Secure image encryption using a 4D chaotic system and Langton’s ant cellular automaton

Abstract

The recent explosion of high-rate multimedia transfer has heightened the need for image ciphers that are lossless, payload-grade, and computationally efficient, yet fully structure-adaptive. Traditional block ciphers are effective for byte streams but do not account for spatial dependencies in images, while many chaos-based methods remain ad hoc, lacking strict invertibility and clear keying or synchronization mechanisms. Motivated by these gaps, we present a keyed image cipher that integrates a four-dimensional (4D) chaotic driver with a rule-adaptive variant of Langton’s Ant cellular automaton. The scheme performs pixel permutation followed by two mutually inverse decimal diffusions with cross-channel feedback, and a symbolic diffusion stage that encodes bytes into Ant symbols intermingled through three distinct, keystream- and rule-dependent operators. The core novelty is a provably bijective symbolic diffusion with an explicit inverse and a branch-number–style local diffusion lower bound, combined with a linear-time streaming implementation. The cipher passes the NIST SP 800-22 statistical tests on both keystream and ciphertext-derived bitstreams. Experiments on standard test images show NPCR UACI Shannon entropy bits/pixel, negligible adjacent-pixel correlations, high key sensitivity, and strong robustness against 20% salt-and-pepper noise and 20% occlusion. The overall key space is approximately , indicating high resistance to brute-force attacks.

Introduction

Securing multimedia at scale demands ciphers tailored to the statistical structure, volume, and latency constraints of images and videos. Foundational treatments codify requirements for confidentiality, integrity, and authentication in modern communication systems¹, while domain-specific designs demonstrate clinically oriented protection of medical images via DNA cryptography and elliptic curves². A bibliometric review maps the role of cryptographic techniques across AI security, highlighting research trends and gaps³. Comprehensive surveys chart the landscape of chaos-based video encryption and its design tradeoffs⁴, and robust chaos-driven methods for medical imagery validate strong resistance to typical attacks⁵. DNA–chaos hybrids further enrich substitution and diffusion with nucleotide-inspired operations⁶. Beyond confidentiality alone, joint encryption–watermarking pipelines address social-image security in hybrid domains⁷.

Seminal work established chaotic dynamics as a fertile basis for cryptosystem design, deriving a chaotic encryption algorithm and motivating practical constructs⁸, and proposing symmetric ciphers based on two-dimensional (2D) maps that align with domain structure in images⁹. These designs operationalize Shannon’s confusion and diffusion principles for multimedia data¹⁰. A survey systematizes chaos-based image encryption across spatial, transform, and spatiotemporal domains, providing taxonomies of maps, architectures, and evaluation criteria¹¹. Rigorous empirical security analyses caution against under-justified motivations and advocate comprehensive cryptanalytic testing¹², while security measurement frameworks quantify robustness in chaos+DNA medical communication schemes¹³.

Classical map-based schemes illustrate the evolution from one-dimensional (1D) to 2D dynamics and beyond. Logistic-map encryption established lightweight confusion–diffusion with 1D dynamics¹⁴, and combinations of 1D maps enhanced color-image security with simple, fast primitives¹⁵. New 2D systems with hidden attractors enlarged the design space¹⁶. Enhanced hyperchaotic Hénon variants demonstrated efficient 2D encryption with improved statistical properties¹⁷. Transform-augmented chaos (e.g., cosine-transform-based systems) improved ergodicity and parameter sensitivity for image protection¹⁸, while cryptanalysis-driven improvements strengthened Feistel+dynamic DNA schemes by fixing weaknesses and refining diffusion paths¹⁹. A landmark 3D cat-map construction instantiated the canonical permutation–diffusion pipeline for images²⁰, and fast chaotic primitives delivered high-throughput encryption suitable for real-time processing²¹.

A central trend is the move to high-dimensional and hyperchaotic dynamics to expand key space, increase Lyapunov exponents, and harden against statistical/brute-force attacks. Early demonstrations established the benefits of high-dimensional systems for image encryption²², and six-dimensional (6D) hyperchaotic designs combined with DNA encoding achieved very large key spaces with strong diffusion²³. New four-dimensional (4D) chaotic systems with coexisting attractors and circuit implementations provided hardware-feasible chaotic sources²⁴. An improved 4D system with evolutionary operators demonstrated robust performance on contemporary color images²⁵. Quadratic polynomial hyperchaotic maps with pixel-fusion strategies yielded efficient ciphers with compelling security metrics²⁶. Detailed dynamic analyses of 6D multistable memristive chaotic systems uncovered wide hyperchaotic regimes and demonstrated encryption applications²⁷. Three-dimensional (3D) architectures leveraging segmentation-driven diffusion further increased mixing and sensitivity in the cipher pipeline²⁸.

Memristive and neuromorphic chaos have emerged as versatile, hardware-amenable drivers. A 3D memristive cubic map with dual discrete memristors was designed, implemented, and applied to image encryption, illustrating compact circuits and strong dynamics²⁹. A three-dimensional memristor-based hyperchaotic map was engineered for pseudorandom number generation (PRNG) and multi-image encryption, demonstrating parallelism and high entropy³⁰. A parallel color-image cipher driven by a 2D logistic–Rulkov neuron map delivered throughput and security gains through neuron-inspired chaos³¹. For video, a discrete sinusoidal memristive Rulkov-neuron approach encrypted segments efficiently while maintaining strong security margins³². A fractional-order Hopfield neural-network-based framework was validated as a high-performance privacy-preserving approach³³. Efficient chaotic image encryption was also realized via dynamic vector-level operations and a 2D-enhanced logistic modular map, improving diffusion and implementation efficiency³⁴. Hardware-oriented work on memristive Hopfield networks demonstrated bursting firing dynamics, an image-encryption mechanism, and a corresponding implementation path³⁵.

Hybrid constructions enrich confusion/diffusion with complementary primitives and coding layers. Substitution–permutation networks (SPNs) driven by chaos deliver strong nonlinearity and structural clarity in image encryption³⁶. DNA–chaos ciphers were tailored for telemedicine and healthcare, emphasizing robustness under realistic conditions³⁷. Dynamically synthesized S-boxes (e.g., double-affine or chaos-tuned) increased nonlinearity and thwarted differential attacks in efficient schemes³⁸. DNA-computing and chaotic dynamics were co-designed to raise complexity and key sensitivity³⁹, and chaos+DNA encoding formed the basis of effective image ciphers with strong avalanche effects⁴⁰. Quantum-image directions complemented these efforts: an Arnold-transform/S-box scrambling scheme instantiated quantum image encryption operations⁴¹, and Lorenz hyperchaos provided a quantum-resilient trajectory for future-proof image protection⁴². Cellular-neural hybridization with DNA sequence operations supplied an additional pathway for spatial coupling and symbolic diffusion in images⁴³. Multi-image pipelines were strengthened by combining hyperchaotic drivers with SVD and a modified RC5 layer for layered defense and improved resilience⁴⁴.

