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，该框架利用SINDy-PI算法直接从数据中辨识混沌映射。与传统依赖预定义映射的加密方案不同，我们的方法从观测数据中学习完整的显式动力学特征——包括交叉项和高阶非线性项。通过在三个不同混沌系统上的验证（Hénon映射、三维Logistic映射和分段线性Lozi映射），证明了该方法的通用性。加密密钥仅由初始条件构成；映射结构本身变为数据相关，从而引入额外安全层。此外，即使初始条件固定，不同的训练数据（例如含有微小的噪声种子）也会生成略有差异的映射，这些映射会产生完全不同的密文（NPCR ≈ 99.6%，UACI ≈ 33.5%）。在Hénon系统上的数值实验显示，该方法具有接近理想的信息熵（≈ 8比特）、可忽略的像素间相关性，并对初始条件极其敏感：10⁻¹⁶的扰动即会导致完全解密失败。该方案能抵御差分攻击和统计攻击，其NPCR和UACI值均与理论理想值吻合。我们的研究结果超越了固定映射的局限，建立了混沌密码学的新范式。