=[H0,H1,…,HN−1](4)

the rank-order mapping produces a permutation P of {0,1,…,N−1} as follows:

1.   Sort H in ascending order and record the original indices idx(j) of the j-th smallest element.

2.   Define the permutation P:{0,…,N−1}→{0,…,N−1} by

P(idx(j))=j,j=0,1,…,N−1(5)

This procedure ensures that each element of the original sequence is assigned a unique rank, which will help us yielding a bijective mapping suitable for later use in constructions of S-boxes.

4  Proposed S-Box Generation Method

This section presents our method for constructing cryptographically strong S-boxes by combining a hybrid Hénon chaotic map with key dependent Brownian perturbations and applying deterministic rank-order mapping [44]. The procedure ensures sensitive dependence on the secret key while mitigating finite precision artifacts inherent to digital chaotic systems.

4.1 Hybrid Hénon-Brownian Chaotic Source

To enhance the unpredictability of the Hénon map in finite precision computations, we inject small key dependent Brownian perturbations into the system. Let ε1,ε2>0 denote small perturbation scales, and let ΔXn,ΔYn be Gaussian increments with mean 0 and variance σ2, generated from a cryptographically secure PRNG seeded by the secret key K. The hybrid system is defined as:

xn′=xn+ε1ΔXn,(6)

yn′=yn+ε2ΔYn,(7)

xn+1=1−a(xn′)2+yn′,(8)

yn+1=bxn′,(9)

where a and b are the standard Hénon parameters, and (x0,y0) are the initial states derived from the secret key. The perturbations ε1ΔXn and ε2ΔYn reduce latent periodic cycles while preserving the map’s chaotic properties.

Fig. 3 illustrates the phase space trajectories of the Hénon map after introducing controlled Brownian perturbations into its iterative process. Each trajectory corresponds to an independent realization generated using different pseudo random seeds while keeping the chaotic parameters fixed. Compared with the classical Hénon attractor under finite precision arithmetic, the perturbed system exhibits increased trajectory dispersion and reduced recurrence of short periodic orbits. This behavior indicates that the Brownian perturbation disrupts deterministic numerical artifacts and mitigates degradation effects commonly observed in digital chaotic implementations.

Figure 3: Trajectory of the Hénon map with Brownian perturbations applied at each step.

4.2 Key Dependent Initialization

The secret key K initializes the system as follows:

1.   Apply a key derivation function (KDF) to K.

2.   Extract the initial states (x0,y0), Hénon parameters (a,b), and perturbation scales (ε1,ε2) from the KDF output.

3.   Seed the PRNG for generating Gaussian increments ΔXn,ΔYn.

Each key produces a unique chaotic trajectory, ensuring that the resulting S-box is key-dependent and distinct.

4.3 Optional 3D Brownian Perturbation

While the Hénon map is two-dimensional, the principle of Brownian perturbation can be extended to three dimensions for applications such as image encryption. 3D Brownian perturbation introduces multidimensional (x,y,z) random variations into the Hénon map iterations to further enhance unpredictability and diffusion. This perturbation can be enabled or disabled depending on the required security level. For S-box generation, 2D perturbations suffice, but this extension demonstrates the method’s flexibility.

4.4 S-Box Construction via Rank-Order Mapping

Using the chaotic trajectory generated by the hybrid system, a bijective S-box is obtained through rank order mapping. Let N=2n denote the S-box size (e.g., N=256 for an 8×8 S-box).

1.   Discard the first N0 iterations of xn as transient to avoid initialization bias (e.g., N0=1000).

2.   Collect the next N values H0,H1,…,HN−1 from the x-coordinate.

3.   Sort H in ascending order and record the original indices idx(j) of the j-th smallest element.

4.   Define the S-box permutation S:{0,…,N−1}→{0,…,N−1}:

S(idx(j))=j,j=0,1,…,N−1.

This method guarantees a bijective, key dependent S-box with strong diffusion properties while mitigating finite precision artifacts in the chaotic system.

4.5 Hénon–Brownian S-Box Generator Algorithm

Algorithm 1 describes the step-by-step procedure employed to construct the Hénon–Brownian S-box generator.