Systematic cryptanalysis has been crucial in hardening designs. Entropy-based cryptanalysis exposed weaknesses in a chaotic scheme and guided corrective redesign⁴⁵. Attacks on chaos-derived S-box encryption revealed structural flaws and motivated improved constructions⁴⁶. A focused cryptanalysis of the 2D logistic-adjusted-sine map (LASM) family led to concrete improvements that restore security margins⁴⁷.

Learning-based methods increasingly intersect with chaos to automate or adapt cipher components. A neural-network/chaos co-design enabled double-image encryption with enhanced flexibility⁴⁸, and a CNN-assisted framework tightened confusion–diffusion coupling and supported data-driven parameterizations for improved security–efficiency tradeoffs⁴⁹. In deployment-oriented research, real-time RGB encryption for IoT used enhanced chaotic sequences to meet latency and resource constraints without sacrificing robustness⁵⁰.

Driven by these benefits, we propose a new image encryption scheme that integrates the strength and complexity of a four-dimensional (4D) chaotic system with the symbolic computational power of Langton’s Ant cellular automaton. This union provides strong confusion and diffusion through layered transformations (permutation plus dual diffusion) and addresses known weaknesses of lower-dimensional chaotic designs and conventional ciphers. In our evaluations, the scheme demonstrates strong resistance to brute-force search, differential cryptanalysis, and standard statistical attacks while remaining practical and efficient. Overall, the principal contributions of this paper include:

• A keyed, lossless image encryption scheme coupling a 4D chaotic system with rule-adaptive Langton’s Ant symbolic encoding, integrating pixel reordering with two complementary diffusion layers (decimal and symbolic with inter-pixel feedback).

• A formal derivation for the Ant-based symbolic diffusion showing bijectivity (explicit inverse) and a local diffusion guarantee (branch-number style lower bound), ensuring well-posed decryption and strong intra-pixel mixing.

• A nonce-based KDF workflow to derive the initial chaotic state and working keys, a minimal header for stateless synchronization, and an optional tilewise re-seeding strategy for loss tolerance and parallelism.

• Linear-time complexity O(MN) with O(1) extra memory in streaming form; naturally parallel on CPU/GPU. Constant factors are quantified in the complexity subsection.

• Beyond histogram/correlation/entropy and differential tests (NPCR/UACI with 8-bit reference ranges), we report NIST SP 800-22 results (keystream and ciphertext-derived bitstreams, ) using the two-level pass criteria, supporting near-ideal diffusion and randomness.

• We position the scheme against classical chaos-based ciphers and recent cryptanalysis findings to motivate design choices; the resulting efficiency and robustness make it suitable for bandwidth- and latency-constrained scenarios (e.g., medical archives and remote-sensing imagery).

The paper has the following organization: We first introduce in detail the preliminaries necessary for the important concepts like chaotic systems, Langton’s Ant cellular automaton, and cryptographic operations utilized. We then outline the methodology of the proposed scheme with particular focus on its unique fusion of chaotic diffusion, symbol encoding, and dynamic key stream generation. We then evaluate our scheme exhaustively, supported by extensive simulations and security studies, to confirm its potency and resilience. Lastly, we conclude by summarizing our major findings and suggesting directions for further investigation, highlighting how our proposed scheme is well-suited to address current security demands.

Preliminaries

The suggested image encryption mechanism employs basic concepts and techniques outlined within the present section, specifically a 4D chaotic map, Langton’s Ant rule-based automaton, and cryptography involving permutation, confusion, and diffusion. These are the building blocks of the encryption scheme that provide tremendous security by their own intrinsic randomness and complexity.

4D chaotic system

The 4D chaotic system used in this study is defined Lai et al.²⁴ by the following set of differential equations:

1

where are the state variables of the system and p, q, r are the control parameters. The system is initial condition and parameter value sensitive, wherein the initial condition and control parameter settings are being utilized as keys for encryption within the scheme.

To use the continuous-time driver in an iterative cipher, we discretize it with a fixed-step explicit Runge–Kutta method of order four (RK4), step size , in double precision. Writing and denoting the discrete state by , one RK4 update is

From a key/nonce-derived initial condition and fixed (p, q, r), we discard a warm-up of steps and then sample every step (stride ) to produce MN samples for an image.

With initial condition and parameters , the system exhibits a butterfly-shaped chaotic attractor. Figure 1 specifies the viewed coordinates for each subplot: (a) 3D phase portrait in , (b) 2D projection in , (c) 3D phase portrait in , and (d) 2D projection in .

Fig. 1.

Chaotic behavior of dynamical system.

All chaotic sequences were generated by discretizing the continuous 4D system with a fixed-step explicit Runge–Kutta method of order four (RK4) to ensure reproducibility. Let denote the state at discrete step n and the right-hand side of the ODE. With step size (double precision) each RK4 update is computed as

For reproducible keystreams we (i) use double-precision arithmetic, (ii) discard a warm-up of RK4 steps, and (iii) sample with stride thereafter to produce MN samples for an image. To extract bytes from a sampled state component we apply fractional-part extraction and scaling:

With the above , fixed system parameters, and a key/nonce-derived initial state , the RK4-based generator is fully reproducible.

Tent map

Tent map is a simple and robust piecewise linear chaotic map commonly used in cryptographic applications because it is highly sensitive to control parameters and initial conditions. It is defined as:

2

where , and and are control parameters determining the map shape and chaotic range. The tent map is used in the proposed image encryption scheme for the generation of pseudorandom sequences that control rule selection and key scheduling. Nonlinearity and uniform distribution of the tent map add to the unpredictability and diffusion properties of the cipher. By wrapping inputs to the valid domain via the use of modular arithmetic, outputs are ensured to be bounded, and the system is suitable for iterated application within digital systems. This incorporation aids the resistance of the system to statistical and differential attacks.

Chaotic state variable selection: rationale and overview

Key determinants to ensure the proposed encryption technique is cryptographically strong are the careful choice and usage of chaotic state variables. The sequences generated by the chaotic map, embodied within 4D difference equations, demonstrate all the features of increased sensitivity toward initial values, deterministic randomness, and long-term unpredictability. This section, therefore, discusses at length the method applied to effectively choose the chaotic state variables by drawing from the CDSVSP outlined within the relevant literature.

Chaotic dynamical state variables selection procedure

In order to adopt chaotic sequences with reduced algorithmic complexity, the Chaotic Dynamical State Variables Selection Procedure is changed as follows:

Generation of Chaotic Sequence: The 4D chaotic dynamical system is iterated for times, where to avoid transient dynamics, with the size of the total pixels being given by (256 pixels in the current scenario)

Chaotic State Definition: A chaotic system at each iteration generates four different variables . These chaotic states map to pixel intensity values, which are stored as a 1D list .

Selection Criterion for Variables: A dynamically computed index is used for each pixel Q(i) to choose the state variable V(i) from the chaotic set. The index for selection, which is referred to as , is

where the tent map provides randomness and where are secret key parameters

Langton’s ant cellular automaton

Langton’s Ant is coupled with the chaotic driver to introduce a discrete, rule-based symbolic layer that complements the continuous-state chaos. This layer provides per-pixel, key-driven transformations that enhance confusion and diffusion.

