The Fractal Spectral Wave Filter: A Quantum-Resilient Cryptographic Primitive Harnessing the Aliasing Collisions

The rapid evolution of Quantum Computing poses a significant threat to the standard mathematical foundations of classical cryptography, prompting the global transition towards Post-Quantum Cryptography (PQC). While NIST is standardising lattice-based algorithms, chaotic cryptography continues to struggle with high-dimensional data, as traditional models require O(N^3) computational time and suffer from catastrophic memory overhead. Building upon prior optimization frameworks designed to eliminate cubic computational bottlenecks in chaotic systems [6], this paper introduces a novel cryptographic framework centered on the Fractal Spectral Wave Filter. By mapping arbitrary two-dimensional spatial data into a one-dimensional array using the Morton Z-order curve, we preserve local data relationships without dense matrix overhead. The system then utilizes the Number Theoretic Transform (NTT) over finite Galois integer rings—forming the core mathematical framework of NIST's PQC standards—for frequency-domain processing. Furthermore, by intentionally omitting standard zero-padding, the resulting unmitigated aliasing collisions act as a secure, lossless fractal scrambler, diffusing data in strict O(N log N) time. Experimental validations demonstrate perfect lossless recovery (MSE of 0.000000) and an optimal Shannon entropy of up to 7.9991 bits/symbol, establishing a new generalized paradigm for quantum-resilient data security.