A chaotic flux cipher based on the random cubic family $f_{c_n}(z)=z^3+c_n z$

Abstract

This paper presents a symmetric stream cipher that utilizes the dynamic properties of random cubic mappings in the complex plane to generate pseudo-random key streams. The system is based on the iterations of the random cubic polynomial $f_n(z)=z^3+c_n z$, where the parameters $c_n$ are chosen randomly from a disc of radius $\delta$ and with center at the origin, aiming to improve the chaotic behaviour and, consequently, the randomness of the generated sequence. The stability of the Julia set under small parameter perturbations, when $\delta < \delta_0\simeq 0.89$, is considered to ensure key consistency in noisy environments, such as 5G networks. On the other hand, for $\delta > 3$, the system exhibits instability and chaos, ideal for generating ultra-secure keys. The Python implementation integrates secure key derivation, robust key stream generation via warmed-up iteration, and an authenticated encryption scheme using the modern cryptographic primitives (\texttt{HKDF} and\texttt{HMAC-SHA-256}), to ensure message integrity and authenticity. Statistical analyses, including chi-square test and entropy calculation, are performed on the output of the key stream generator to evaluate its randomness and distribution. In addition, a complete statistical validation, compliant with \texttt{NIST SP 800-22} standards in modern cryptography, was performed to enhance the proposed system's credibility.