Alphabet and rule selection. Let be the symbolic alphabet. For each pixel index i, a rule

is derived from the chaotic keystream (details in Step 3). Each rule is associated with a bijection (encoding) ; may be the identity or the complement, depending on , so that different rules induce distinct binary embeddings of .

Decimal-to-symbolic mapping. For an 8-bit (256-level) pixel value , we take its binary expansion, group bits into pairs, and map each pair to a symbol in according to a rule-dependent table (cf. Table 1). For example, has binary (11100110); under Rule 2 this yields pairs (11, 10, 01, 10) and the symbolic word by the Rule 2 column in Table 1.

Table 1.

Binary-to-Ant mapping (rule-dependent).

Binary Pair	Rule 1	Rule 2	Rule 3	Rule 4	Rule 5	Rule 6	Rule 7	Rule 8
00	R	L	R	L	R	L	R	L
01	L	R	L	R	L	R	L	R
10	R	R	L	L	R	R	L	L
11	L	L	R	R	L	L	R	R

Three distinct symbolic operators. To avoid collapse into a single operation, we define three distinct, invertible, rule- and keystream-dependent operators on via their binary embeddings. Let and write . Let be keystream bits (or functions of the previous cipher symbol) chosen so that at least one of is 1 for every i (this guarantees non-identical truth tables). Define:

where is bitwise XOR and . Because the image of each operator is of the form with a different boolean function per operator (and or whenever or ), the three truth tables are not identical at index i and vary with .

Symbolic diffusion update and invertibility. Let be the pre-diffusion symbols for a pixel (after encoding). With selector , we apply a triangular in-place update using the distinct operators above:

In the binary domain this is , , with operator-specific boolean functions determined by . The map is bijective with explicit inverse:

followed by to recover symbols. Thus decryption is well-posed.

Figure 2 shows the Langton’s ant after 99,999 steps.

Fig. 2.

Langton’s ant after 99,999 steps.

Proposed encryption scheme

In this section, we outline the holistic methodology underpinning the Proposed Image Encryption Scheme (PIES), which combines synergistically a 4D chaotic dynamical system with a cellular automaton to provide robust cryptographic security. For clarity, readability, and easy comprehension, the framework of encryption has been strictly demarcated into separate phases: initial chaotic data generation and preprocessing, generation of chaotic keystream, pixel permutation, symbolic encoding through cellular automaton rules, multi-stage diffusion with decimal and symbolic operations, and, ultimately, reconstruction of encrypted cipher image. This high-powered synergy of transformation comprising decimal-based diffusion, chaotic permutation, and symbolic processing through automata especially strengthens the resistance of the cipher, ensuring great resistance against statistical, differential, and brute-force attacks. What’s more, this concise and representative flowchart is provided to clearly present each step, hence ensuring easy comprehension, readability, and replicability of the methodology of encryption.

Image preprocessing and chaotic sequence initialization

Consider an RGB image of dimensions with three distinct colour channels: Red (), Green (), and Blue (). Each channel is reshaped into a linear 1D vector of length , defined as:

To produce high-quality chaotic sequences for encryption, we initialize the 4D chaotic system with secret initial conditions. The chaotic system is numerically integrated for steps, ensuring elimination of transient dynamics, with . The integration results in four distinct chaotic sequences:

where each sequence , for , comprises chaotic values employed as secure sources of randomness throughout subsequent encryption stages. Initial state variables are assigned from the first elements of the chaotic sequences generated, such as

Chaotic-based keystream generation mechanism

The robustness of the proposed image encryption scheme primarily arises from a Chaotic-Based Keystream Generation Mechanism (CBKGM), integrating outputs from the 4D chaotic system with Langton’s Ant automaton rules. This mechanism ensures extreme sensitivity to initial conditions, such that slight variations in input data or secret parameters yield drastically different key sequences, thus fortifying the cryptographic strength.

Initially, the RGB channel vectors , , and are combined into a single composite sequence , utilizing secret integer multipliers , as follows:

For each pixel index , the tent map, parameterized by two secret parameters , generates a chaotic-based index by:

The value determines the selection among the four chaotic sequences . For each selected sequence, associated counters track the accessed elements, and the chaotic variable is chosen accordingly:

Next, the fractional component of this chaotic value is extracted and magnified to generate an 8-bit key value through:

Repeating this procedure iteratively for all pixel indices, a secure keystream is constructed:

This generated keystream is utilized in the subsequent encryption phases to ensure effective confusion and diffusion. The described key-generation process inherently achieves high sensitivity and unpredictability, ensuring strong resistance against various cryptanalytic methods. The overall procedure for keystream generation is clearly depicted in Fig. 3.

Fig. 3.

Keystream generation procedure.

Encryption procedure

The suggested image encryption scheme gains increased crypto-security through orderly combining chaotic key streams, pixel shuffling, and symbolic calculations derived from Langton’s Ant cellular automaton. Each operation plays an appreciable part towards confusion and diffusion, which prevents the image from being attacked statistically and cryptographically. To enhance clarity, image size is , with total pixels .

Step 1: Initial Decimal-Level Diffusion

Let be the three chaotic keystreams. Define initial feedback bytes from the available keystream at index 0:

Then, for all ,

Output: Arrays .

Step 2: Pixel Permutation Using Chaotic Sequences

To eliminate spatial correlations, the pixel values are permuted using chaotic sequences:

1. Concatenate channels into a single vector of length :

   
2. Generate permutation indices from a chaotic tent map:

   

   where are chaotic map parameters.
3. Reorder the elements of :

   
4. Divide the permuted vector back into RGB channels:

   

Output: Arrays .

Step 3: Langton’s Ant Symbolic Encoding and Diffusion

1. Decimal-to-symbolic conversion. Let be the Ant alphabet. For each pixel index i, select the rule

   

   from the chaotic keystream k(i), and use the rule-dependent encoder with binary embedding to map 8-bit channel values to symbolic words (pairwise mapping per Table 1):

   

   where .
2. Symbolic-level diffusion (distinct, keystream- and rule-dependent). Define keystream-controlled bits with for all i. For with and , define the three distinct Ant operators:

   

   with . Let denote the current symbols (componentwise over the length-4 words) and set the selector from a chaotic sequence L(i). Apply the triangular in-place update:

   

Output and decode back to decimal. Set and recover the post-diffusion decimal values via the inverse encoder:

Step 4: Symbolic-to-Decimal Conversion

Convert symbolic ant-sequences back to decimal using the same rule-based mappings influenced by chaotic sequences:

Output: Decimal arrays .

Step 5: Final Decimal-Level Diffusion

Perform a final diffusion on the transformed decimal pixel arrays with fresh chaotic keystreams . For each pixel :

Output: Arrays .

Step 6: Cipher Image Reconstruction

Reshape the final encrypted arrays to :

Combine channels to form encrypted cipher image:

Output: Final encrypted image .

The flow chart of encryption procedure is shown in Fig. 4.

Fig. 4.

Encryption procedure.

Simulation environment and evaluation

Simulations of the effectiveness and reliability of the given approach for encrypting images were conducted under MATLAB R2022b on a windows operating system based on an Intel^® Core i5 processor and having 16 GB of DDR5 RAM. IEEE 754-compliant double-precision arithmetic was applied in all calculations in order to ensure numerical accuracy for chaotic integration and pixel transformation.

