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A strongly connected graph-based digital chaotic system and its application to bidirectional-diffusion image encryption

This paper proposes a strongly connected graph-based digital chaotic system constructed via an inverse approach to eliminate finite precision effects and ensure rigorous chaos properties, which is then applied to a bidirectional-diffusion image encryption algorithm that demonstrates robust security against statistical and differential attacks.

Original authors: Ziting Xie, Qianxue Wang, Jiayu Lin, Yichun Huang

Published 2026-08-31

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CC BY 4.0

Original authors: Ziting Xie, Qianxue Wang, Jiayu Lin, Yichun Huang

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the digital age, images are more than just pictures; they are dense carriers of information, flowing constantly through open networks from medical scanners to remote sensors. Protecting these images requires more than just hiding them; it requires transforming them into something that looks like random noise, making it impossible for an outsider to guess the original content. For decades, scientists have looked to chaos theory for a solution. Chaos describes systems that are deterministic—meaning they follow strict rules—but appear random and are incredibly sensitive to tiny changes. If you change the starting conditions of a chaotic system by the smallest amount, the outcome diverges wildly, a property essential for scrambling data securely. However, when these chaotic systems are run on real computers, which have limited precision, they often lose their chaotic nature and fall into predictable loops, creating a vulnerability that hackers can exploit.

To solve this, researchers at the Guangdong University of Technology have developed a new way to build a digital chaotic system that avoids these pitfalls entirely. Instead of relying on the floating-point numbers that computers struggle to handle perfectly, they constructed a system based on a specific type of mathematical map called a strongly connected graph. Imagine a city where every intersection is connected to every other intersection by at least one road, ensuring that no matter where you start, you can eventually reach any destination. The researchers designed their system to function like this map, but with a crucial twist: they ensured the entire network forms a single, unbroken loop that visits every possible state before repeating, while also adding random shortcuts to make the path unpredictable. This design guarantees that the system remains chaotic and unpredictable even when running on standard digital hardware, effectively eliminating the "finite precision effect" that has weakened many previous encryption methods.

Using this robust chaotic system, the team created a new image encryption algorithm that works in two main stages: scrambling and diffusion. First, the algorithm shuffles the pixels of an image, not just by moving whole pixels around, but by also twisting the individual bits of data that make up each pixel's color. This dual-layer scrambling breaks the natural patterns and correlations found in ordinary photos. Next, the system applies a bidirectional diffusion process. This means the algorithm spreads the influence of every single pixel change across the entire image in both forward and backward directions simultaneously. If an attacker were to change even a single pixel in the original image, this process ensures that the resulting encrypted image would look completely different, with nearly every pixel altered. The researchers proved mathematically that their system is sensitive to initial conditions, meaning that a tiny difference in the starting key leads to a massive difference in the output, a requirement for any secure cipher.

The results of their tests were striking. When they encrypted standard test images, the resulting files looked like static on a television screen, with no visible traces of the original picture. Statistical analysis showed that the encrypted images had a perfectly uniform distribution of colors, making it impossible to infer any information about the original image through statistical attacks. Furthermore, the system demonstrated a vast key space, meaning there are more than 2 to the power of 128 possible keys, a number so large that a brute-force attack would be practically impossible. The researchers also tested the system against differential attacks, where an attacker tries to find patterns by comparing encrypted versions of slightly different images. The new algorithm showed that a change in just one pixel of the original image caused a change in nearly 99.6% of the pixels in the encrypted image, with the average intensity of change matching theoretical ideals. Even when tested with extreme inputs, such as a completely black or completely white image, the system produced encrypted results that were indistinguishable from random noise.

This work represents a significant step forward in securing digital images. By moving away from traditional floating-point calculations and grounding the chaotic system in a rigorous graph-theoretic structure, the researchers have created an encryption method that is both theoretically sound and practically efficient. The algorithm avoids the common weaknesses of previous chaotic ciphers, such as short cycles and predictable patterns, while maintaining high speed and low computational cost. The findings suggest that this approach could be widely adopted for securing sensitive data in fields ranging from medical imaging to the Internet of Things, providing a reliable shield against increasingly sophisticated cyber threats. The study confirms that by carefully designing the underlying mathematical structure, it is possible to create digital chaos that is as unpredictable as nature itself, yet perfectly reliable for the machines that run our world.

Technical Summary: A Strongly Connected Graph-Based Digital Chaotic System and Its Application to Bidirectional-Diffusion Image Encryption

Problem Statement
The paper addresses critical vulnerabilities in existing chaos-based image encryption schemes, specifically focusing on three main issues:

Finite Precision Effect: Traditional chaotic systems often rely on floating-point arithmetic, which leads to dynamical degradation and short-period orbits when implemented on digital hardware with finite precision.
Structural Security Flaws: Many existing algorithms employ homogeneous operations (e.g., pure XOR or modular arithmetic) that are susceptible to linearization and differential cryptanalysis. Furthermore, some schemes rely on superficially complex operations (like DNA coding) that do not fundamentally enhance nonlinearity.
Unidirectional Sensitivity: Conventional designs often emphasize the sensitivity of ciphertext to plaintext changes but neglect the reverse sensitivity (plaintext sensitivity to ciphertext), leaving systems vulnerable to chosen-ciphertext attacks.

