Image Encryption via Data-Identified Discrete Chaotic Maps

TL;DR

This paper introduces a novel data-driven image encryption method that learns chaotic maps directly from data, enhancing security by making the encryption structure data-dependent and highly sensitive to initial conditions.

Contribution

It develops a framework that identifies explicit chaotic maps from observational data using SINDy-PI, moving beyond fixed maps in chaos-based cryptography.

Findings

Successfully applied to multiple chaotic systems demonstrating generality.

Achieves high information entropy and low pixel correlation.

Exhibits extreme sensitivity to initial conditions, ensuring security.

Abstract

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{\'e}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{\'e}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.