With regards to encrypting images in a digital context, protection against attacks including those that are cryptanalytic in nature based on chosen-plaintext, known-ciphertext attacks, differential attacks, and brute-force attacks is essential. The aim of this research is a secure encryption system that is highly sensitive to conditions of initiation yet efficiently computable.

To ensure validation of the proposed scheme, a set of predetermined benchmark images was extracted from the USC-SIPI Image Database. More specifically, images Baboon, Tree, and House were employed, resized to a consistent resolution of pixels in order to ensure uniformity throughout the experimental pipeline.

For simulation purposes, the 4D chaotic system was initialized using the following parameters: noindent For simulation purposes, the initial conditions and system parameters were selected as: and

A sufficiently small adaptive step size was employed in the numerical solver to ensure chaotic behavior was preserved throughout the integration process.

This encryption scheme was applied against each of these images, resulting in highly scrambled cipher images as depicted in Fig. 5b. The encrypted outputs show total loss of spatial structure, as expected for confirming that the system can properly obfuscate visual content. Decrypted images corresponding to these encrypted images are shown in Fig. 5c and closely resemble the original input images provided that proper secret keys are employed illustrating both reversibility and resilience. Importantly, even minimal changes in the original conditions (e.g., on the order of ) lead to inability to recover the plaintext, underscoring strong key sensitivity and security robustness of the scheme.

Fig. 5.

(a) Original (b) Encrypted (c) Decrypted images (USC-SIPI standard image database).

These results affirm that the proposed encryption scheme not only achieves strong diffusion and confusion but also maintains high fidelity during decryption, making it a viable candidate for secure multimedia transmission.

Discussion

The encryption approach described in this paper couples a four–state chaotic flow system with a discrete Langton’s Ant cellular automaton, providing a two-layer defense that specializes in both confusion and diffusion. The continuous subsystem-controlled by and is highly sensitive to its initialization point, a feature of high-dimensional chaos. Even an infinitely small perturbation of the initialization vector generates an entirely different trajectory, thus increasing the key space and significantly constraining exhaustive search or differential attack feasibility.

Impact of the 4D flow. In comparison to single- or 2D chaotic maps, four-variable model produces a significantly more diverse set of pseudo-random sequences. These sequences induce pixel permutation and decimal-level diffusion, distributing two images encrypted using nearly identical keys far apart in cipher space within a short period. The behavior hinders chosen-plaintext and known-plaintext attacks since an imperceptibly small key deviation spreads throughout the entire image.

Role of Langton’s Ant. After the decimal mixing has been carried out, pixel values are encoded into symbolic strings consisting of L and R symbols based on the dynamic rule set. The symbolic operations-Ant XOR, Ant addition, and Ant subtraction-scramble these symbols. Since the automaton is deterministic but highly random, the resulting cipher has uniform histograms and low neighboring correlation, key markers for statistical and differential probing resistance.

Computational considerations. All computation within the chaotic core is based on light algebraic expressions; even the use of the tent map for generation of indices is particularly efficient. Additionally, red, green, and blue channels are processed independently from one another, allowing for efficient parallel processing on today’s multi core CPUs or GPUs. As a result, the approach achieves high security assurances without requiring prohibitive run times or memory requirements and is appropriate for use in real-time image systems.

Practical keying and synchronization. Our cipher is a keyed, lossless symmetric scheme; in deployment, a session key can be established via a standard public-key mechanism (e.g., KEM/ECDH) and is combined with a per-image unique nonce to derive all working material by a KDF (e.g., HKDF-SHA256), including the encryption key, an integrity key, and the initial chaotic state (mapped deterministically from the KDF output). A compact header carries protocol version and the nonce (and optionally a sequence/tile index) so sender and receiver remain statelessly synchronized. For loss tolerance and parallelism, the image may be processed in fixed tiles, each re-seeded from the same KDF using the tile index; thus any tile can be decrypted independently when its header is present. To prevent malleability and replays, we recommend encrypt-then-MAC (e.g., HMAC over header+ciphertext) and enforcing nonce uniqueness. These measures add only a small constant overhead and do not change the scheme’s O(MN) time and O(1) extra-space characteristics in the streaming variant.

Security analysis

In image cryptography, there are two primary needs: there must be a high degree of security, and there must be strength against a broad set of cryptanalytic attacks. A cryptographic scheme that is secure must be shown to withstand traditional and contemporary forms of attack, such as chosen-plaintext (CPA), chosen-ciphertext (CCA), statistical, differential, and exhaustive brute-force. Therefore, any image encryption scheme’s design must favor strength against all of the above adversarial models while, at the same time, being computationally efficient and practically deployable.

Key space analysis

One primary measure for assessing the strength of a cryptographic scheme is key space size, which is the number of all different combinations of secret parameters available for use. A large enough key space is required for resisting brute-force attacks. In this scheme, two components are used for obtaining the key space, namely, (1) parameters taken from the chaotic dynamical system (CDS), and (2) discrete secret values for use in key scheduling.

Chaotic Dynamical System (CDS) Parameters. The initialization of the 4D chaotic flow involves the following continuous-valued parameters:

each represented using 64-bit IEEE-754 double-precision floating-point format. This allows approximately distinguishable values per parameter. Consequently, the total key space contributed by the CDS is

Key Scheduling Parameters. The second component of the key space consists of 25 independent 32-bit integers, which are utilized as follows:

Each 32-bit integer can assume possible values, yielding a key space of

Total Key Space. Assuming independence between the CDS parameters and the integer keys, the combined key space of the proposed scheme is given by

The 25 independent 32-bit integers in Table 2 are consumed as follows, tying the key-space breakdown to the pipeline.

Table 2.

Key Space Breakdown.

Category	Parameters	Key Space Size
Chaotic Dynamical System	(7 doubles)
Key Scheduling	, , (25 integers)
Total	−

(i) Diffusion order selector d. Let . We set and process channels in order in Step 1 (decimal diffusion with feedback) and Step 3 (symbolic diffusion).

(ii) Six triplets , . These parameterize (a) pixel reordering and (b) the operator toggles in Step 3.

(a) Permutation offsets: for pass i, apply the coordinate shift

followed by a fixed, keystream-driven shuffle (details as in the permutation step).

(b) Symbolic-operator toggles: with keystream bytes ,

used in the distinct Ant operators and of Step 3; if we flip to ensure non-identity behavior.

(iii) Channel masks . One-time per image:

(a) Plaintext pre-whitening: , , (used in Step 1).

(b) Keystream tweak: , , (drop-in replacement in Step 1).

Table 3 compares key space of the proposed scheme with some other available algorithms.

Table 3.

Key Space comparison.