4.6 Toy Example: 4 × 4 S-Box (16 Elements)

To illustrate the procedure of Algorithm 1, we consider a small-scale example with N=16:

(x0,y0)=(0.10,0.30),a=1.4,b=0.3,ε1=ε2=10−2,N0=2,N=16

After discarding N0 transient iterations, the Hénon Brownian system produces the following x-sequence:

H=[0.72,−0.34,0.11,0.93,−0.15,0.58,0.46,−0.09,0.67,−0.44,0.25,0.81,0.39,0.12,0.56,0.05].

Applying rank-order mapping, we obtain the permutation index array:

idx=[9,1,4,7,15,2,13,10,12,6,14,5,8,0,11,3].

Finally, the resulting 4×4 S-box is:

S=[13,1,5,15,2,11,9,3,12,0,7,14,8,6,10,4].

This example demonstrates the same procedure as Algorithm 1, but on a smaller scale for clarity. The full 8×8 S-box is obtained by following identical steps with N=256.

4.7 Discussion on Proposed Method

We now discuss how parameters are chosen in practice and why these choices contribute to both functionality and security of our Proposed method.

4.7.1 Parameter Selection

Each stage of the construction is governed by parameters that balance chaotic richness with numerical stability:

1.   Hénon map parameters (a,b): The canonical values a=1.4 and b=0.3 keep the map within a strongly chaotic regime. Minor variations may be derived from the secret key, provided the system remains chaotic.

2.   Perturbation scales (ε1,ε2): Small perturbations (10−3–10−2) preserve the geometry of the Hénon attractor while introducing stochastic “zigzag” deviations, thereby reducing the risk of short cycles in finite-precision computations.

3.   Transient length N0: Discarding the first N0=50,000 iterations (for an 8×8 S-box) eliminates bias from initial conditions and ensures that the system samples the attractor’s invariant distribution.

4.   Sequence length N: For an n-bit S-box, N=2n outputs are drawn from the perturbed trajectory. Applying rank-order mapping to this sequence yields a bijective permutation of integers.

5.   PRNG seeding: The Gaussian increments used in perturbation are generated from a pseudo random number generator seeded with the secret key, ensuring both reproducibility and key dependence.

4.7.2 Remarks and Implementation Guideline

The proposed framework admits several natural extensions and practical considerations. Larger S-boxes can be generated by increasing the sequence length N, while multiple S-boxes may be obtained by exploiting different chaotic coordinates. Furthermore, the perturbation principle is not restricted to two dimensions; its extension into three dimensions enables stronger shuffling mechanisms in multimedia and image encryption applications. In all such cases, parameter values must be tuned to preserve both chaotic behavior and the small perturbation scales that underpin security.

From an implementation perspective, double-precision floating point arithmetic is recommended in order to maximize the effective state space and reduce numerical artifacts arising from finite precision. When applying rank-order mapping, a stable sorting procedure should be used to ensure consistency in the presence of potential ties. Finally, if the S-box is intended for high-assurance scenarios where secrecy is critical, the Gaussian increments driving perturbations should be generated by a cryptographically secure PRNG, ensuring resistance against adversarial reconstruction attempts.

4.7.3 S-Boxes Generated Using Proposed Scheme

In the proposed scheme, we present generated S-boxes. These S-Boxes provide better resistance against cryptanalytic attacks. The constructed S-Boxes are presented in Tables 1 and 2.

5  Results and Comparisons

We generated a number of 8×8 S-boxes using the proposed HénonBrownian methodology with different random keys and computed their cryptographic properties. In this section, we present the results for one representative S-box instance generated by a particular key. Furthermore, we compare these metrics with those of well-known existing S-boxes to highlight the effectiveness of the proposed construction.

5.1 S-Box Generation Time Analysis

The computational efficiency of our proposed Hénon Brownian S-box generation method was evaluated by measuring the time taken to generate each S-box for a set of different keys. Fig. 4 shows the generation times for 20 representative keys. The Fig. 4 indicates that the time remains fairly consistent across different keys, with minor variations due to the random initialization in the Hénon Brownian process. The average generation time per S-box is approximately 0.01 s, demonstrating that our method is computationally efficient and suitable for practical cryptographic applications.

