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 $δ$ 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 $δ< δ_0\simeq 0.89$, is considered to ensure key consistency in noisy environments, such as 5G networks. On the other hand, for $δ> 3$, 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.

翻译：本文提出一种对称流密码，利用复平面中随机三次映射的动态特性生成伪随机密钥流。该系统基于随机三次多项式$f_n(z)=z^3+c_n z$的迭代，其中参数$c_n$从半径为$\delta$且中心位于原点的圆盘中随机选取，旨在增强混沌行为，进而提升生成序列的随机性。当$\delta < \delta_0 \simeq 0.89$时，考虑Julia集在微小参数扰动下的稳定性，以确保在噪声环境（如5G网络）中密钥的一致性。另一方面，当$\delta > 3$时，系统呈现不稳定与混沌特性，适用于生成超安全密钥。Python实现集成了安全密钥派生、通过预热迭代生成稳健密钥流，以及基于现代密码学原语（\texttt{HKDF}和\texttt{HMAC-SHA-256}）的认证加密方案，以保障消息完整性与真实性。对密钥流生成器的输出执行了统计分析（包括卡方检验和熵计算），以评估其随机性与分布。此外，按照现代密码学中\texttt{NIST SP 800-22}标准完成了完整的统计验证，以增强所提出系统的可信度。