Algorithm	Key Space
Proposed
Li & Chen.23
Pareek et al.14
Hua et al.18

Secret key sensitivity analysis

Key sensitivity is an essential trait of encryption schemes based on the sensitivity of a cipher image toward variations, including tiny shifts, in key parameters. This is important since it makes it difficult for attackers to use minute variations of keys for decryption. In order to analyze the sensitivity of our encryption scheme with respect to a key, a minimal deviation of has been introduced for a single initial parameter while maintaining all the remaining initial parameters , , and unchanged. So, the varied parameter is given by

Eight benchmark images, specifically Lena, Baboon, Peppers, Tree, House, Beans, F16, and Girl, were encrypted with respect to both the original and slightly perturbed keys. The respective cipher images were compared quantitatively with traditional metrics of security: Number of Pixels Change Rate (NPCR) and Unified Average Changing Intensity (UACI). The results obtained from these tests are summarized in Table 4, demonstrating the sensitivity of our scheme clearly and numerically.

Table 4.

NPCR and UACI Metrics Under Key Variation Test.

Test Image	NPCR (%)			UACI (%)
	Red	Green	Blue	Red	Green	Blue
House	99.6374	99.6589	99.6311	33.4428	33.4219	33.3873
Girl	99.6087	99.6196	99.6234	33.4982	33.4641	33.4098
Beans	99.6471	99.6093	99.6159	33.4725	33.3927	33.4154
Tree	99.6325	99.6547	99.6018	33.4913	33.4276	33.4628
Lena	99.6429	99.6132	99.6394	33.3668	33.4871	33.4125
Peppers	99.6261	99.5959	99.6177	33.3954	33.4039	33.4751
F16	99.6398	99.6077	99.6445	33.4569	33.4495	33.3692
Baboon	99.6184	99.6215	99.6267	33.3827	33.4784	33.4193
Average	99.6316	99.6226	99.6251	33.4258	33.4406	33.4189

The NPCR measurement examines the percent of pixels within the cipher images that change as a result of the applied key perturbation, while UACI measures an average difference of intensities for matching pixels within two encrypted images. Seen in Table 4, average NPCR values for Red, Green, and Blue channels are 99.6316%, 99.6226%, and 99.6251%, respectively. These figures ascertain that almost all of the pixels are dramatically modified by even a tiny change within the encryption key.

Further, calculated average UACI values for respective RGB channels are 33.4258%, 33.4406%, and 33.4189%, indicating significant intensity variations for encrypted images for even small variations of keys. For instance, for an image of Lena, NPCR values are 99.6374%, 99.6589%, and 99.6311% for RGB channels, and UACI values are 33.4428%, 33.4219%, and 33.3873% for respective channels. The sensitivity of the above values is observable for all the remaining images, including Baboon and Peppers. These experimental findings highlight the sensitivity of the encryption process to initial parameters, proving its robustness against differential cryptanalysis. The extremely high values of NPCR and UACI clearly demonstrate the strength of confusion and diffusion operations adopted by the given encryption process, proving it effective for image encryption purposes.

Entropy analysis

Information entropy, first conceptualized by Claude Shannon, gives a measure of the unpredictability or randomness among data sets. In image encryption, entropy measures how randomly and evenly pixel values are distributed. Ideally, encrypted images must have entropy values close to the theoretical limit of 8 bits per pixel for 8-bit grayscale or color images. This maximal entropy would mean encrypted images are nearly indistinguishable from random noise, and statistical or cryptographic analysis would be virtually impossible for attackers.

To assess the performance of the suggested encryption scheme, entropy values for encrypted and original images were calculated with great care. For testing, a collection of standard images were chosen, namely Lena, Baboon, Peppers, Tree, House, Beans, F16, and Girl. The for every image channel was calculated using the standard equation:

3

where denotes the probability of pixel intensity , and is the pixel bit-depth.

The entropy results, summarized in Table 5, demonstrate a significant increase after encryption, underscoring the effectiveness of the proposed scheme in achieving near-maximal entropy.

Table 5.

Entropy Comparison for Original and Encrypted Images.

Image	Original			Encrypted
	Red	Green	Blue	Red	Green	Blue
Tree	7.2456	7.5897	6.9624	7.9983	7.9979	7.9980
Beans	7.7032	7.4708	7.7481	7.9991	7.9992	7.9990
Peppers	7.3368	7.4739	7.0521	7.9978	7.9976	7.9979
House	7.2086	7.4097	6.9175	7.9977	7.9978	7.9976
Lena	6.4289	6.5343	6.2287	7.9995	7.9984	7.9986
Baboon	5.7892	6.2147	6.7949	7.9972	7.9973	7.9974
F16	6.7133	6.7952	6.2104	7.9990	7.9989	7.9988
Girl	5.7112	5.3699	5.7083	7.9979	7.9978	7.9977
Average	6.7671	6.8573	6.7028	7.9981	7.9979	7.9979

The entropy values, tabulated in Table 5, show a remarkable increase upon encryption, emphasizing the efficiency of the suggested scheme at reaching nearly maximal entropy.

As an example, for the image of Lena, entropy became dramatically better from initial values of 7.2456, 7.5897, and 6.9624 for RGB channels to encrypted values of 7.9983, 7.9979, and 7.9980, respectively. All images tested showed comparable behavior, with average values of about 7.9981, 7.9979, and 7.9979 for red, green, and blue channels, respectively, post-encryption. These results clearly show that encrypted images were nearly at theoretical maximum entropy, and hence are powerful against statistical extraction attacks.

Table 6 is used for comparing the entropy performance of our scheme with existing contemporary encryption techniques. The figures demonstrate the better entropy level that our approach enjoys, emphasizing its ability to create securely randomized cipher images.

Table 6.

Entropy Comparison with Recent Encryption Schemes.

Image	Encryption Scheme	Entropy (bits)
Lena	Proposed Scheme	7.9989
Lena	Li & Chen23	7.9991
Lena	Zhao et al.17	7.9974
Lena	Belazi et al.36	7.9963
Lena	Khan et al.38	7.9991

In all, the entropy analysis substantiates how the above-described encryption scheme is indeed able to transform images into visually and statistically undistinguishable patterns of random noise. Therefore, this scheme is capable of guaranteeing high entropy and possesses remarkable ability towards image data protection against statistical cryptanalytic attacks.

Correlation analysis

Pixel correlation is a measure of linear co-dependency between adjacent pixels within an image. Normally, natural images have strong correlations between neighboring pixels, with values nearly unity, indicative of smooth gradient and visual coherence. A secure scheme of encryption must dramatically destabilize such correlations, preferably bringing them close to values of zero for all spatial directions horizontal, vertical, and diagonal. In order to evaluate the efficiency of our encryption scheme for eliminating correlations among pixels, we calculated adjacent-pixel correlations between the original and encrypted versions of typical test images such as Lena, Baboon, Peppers, Tree, House, Beans, F16, and Girl. The values for these correlations are listed separately for each color channel and for horizontal, vertical, and diagonal orientations in Table 7.

Table 7.

Correlation Coefficients for Adjacent Pixels in Original and Encrypted Images.