Figure 4: Generation time for Hénon Brownian S-boxes across 20 different keys.

5.2 Cryptographic Properties of the Proposed S-Box

The strength of any S-box is determined by its ability to withstand cryptanalytic attacks. To this end, the proposed S-box has been evaluated against the standard set of cryptographic properties that ensure confusion, diffusion, and resistance to both linear and differential attacks.

Bijectivity:

By construction, the S-box is a permutation of 0–255, hence bijective (one-to-one and onto). It can be seen in Fig. 5 that all 256 output values are unique and every input maps to a unique output. There were no fixed points in our S-box (i.e., no input x such that S(x)=x) and no opposite fixed points (S(x)=x¯) either, which eliminates certain weak linear structures.

Figure 5: Bijectivity analysis of the proposed S-box.

Nonlinearity:

The nonlinearity of an S-box is measured as the minimum Hamming distance of its output Boolean functions from all affine functions [9]. For an 8×8 S-box, there are 8 Boolean output functions. We computed the Walsh–Hadamard spectrum of each output bit to find its nonlinearity. Our S-box’s minimum nonlinearity among the 8 bits is 106, and the average nonlinearity is 110.5. This is good enough, the theoretical maximum for 8-bit S-boxes is 112, achieved by AES. Achieving 106 means the S-box’s outputs are highly nonlinear and well balanced. This strong NL directly indicates resistance to linear cryptanalysis, since any linear approximation holds only slightly better than random guessing. The Walsh–Hadamard spectrum of an output bit is illustrated in Fig. 6.

Figure 6: Walsh–Hadamard spectrum of an output bit.

Strict Avalanche Criterion (SAC):

We tested SAC by flipping each input bit and observing the change in output. The average number of output bits that change was 3.994 out of 8 (49.925 percent), which is essentially the ideal 50 percent [45]. Each of the 8 input bits individually showed close to 50 percent avalanche effect, with no weak input bit. Satisfying SAC means a one-bit difference in input leads to a completely different output, ensuring strong diffusion as shown in Fig. 7.

Figure 7: Strict Avalanche Criterion (SAC) results of the proposed S-box.

Bit Independence Criterion (BIC):

BIC refines the SAC by additionally testing whether output bits change independently when a single input bit is flipped [46]. We evaluated the BIC–SAC values by computing correlations between the avalanche patterns of different output bits. All correlation values were found to be very close to zero (|ρ|<0.05), which indicates excellent statistical independence among output bits. This ensures that no exploitable linear or multi-bit dependencies exist, thereby strengthening resistance against advanced differential and linear cryptanalysis techniques as shown in (Fig. 8).

Figure 8: BIC–SAC analysis of the proposed S-box.

Comparison

To validate the effectiveness of the proposed S-boxes, we compared their cryptographic performance with several state-of-the-art S-boxes reported in recent literature. Table 3 presents a comprehensive comparison in terms of NL, SAC, BIC, and DU. The proposed S-boxes achieve high average nonlinearity (around 106–106.5) and well-balanced SAC values close to the ideal 0.5, while maintaining a low maximum differential probability of 0.0469. These results demonstrate that our S-boxes not only provide strong confusion and diffusion properties but also surpass many existing constructions, highlighting their robustness against classical cryptanalytic attacks.

6  Application in Image Encryption

To demonstrate the practical utility of the proposed S-box, we apply it to image encryption. Digital images contain a high degree of correlation among adjacent pixels, which makes them vulnerable to statistical attacks if not sufficiently diffused and confused. The nonlinearity and key-dependence of our S-box design provide an effective mechanism for breaking this correlation and achieving secure encryption.

Let D=[dij] represent a grayscale image with a rows and b columns, where each pixel dij is represented over the symbol set {0,1,…,2m−1}. The encryption procedure using the proposed S-box generator is as follows:

1.    S-box generation: Generate a collection S of m×m S-boxes using the proposed Hénon Brownian method. Let ρk denote the k-th S-box in the set S.

