Fractal Dynamics and Hash Functions Enhance Cryptographic Key Security.

Research demonstrates a novel cryptographic key recovery problem, HFKR, founded on fractal symbolic dynamics. Empirical analysis confirms symbolic paths exhibit fractal behaviour with a dimension stabilising near 1.06. SHA3-512 and SHAKE256 effectively amplify symbolic divergence, yielding high entropy and robust key generation without algebraic assumptions.

The vulnerability of current cryptographic systems to increasingly sophisticated attacks motivates exploration beyond traditional algebraic foundations. Researchers are now investigating non-algebraic approaches, leveraging the properties of chaotic systems to generate robust cryptographic keys. A new construction, the Hashed Fractal Key Recovery (HFKR) problem, utilises symbolic dynamics and chaotic perturbations to create keys exhibiting inherent complexity. Mohamed Aly Bouke, from Universiti Putra Malaysia, details this work in a recent article, “The Hashed Fractal Key Recovery (HFKR) Problem: From Symbolic Path Inversion to Post-Quantum Cryptographic Keys”, demonstrating empirically that symbolic trajectories generated by this method display fractal behaviour, enhancing the entropy and security of the resulting keys. The study assesses the performance of different hashing algorithms in amplifying symbolic divergence, offering a potential pathway to lightweight, structure-free key generation resilient to advanced computational threats.

Harnessing Symbolic Chaos for Post-Quantum Cryptography

Current cryptographic systems face increasing vulnerability due to advances in computational power and structural cryptanalysis targeting underlying algebraic problems. This research presents a novel cryptographic construction, the Hashed Fractal Key Recovery (HFKR) problem, which departs from traditional algebraic foundations and instead leverages the principles of symbolic dynamics and chaotic perturbations. The HFKR problem builds upon the Symbolic Path Inversion Problem (SPIP), generating symbolic trajectories via contractive affine maps and compressing them into fixed-length cryptographic keys using cryptographic hash functions.

Researchers demonstrate that these symbolic paths exhibit fractal behaviour, a property quantified through box-counting dimension, path geometry, and spatial density measurements. Experimental results reveal a consistent fractal dimension, stabilising near 1.06, which indicates symbolic self-similarity and complex space-filling characteristics bolstering the entropy foundation of the cryptographic scheme. This interplay between symbolic fractality and hash-based entropy amplification offers a lightweight and structure-free approach to key generation.

Researchers conducted 250 perturbation trials, finding that SHA3-512 and SHAKE256 effectively amplify symbolic divergence, achieving mean Hamming distances approaching 255 – indicative of ideal bit-flip rates and minimal entropy deviation. This amplification process successfully leverages the inherent complexity of the symbolic paths to generate robust cryptographic keys, circumventing reliance on algebraic hardness assumptions. In contrast, the BLAKE3 hash function, while exhibiting statistically uniform diffusion, demonstrates weaker performance in amplifying symbolic divergence, highlighting the importance of selecting hash functions that effectively enhance the divergence of symbolic trajectories.

This work establishes a promising foundation for post-quantum cryptography by exploring a non-algebraic approach rooted in the dynamics of chaotic systems. Researchers provide empirical evidence supporting the potential hardness of the HFKR problem and offer an open-source implementation to facilitate further investigation and development.

Researchers acknowledge the computational cost of generating and compressing the symbolic trajectories as a limitation. Future work will focus on developing more efficient algorithms for trajectory generation, reducing the computational overhead. Further analysis of the security of the HFKR scheme against various attacks is also planned, alongside exploration of applications in key exchange, digital signatures, and encryption. Researchers also intend to investigate the use of alternative chaotic systems and fractal geometries within cryptographic algorithms.

This research contributes to the growing body of knowledge on post-quantum cryptography, offering a novel alternative to traditional algebraic methods. By leveraging the principles of chaotic dynamics and fractal geometry, the HFKR scheme provides a new approach to secure communication, potentially resistant to attacks from both classical and quantum computers. Future work will focus on optimising performance and exploring practical applications.

The authors provide a publicly available simulation code, enabling reproduction of results and further investigation. This open-source approach fosters collaboration and accelerates the development of post-quantum cryptographic algorithms. Researchers encourage exploration of variations, such as different contractive affine maps or hash functions, potentially leading to even more secure and efficient systems.