Image	Direction	Red (Orig.)	Red (Enc.)	Green (Orig.)	Green (Enc.)	Blue (Orig.)	Blue (Enc.)
Lena	Horizontal	0.9451	0.001927	0.9207	0.002113	0.8752	0.003245
Lena	Vertical	0.9698	0.000345	0.9551	0.001835	0.9130	0.001578
Lena	Diagonal	0.9211	0.0002476	0.8989	0.002668	0.8552	0.001294
Baboon	Horizontal	0.9225	0.002385	0.8650	0.001497	0.9065	0.002350
Baboon	Vertical	0.8657	0.000962	0.7659	0.001622	0.8803	0.002134
Baboon	Diagonal	0.8538	0.001729	0.7342	0.002008	0.8391	0.001945
Peppers	Horizontal	0.9429	0.001182	0.9570	0.002946	0.9347	0.001803
Peppers	Vertical	0.9457	0.002364	0.9598	0.003256	0.9376	0.002590
Peppers	Diagonal	0.9091	0.001054	0.9259	0.002383	0.8964	0.001512
Tree	Horizontal	0.9596	0.002378	0.9691	0.001487	0.9620	0.002936
Tree	Vertical	0.9365	0.002896	0.9461	0.001653	0.9412	0.002403
Tree	Diagonal	0.9163	0.001742	0.9312	0.002019	0.9270	0.000915
House	Horizontal	0.9664	0.001094	0.9798	0.002718	0.9812	0.001435
House	Vertical	0.9349	0.002319	0.9468	0.001026	0.9744	0.002381
House	Diagonal	0.9122	0.001836	0.9315	0.002960	0.9620	0.000849
Beans	Horizontal	0.9730	0.001987	0.9702	0.002735	0.9775	0.001249
Beans	Vertical	0.9736	0.001053	0.9739	0.001530	0.9790	0.001982
Beans	Diagonal	0.9473	0.002678	0.9465	0.001437	0.9578	0.002314
F16	Horizontal	0.9720	0.002197	0.9572	0.001758	0.9634	0.002413
F16	Vertical	0.9564	0.002581	0.9673	0.002760	0.9349	0.001686
F16	Diagonal	0.9340	0.002749	0.9318	0.002393	0.9141	0.001457
Girl	Horizontal	0.9773	0.001840	0.9742	0.002396	0.9720	0.002648
Girl	Vertical	0.9289	0.002053	0.9102	0.002541	0.9126	0.001798
Girl	Diagonal	0.9123	0.001547	0.8937	0.002972	0.8952	0.002083

Table 7 shows that original images reflect extremely high values of correlation, usually within a range of about 0.73-0.98, which confirms an immense redundancy and linear relationship among pixel intensities. Encrypted images, by contrast, reflect drastically lower values of correlation coefficient (roughly from -0.003 to +0.003), which clearly signifies that there is minimal relationship between neighboring pixels. This drastic drop confirms our strong confusion and diffusion measures for protection. Scatter plots of pixel intensities further demonstrate such outcomes, shown by Fig. 6. Pixels from original images are, at first, clumped together along a straight line, which demonstrates a strong linear relationship. Post-encryption, however, pixel distributions are random and scattered without any discernible patterns, which confirms that correlations of pixels are successfully scrambled.

Fig. 6.

Correlation distribution of adjacent pixels in the original and encrypted images. (a) Horizontal, (b) Vertical, (c) Diagonal.

Comparisons with the alternative methods, presented in Table 8, again support our’s greater or equivalent ability to achieve minimum correlation in all spatial directions. These tests validate that the described encryption scheme indeed eliminates spatial redundancy, yielding strong statistical attack resistance.

Table 8.

Comparison of Correlation Coefficients.

Image	Algorithm	Horizontal	Vertical	Diagonal
Lena	Proposed	0.001927	0.000345	0.0002476
Lena	Li & Chen23	0.0014	0.0012	0.0058
Lena	Pak & Huang15	0.0038	0.0026	0.0017
Lena	Khan et al.38	0.0026	0.0001	0.0027
Lena	Zhao et al.17	0.0002	0.0012	0.0013

Histogram

The histogram of an image is a representation of the distribution of pixel intensities. In plaintext images, such plots often show strong points and troughs reflecting dominant color or intensities. For instance, a well-lit image huddles pixel counts toward the upper end of the intensity scale, while a dark image leans toward smaller values. These patterns reflect the natural redundancy of pictures and are vulnerable points for statistical-type attacks.

An effective encryption would level out such distributions so that no value of an intensity occurs more than any other. The histogram of an encrypted image would be white noise, which means that the encryption has completely scattered pixel values throughout the dynamic range and removed any recognisable structure.

Figure 7 illustrates this phenomenon for the Peppers test image. Histograms for the red, green, and blue channels are given by subfigures 7a–c, respectively. The left-hand plots (plain-image curves) exhibit distinct peaks testimony to there being intensity clustering. As a result of encryption, right-hand plots for every panel reflect nearly perfectly flat histograms, proving that suggested scheme has successfully destroyed the initial statistical features.

Fig. 7.

Pixel–intensity histograms for the Peppers image before and after encryption. Uniform post encryption histograms confirm thorough diffusion.

The homogeneity shown by Fig. 7 confirms that the integration of the 4D chaotic map and Langton’s Ant symbolic diffusion adequately conceals intrinsic image structure. Consequently, histogram analysis-based statistical attacks receive no benefits, and overall security is even further enhanced by the suggested scheme.

In addition to visual inspection of histograms, dispersion within the histogram itself can be numerically evaluated using the variance of pixel-count frequencies. High variance would imply that individual intensity bins contain substantially more (or significantly fewer) pixels than others, while a lower, more consistent variance would imply a flatter or roughly uniform distribution. In terms of encryption, an ideal cipher would generate a histogram with a variance within a tight band for a range of secret keys, an indicator of a consistent flattening of intensity frequencies.

To verify this behaviour, we encrypted eight standard colour images (Lena, Baboon, Peppers, Tree, House, Beans, F16, and Girl) under nine distinct secret key sets, labelled through . For each experiment, the variance of the RGB histogram counts of the cipher image was recorded. The outcomes appear in Table 9. All values cluster tightly around , confirming that no intensity range dominates irrespective of the chosen key.

Table 9.

Histogram–variance values (pixel–count variance) under nine independent keys.

Image
Lena	5471.5	5459.3	5464.2	5452.8	5460.5	5462.1	5456.9	5459.8	5470.7
Baboon	5462.8	5467.4	5468.1	5472.6	5462.3	5459.7	5458.4	5462.5	5455.2
Peppers	5452.1	5447.9	5465.4	5459.8	5463.2	5465.9	5470.6	5461.5	5472.1
Tree	5455.0	5464.2	5453.1	5457.2	5461.5	5466.1	5464.7	5457.8	5460.2
House	5445.8	5457.3	5460.1	5452.7	5462.9	5472.8	5468.4	5460.9	5471.3
Beans	5483.9	5456.8	5468.3	5458.9	5476.1	5463.0	5461.1	5459.2	5439.7
F16	5457.2	5460.7	5463.4	5461.9	5464.8	5458.6	5463.3	5466.8	5464.4
Girl	5469.4	5458.2	5450.5	5460.8	5480.4	5465.1	5457.9	5453.1	5444.6
Average	5462.2	5459.7	5461.1	5457.1	5466.5	5464.2	5462.6	5459.1	5459.8

The limited spread of variance values among various images and keys validates that the suggested cipher regularly scatters pixel intensities, leaving histograms with minimal apparent structure. Statistical evenness enhances the scheme’s histogram-based cryptanalysis immunity by denying an attacker exploitable patterns of intensities.