Methodology
The authors propose a novel image encryption framework built upon a Three-Dimensional Digital Chaotic System (3D-DCS) constructed via a strongly connected graph inverse approach. The methodology is divided into system construction and encryption algorithm design:

System Construction (Spatiotemporal Decoupling):
- Graph-Based Inverse Approach: Instead of random loop construction after state traversal, the system constructs a full-state closed loop to guarantee strong connectivity and randomly adds transition edges to enhance complexity. This is derived inversely from a directed state transition graph.
- Theoretical Guarantee: The system is designed in a single-bit Boolean space ( $N=1$ ) using only bitwise operators (AND, OR, NOT). Citing Theorem 1, the authors assert that if the graph is strongly connected at $N=1$ , it remains strongly connected for any finite precision $N > 1$ , thereby circumventing finite precision degradation.
- Spatiotemporal Decoupling: The system separates spatial topology (static strongly connected graph) from temporal evolution (controlled by random sequences). This avoids the timing convergence issues of strongly coupled systems and allows for high-speed parallel hardware implementation.
- Sensitivity Proof: The paper provides a rigorous mathematical proof (Theorem 2) that the system satisfies Devaney's definition of chaos, specifically establishing a lower bound for the sensitivity constant ( $\delta$ ) and proving that minute perturbations in initial conditions or control sequences lead to maximum Hamming distance divergence within finite steps.
Encryption Algorithm Design:
- Keystream Generation: The 3D-DCS is driven by control sequences generated via the ISAAC pseudorandom number generator, initialized by a 128-bit key ( $p_1, p_2, p_3, p_4$ ).
- Multi-Dimensional Scrambling:
  - Pixel-Level: The image is flattened and permuted using index sequences derived from sorting chaotic sequences.
  - Bit-Level: Pixel values undergo cyclic bit-shifting based on a chaotic sequence modulo 8.
- Bidirectional Nonlinear Diffusion:
  - A forward diffusion phase (left-to-right) and a backward diffusion phase (right-to-left) are employed.
  - The diffusion equations utilize heterogeneous operations (XOR, modular addition, and time-delayed feedback involving plaintext, key, and previous ciphertext states). This design aims to break algebraic equivalences in the differential domain and ensure bidirectional sensitivity.

Key Contributions

Optimized Construction Strategy: The paper introduces an optimized method for constructing digital chaotic systems that ensures strong connectivity through a full-state closed loop, significantly improving derivation efficiency and state transition unpredictability compared to random loop methods.
Rigorous Sensitivity Analysis: Unlike many prior works that merely claim chaos, this paper provides a constructive mathematical proof of sensitivity to initial conditions and explicitly derives the sensitivity constant, establishing a theoretical foundation for the system's avalanche effect.
Bidirectional Sensitive Architecture: The encryption scheme integrates a bidirectional diffusion mechanism with heterogeneous operations. This specifically targets the "unidirectionality" vulnerability, ensuring that the system is sensitive to changes in both plaintext and ciphertext, thereby resisting chosen-ciphertext and differential attacks.

Experimental Results
The proposed algorithm was evaluated using standard test images (Baboon, Fruits, Yacht, Monarch) and extreme inputs (all-black/all-white images). Key performance metrics include:

Key Space: The algorithm utilizes a 128-bit key space ( $2^{128}$ ), deemed sufficient to resist brute-force attacks.
Statistical Analysis: Encrypted images exhibit uniform histograms and information entropy values close to the theoretical maximum of 8 (e.g., 7.999+), indicating high randomness.
Correlation: Adjacent pixel correlation coefficients in horizontal, vertical, and diagonal directions were reduced to near-zero values (approx. 0.001), effectively breaking spatial correlations.
Differential Attack Resistance: The Number of Pixels Change Rate (NPCR) and Unified Average Changing Intensity (UACI) values closely matched theoretical ideals (NPCR $\approx$ 99.61%, UACI $\approx$ 33.46%), demonstrating strong sensitivity to single-pixel changes.
Randomness Testing: The ciphertext binary streams passed all 15 categories of the NIST SP 800-22 randomness test suite with high pass rates.
Avalanche Effect: Single-pixel modification tests confirmed that bidirectional diffusion allows a change in one pixel to propagate to the entire ciphertext, achieving ideal NPCR/UACI values rapidly.

Significance and Claims
The paper claims that the proposed method offers a robust solution for secure image transmission by fundamentally addressing the finite precision limitations of traditional chaotic systems through a graph-theoretic, integer-domain approach. By rigorously proving the system's chaotic properties and designing a bidirectional diffusion mechanism, the authors assert that the algorithm effectively eliminates structural vulnerabilities found in previous schemes (such as linearization and unidirectional sensitivity). The use of bitwise operations and the absence of floating-point arithmetic make the system particularly suitable for implementation in resource-constrained hardware environments, such as FPGAs, DSPs, and IoT terminals, ensuring deterministic and reproducible cryptographic performance.

Technical Summary: A Strongly Connected Graph-Based Digital Chaotic System and Its Application to Bidirectional-Diffusion Image Encryption

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