Limit Properties at Critical Indices of Linear Canonical Riesz Potentials and Their Applications to Security of Multi-Image Encryption

2026-05-02

AI Summary

This work addresses the convergence issues of the classical Riesz potential at critical exponents when applied to discrete image functions, which limit both security and efficiency in image encryption. The authors propose the Linear Canonical Riesz Potential (LCRP) and, for the first time, reveal its distinct limiting behaviors between Schwartz functions and discrete image functions at critical exponents. Building upon the LCRP and its inverse operator, they develop an asymmetric cascaded multi-image encryption framework that integrates the linear canonical transform, Riesz potential theory, and the linear canonical Laplacian operator. Experimental results demonstrate that the proposed method significantly outperforms existing approaches in sensitivity, statistical properties, and robustness against noise and occlusion attacks, while also surpassing fractional-order Riesz potential schemes in single-image encryption performance.

Abstract

In this article we introduce the linear canonical Riesz potential (for short, LCRP) and give its symbol in terms of linear canonical transforms. Driven by image processing, we establish the convergence/divergence of these LCRPs for different kinds of functions. Concretely, for grating functions, we prove that their classical Riesz potentials diverge, whereas their LCRP converge due to the key role of chirp functions. For the characteristic function ${\mathbf 1}_P$ of a convex polygon $P$, we show that the limit of its Riesz potential at any non-boundary point $\boldsymbol{x}$ equals ${\mathbf 1}_P(\boldsymbol{x})$, but its limit at the boundaries differ from ${\mathbf 1}_P$, while it is known that, for any Schwartz function $f$, the limit of its Riesz potential at any point $\boldsymbol{x}$ always equals $f(\boldsymbol{x})$. Based on these and the inverse operator of the LCRP (namely the linear canonical Laplacian operator), we propose an asymmetric cascaded LCRP method for the multi-image encryption and create an efficient and secure cryptosystem. Systematic security evaluations, including sensitivity, statistical, noise attack, and occlusion attack analyses, demonstrate its robustness and its security. Even for a single image, the proposed method is more efficient than the known encryption approach based on the fractional Riesz potential. The novelty of these results lies in that the convergence and the divergence of LCRTs at the critical indices, respectively, for ``good" Schwartz functions and for ``bad" discrete image functions essentially affect the security of image encryption and decryption.

Problem

• linear canonical Riesz potential
• critical indices
• limit properties
• multi-image encryption
• convergence and divergence

Innovation

• Linear Canonical Riesz Potential
• Convergence at Critical Indices
• Asymmetric Image Encryption
• Linear Canonical Laplacian
• Multi-image Cryptosystem