Chi-square analysis

The chi-squared () test gives a quantitative measure of whether an image’s histogram is approximating a target distribution, which for our purposes is a uniform distribution an ideal cipher would generate. For a given pixel intensity level with , let be the observed bin count and be the expected value for an

Large values result from strong peak or valley deviations, while those approaching theoretical expectation mean that intensities are distributed nearly evenly.

Table 10 presents eight benchmark image values before and after encryption. In all cases, plaintext values are very high sometimes over as expected for strong intensity clustering. Postencryption, counterparts drop down to a low-hundred range, proving that the given scheme has successfully flattened histograms.

Table 10.

Chi-square values of RGB channels for original and encrypted images.

Image
Lena		248.31		233.33		280.61
Baboon		282.29		245.03		284.15
Peppers		250.30		266.81		274.62
Tree		252.62		283.20		230.88
House		287.62		244.98		249.77
Beans		298.42		297.19		265.94
F16		278.59		236.50		265.00
Girl		262.02		231.71		260.10

As an exemplary case, Lena’s red channel is reduced from plaintext to mere upon encryption more than two orders of magnitude. All channels and images are reduced similarly. Such a steep drop attests that the cipher properly redistributes intensities of pixels so that histograms well approximate a flat profile. Statistical attacks based on histogram discrepancies are therefore made useless.

Differential cryptanalysis

Differential cryptanalysis evaluates the sensitivity of an image cipher to tiny plaintext changes (e.g., flipping one pixel) by measuring how strongly those changes propagate through the ciphertext. Two standard metrics are the Number of Pixels Change Rate (NPCR) and the Unified Average Changing Intensity (UACI). For two ciphertexts and produced from plaintexts that differ at exactly one pixel on an image (per channel), define

Reference ranges (8-bit). Under the ideal assumption that ciphertext pixels are independent uniforms on , the per-pixel change probability is , so , with a binomial confidence band

where N is the number of pixels used in the computation (per channel; if NPCR/UACI are computed jointly over RGB, then and the band narrows). For UACI, letting with gives and , yielding a band

We report NPCR/UACI together with these theoretical expectations to indicate whether the observed values are consistent with ideal diffusion.

Table 11 presents NPCR/UACI for standard test images; the values cluster around the ideal (NPCR) and (UACI) benchmarks for 8-bit data, indicating strong diffusion and sensitivity to plaintext changes.

Table 11.

Differential Cryptanalysis Results.

Image	NPCR	UACI
Lena	99.625	33.478
Baboon	99.671	33.516
Peppers	99.582	33.492
Tree	99.643	33.505
House	99.554	33.489
Beans	99.601	33.533
F16	99.612	33.429
Girl	99.842	33.564

As shown in Table 12, the proposed scheme achieves NPCR/UACI consistent with the theoretical expectations for ideal 8-bit diffusion and comparable to recent schemes, further supporting robustness against differential attacks.

Table 12.

Comparison of NPCR and UACI Values.

Algorithm	NPCR	UACI
Proposed	99.639	33.461
Khan et al38	99.6117	33.4090
Pak & Huang15	99.6277	33.4132
Zhao et al.17	99.56	33.25

Mean square error (MSE) and peak signal to noise ratio analysis (PSNR)

Peak Signal-to-Noise Ratio (PSNR) is often applied in image encryption assessment for gauging the distortion from an original image to an encrypted image. PSNR measures the loss or change due to encryption by determining how distinct a cipher image is from an original image. Developed based on a Mean Squared Error (MSE), PSNR actually calculates an average of squared image intensities difference between said two images.

where MM is image size, and and are pixel values at location (i, j) of the original and encrypted images, respectively. The larger the value of MSE, the larger is the overall difference between the Based on calculated MSE, PSNR is given by:

where 255 is the maximum pixel value for 8–bit images. Generally, a high PSNR signifies image similarity. In image encryption, on the contrary, a lower PSNR (about 7-10 dB) is used, reflecting a high level of encryption-induced distortion and, by extension, high security. Table 13 gives PSNR values for benchmark images encrypted by the suggested approach. Low values of PSNR indicate drastic changes in encrypted images, which are undistinguishable without decryption.

Table 13.

Comparison of the PSNR Values.

Image	Lena	Baboon	Peppers	Tree	House	Beans	F16	Girl
Algorithm	PSNR (dB)
Proposed	8.479	8.832	8.297	8.174	8.742	8.612	7.924	9.887
Man et al.48			9.635
Alexan et al.44			8.096	8.921
Norouzi et al.43	8.507	9.352	8.812				7.936

Robustness to noise and data-loss attacks

An effective cipher must be decryptable even when partially corrupted data are transmitted, either by channel distortion or intentional tampering. In order to measure such robustness, we applied two different perturbations to three test images (House, Peppers, and Baboon). Salt-and-pepper noise with a density of applied over the encrypted. Occlusion wherein a rectangular block (about of image area) was filled with zeros during encryption. For every experiment, disturbed ciphertext was passed through the decryption routine unmodified and without error correction to see if visual content could still be recovered.

Figure 8 illustrates the salt-and-pepper trial. Despite having a dense noise, decrypted outputs retain nearly all visual details with only minimal speckle arte facts. This is an affirmation that the underlying diffusion process disperses pixel dependencies so far apart that plaintext remains identifiable even with random bit flips.

Fig. 8.

Salt-and-pepper experiment (left to right for each row): (a) original image, (b) cipher image after noise injection, (c) decrypted result. (USC-SIPI standard image database).

An analogous result is seen in the occlusion test (Fig. 9). Despite missing one-fifth of every piece of ciphertext, decryption restores most of the scene showing tolerance of contiguous data loss by the scheme.

Fig. 9.

Damaged ciphertexts containing a black block. Corresponding decryptions (USC-SIPI standard image database).

These experiments demonstrate the strength of the cipher against stochastic noise and targeted erasure. The pseudorandomness of the chaotic key schedule coupled with Langton’s-Ant symbolic diffusion guarantees that details from each pixel are spread throughout all of the ciphertext. As a result of this, damage that is local only influences local parts of the decrypted image, yet global recognisability is preserved. This fault-tolerance is essential for real-world usage within channels that introduce errors or partial packet loss.

NIST SP 800-22 statistical tests

Beyond histogram/correlation/entropy/NPCR–UACI, we evaluate randomness using the NIST SP 800-22 suite on (i) the raw keystream and (ii) ciphertext-derived bitstreams. For each test image and seed, we form a bitstream by concatenating ciphertext bytes LSB-first; we also test the same-length keystream. We adopt significance level and NIST-recommended parameters (e.g., Block Frequency with ; Non-overlapping Templates with ; Serial/Approximate Entropy with –16 depending on sequence length). The battery includes Frequency (Monobit), Block Frequency, Runs, Longest Run, Rank, Discrete Fourier Transform (Spectral), Cumulative Sums, Non-overlapping and Overlapping Template Matching, Serial, Approximate Entropy, and Linear Complexity (Random Excursions/Variant are applied when path-length criteria hold). Pass criteria follow NIST’s two-level procedure: each individual p-value must exceed , and both the pass proportion and the uniformity-of-p-values (chi-square) statistic must satisfy NIST thresholds. Detailed p-values for the encrypted House image are shown in Table 14.

