In this work, we propose a data-driven image encryption framework that identifies chaotic maps directly from data using the SINDy-PI algorithm. Unlike conventional encryption schemes relying on predefined maps, our method learns the full explicit dynamics -- including cross-terms and higher-order nonlinearities -- from observational data. The validity of this approach is verified on three distinct chaotic systems: the H{é}non map, the three-dimensional logistic map, and the piecewise-linear Lozi map, demonstrating its generality. The encryption key consists solely of initial conditions; the map structure itself becomes data-dependent, introducing an extra layer of security. Moreover, even when the initial conditions are fixed, different training data (e.g., with a tiny noise seed) lead to slightly different maps, which produce completely different ciphertexts (NPCR $\approx 99.6\%$, UACI $\approx 33.5\%$). Numerical experiments on the H{é}non system show near-ideal information entropy ($\approx 8$ bits), negligible inter-pixel correlation, and extreme sensitivity to initial conditions: a perturbation of $10^{-16}$ causes total decryption failure. The scheme resists both differential and statistical attacks, with NPCR and UACI values matching theoretical ideals. Our results establish a new paradigm for chaos-based cryptography beyond fixed maps.

翻译：本文提出一种数据驱动的图像加密框架，通过SINDy-PI算法直接从数据中辨识混沌映射。与依赖预定义映射的传统加密方案不同，本方法从观测数据中学习完整的显式动力学特性——包括交叉项和高阶非线性项。该方法的通用性通过三种不同混沌系统得到验证：Hénon映射、三维Logistic映射和分段线性Lozi映射。加密密钥仅由初始条件构成；映射结构本身具有数据依赖性，从而引入额外安全层。此外，即便初始条件固定，不同训练数据（如包含微小噪声种子）也会导致映射存在细微差异，进而产生完全不同的密文（NPCR≈99.6%，UACI≈33.5%）。基于Hénon系统的数值实验表明，该方法可实现接近理想值的信息熵（≈8比特）、可忽略的像素间相关性，以及对初始条件的极端敏感性：10⁻¹⁶的扰动即导致完全解密失败。该方案能抵御差分攻击和统计攻击，其NPCR与UACI值均匹配理论理想值。本研究结果为超越固定映射的混沌密码学建立了新范式。