The Hashed Fractal Key Recovery (HFKR) Problem: From Symbolic Path Inversion to Post-Quantum Cryptographic Keys

Abstract: Classical cryptographic systems rely heavily on structured algebraic problems, such as factorization, discrete logarithms, or lattice-based assumptions, which are increasingly vulnerable to quantum attacks and structural cryptanalysis. In response, this work introduces the Hashed Fractal Key Recovery (HFKR) problem, a non-algebraic cryptographic construction grounded in symbolic dynamics and chaotic perturbations. HFKR builds on the Symbolic Path Inversion Problem (SPIP), leveraging symbolic trajectories generated via contractive affine maps over $\mathbb{Z}^2$, and compressing them into fixed-length cryptographic keys using hash-based obfuscation. A key contribution of this paper is the empirical confirmation that these symbolic paths exhibit fractal behavior, quantified via box counting dimension, path geometry, and spatial density measures. The observed fractal dimension increases with trajectory length and stabilizes near 1.06, indicating symbolic self-similarity and space-filling complexity, both of which reinforce the entropy foundation of the scheme. Experimental results across 250 perturbation trials show that SHA3-512 and SHAKE256 amplify symbolic divergence effectively, achieving mean Hamming distances near 255, ideal bit-flip rates, and negligible entropy deviation. In contrast, BLAKE3 exhibits statistically uniform but weaker diffusion. These findings confirm that HFKR post-quantum security arises from the synergy between symbolic fractality and hash-based entropy amplification. The resulting construction offers a lightweight, structure-free foundation for secure key generation in adversarial settings without relying on algebraic hardness assumptions.