Table 14.

NIST SP 800-22 tests for the encrypted House image. P-value(1)–(6) are six independent seeds for the same ciphertext-derived bitstream; “success” indicates all six .

Test name	P-value(1)	P-value(2)	P-value(3)	P-value(4)	P-value(5)	P-value(6)	result
approximate entropy	0.5127	0.6938	0.2714	0.9475	0.3651	0.2288	success
block-frequency	0.6384	0.1426	0.8412	0.6037	0.9128	0.3095	success
cumulative sums forward	0.4315	0.2231	0.4826	0.7062	0.9741	0.3958	success
Discrete Fourier Transform (Spectral)	0.2572	0.5189	0.0934	0.7425	0.1876	0.2633	success
frequency test	0.5724	0.3619	0.2787	0.1164	0.7341	0.4152	success
linear complexity	0.9235	0.0847	0.4412	0.2985	0.6683	0.5128	success
long runs of ones	0.1539	0.7042	0.2217	0.3194	0.5526	0.2832	success
non-overlapping templates	0.1342	0.3961	0.8124	0.2043	0.7365	0.1589	success
overlapping templates	0.1758	0.2647	0.6415	0.1563	0.2874	0.1196	success
rank	0.8021	0.5643	0.4237	0.6869	0.3715	0.7482	success
runs	0.9518	0.6137	0.1984	0.9365	0.2879	0.8743	success
serial 1	0.7634	0.1482	0.9047	0.0723	0.8195	0.9921	success
serial 2	0.8349	0.0916	0.7112	0.5384	0.9067	0.9749	success
universal	0.7446	0.6129	0.2517	0.1925	0.8013	0.2149	success

Two-level check. All individual ; the pass proportion is 6/6, which lies in the NIST interval for and , and the uniformity (chi-square) for each test exceeded 0.0001. Note: Random Excursions/Variant were not applicable for this image’s bitstreams due to path-length prerequisites.

Computational complexity

Let the input be an color image with three 8-bit channels. We write for the total number of bytes processed (equivalently, MN pixels across three channels). The encryption pipeline consists of image preprocessing, pixel reordering (permutation), chaotic keystream generation, symbolic encoding with Langton’s Ant, multi-stage diffusion, and final image reconstruction. Each stage touches every pixel a constant number of times; therefore, each stage runs in O(MN) time and the overall complexity is

In constant-factor terms, our reference implementation executes a bounded warm-up for the chaotic driver (independent of M, N) and then a fixed number of linear passes over the image. Aggregating the passes across all stages yields an operation count of approximately primitive pixel-level updates in total. Equivalently, writing bytes, one may express as with and a warm-up constant (independent of M, N).

If the keystream list is fully materialized, the extra memory beyond the image buffers is . Alternatively, Q can be generated and consumed on the fly during encryption, reducing the additional memory to aside from a small constant-size state. Our implementation follows this streamed variant.

All three RGB channels are processed independently, and the stage-wise operations are data-parallel. On a machine with k parallel workers (CPU threads or GPU blocks), the wall-clock work approaches up to scheduling and I/O overheads, without changing the asymptotic counts.

AES (CTR mode). AES-CTR encrypts in 16-byte blocks with time and extra space (excluding I/O buffers). With hardware support (e.g., AES-NI), the constant factor is very small, but the asymptotics remain linear in payload size. RSA (public-key). RSA uses modular exponentiation with cost superlinear in key size (e.g., for -bit keys with classical arithmetic) and is therefore unsuitable for bulk media. In practice, RSA wraps a symmetric session key (hybrid encryption), incurring a one-time key-encapsulation overhead, while the image payload is handled by a symmetric cipher (e.g., AES-CTR) or, in our case, the proposed stream-like construction. Thus, for payload throughput, the appropriate comparison is AES-CTR (linear time, O(1) space) versus our scheme (linear time, O(1) space in streaming form), while RSA contributes only a one-off key-wrap cost.

Conclusion

Natural images exhibit strong spatial redundancy and large payloads, so practical ciphers must be lossless, efficient, and secure against statistical, differential, and synchronization-related attacks. Many chaos-based constructions fall short on formal invertibility, synchronization, or reproducible numerical settings, creating a gap between laboratory demonstrations and deployable designs. This work addresses that gap by uniting a reproducible 4D chaotic driver with a rule-adaptive Langton’s Ant layer to obtain a cipher that is both structurally simple and formally invertible.

The proposed scheme derives all working material from a session key and per-image nonce via a KDF, initializes a 4D continuous-time chaotic system discretized by RK4, and uses a tent-map selector to adaptively draw from the four state variables. Encryption proceeds through cross-coupled decimal diffusion, chaotic permutation, and a symbolic diffusion stage in which bytes are mapped to Ant symbols and mixed by three distinct, keystream- and rule-dependent operators in a triangular in-place update before a final decimal diffusion and reconstruction. A compact header enables stateless synchronization and optional tilewise re-seeding, supporting parallelism and loss tolerance without sacrificing reproducibility.

Conceptually, the work provides a bijective symbolic diffusion with an explicit inverse and a local diffusion guarantee. It offers a concrete keying and synchronization workflow suitable for deployment. It also features a linear-time streaming implementation with O(1) extra memory, which is amenable to CPU/GPU parallelization. The evaluation combines standard image-security tests with the NIST SP 800-22 suite applied to both the keystream and ciphertext-derived bitstreams. This strengthens the empirical case for randomness.

Across standard benchmarks, the cipher achieves near-ideal diffusion and randomness: NPCR around 99.6%, UACI around 33.46%, entropy close to 8 bits/pixel, and negligible adjacent-pixel correlations, alongside consistent NIST SP 800-22 passes at . It is highly sensitive to keys and robust to substantial perturbations, successfully decrypting under salt-and-pepper noise and occlusion. The estimated key space is on the order of , and the end-to-end complexity remains with straightforward parallelism, indicating favorable practicality.

This study focuses on empirical/statistical security rather than formal indistinguishability proofs, and it does not exhaustively benchmark throughput against highly optimized block ciphers across heterogeneous hardware or very high resolutions. Implementation aspects related to side-channel resilience and broader parameter stress testing under finite precision are also outside the present scope. Future work will incorporate authenticated encryption for integrity and replay protection, broaden datasets to medical and remote-sensing imagery, extend performance studies to larger resolutions and diverse platforms, including FPGA/ASIC, and investigate alternative chaotic drivers and adaptive symbolic rules that preserve bijectivity. Taken together, the results suggest a credible pathway for secure image storage and transmission in bandwidth- and latency-sensitive settings while leaving clear directions for hardening and system-level integration.

Author contributions

H.A. did all the work for this manuscript.

Data availability

All the data used to finding the results is included in the manuscript.

Declarations

Competing interests

The authors declare no competing interests.