2.    Selection matrix: Construct an (a×b) matrix R=[rij] such that each rij is an integer in the range [1,|S|], chosen via a key-dependent pseudo-random generator.

3.    Pixel substitution: For each pixel dij in the plain image, compute the corresponding encrypted pixel:

eij=ρrij(dij),

thereby obtaining the ciphertext image E=[eij].

The decryption process follows the same procedure in reverse using the identical secret key. Since each S-box in the set S is bijective, its inverse ρk−1 exists and can be efficiently computed. For each encrypted pixel eij, the original pixel is recovered as

dij=ρrij−1(eij),

Fig. 9 illustrates the plain, encrypted, and decrypted images, demonstrating that the proposed scheme produces noise-like ciphertext while allowing exact recovery of the original image without any loss of visual information.

Figure 9: Visual comparison of plain, encrypted and decrypted images using the proposed S-box.

In addition to visual inspection, a quantitative analysis was carried out to evaluate the correlation and entropy of the encrypted images. Table 4 reports the correlation coefficients of adjacent pixels in horizontal, vertical, and diagonal directions, along with entropy values, for both plain and encrypted images. The plain images exhibit high pixel correlation, which is expected due to the natural redundancy in images. However, after encryption, the correlation drops close to zero in all directions, indicating effective removal of statistical dependencies. Furthermore, the entropy values of the encrypted images approach the ideal value of 8, confirming that the proposed scheme produces cipher texts with high randomness and resistance against statistical attacks.

7  Conclusion and Future Work

In this work, we presented a novel methodology for constructing S-boxes with strong cryptographic properties. The proposed S-boxes were thoroughly evaluated using well established criteria such as Bijectivity, NL, SAC and BIC. In addition, their practical effectiveness was demonstrated through image encryption experiments. The visual inspection of encrypted images, together with correlation and entropy analysis, confirmed that the proposed scheme achieves high confusion and diffusion, produces cipher texts with noise like statistical distributions, and effectively removes redundancy from plain images. These results highlight the robustness of the proposed design against classical cryptanalytic and statistical attacks. Additionally, the high key sensitivity ensures that minimal variations in the secret key yield completely different S-boxes and encrypted outputs, further strengthening resilience against key related attacks.

Although the proposed approach shows promising results, there remain several interesting directions for future research. One avenue is to investigate adaptive or key-dependent S-box generation mechanisms, which can dynamically modify substitution layers to resist structural attacks more effectively. Another direction is to explore the integration of the proposed S-boxes into full fledged block and stream cipher architectures, evaluating performance metrics such as throughput, latency, and energy consumption in practical implementations. Moreover, a systematic study on the robustness of the proposed S-boxes against emerging attack paradigms, including algebraic attacks, and machine learning assisted cryptanalysis, can provide additional confidence in their security. Finally, extending the methodology to lightweight and resource constrained environments can ensure that strong cryptographic properties are achieved without compromising efficiency.

Acknowledgement: The authors gratefully acknowledge the support provided by Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Funding Statement: This research was funded by Princess Nourah bint Abdulrahman University Researchers Supporting Project Number PNURSP2026R500, Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Author Contributions: Walaa Alayed: conceptualization, validation, writing—review and editing, funding acquisition. Asad Ur Rehman: methodology, formal analysis. M. Awais Ehsan: methodology, software, investigation, writing—original draft preparation. Waqar Ul Hassan: validation, visualization, writing. Ahmed Zeeshan: conceptualization, formal analysis, supervision, writing—review and editing. All authors reviewed and approved the final version of the manuscript.

Availability of Data and Materials: Data will be available on request. Ahmed Zeeshan (ahmad.zeeshan@iiu.edu.pk). Datasets used are publicly available.

Ethics Approval: This study does not involve human participants, animals, or any clinical trials. All image data used in the experiments are publicly available benchmark images; therefore, ethical approval was not required.

Conflicts of Interest: The authors declare no conflicts of interest.

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