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

TL;DR

This paper introduces the linear canonical Riesz potential (LCRP), analyzes its convergence properties, and applies it to develop a secure multi-image encryption system with demonstrated robustness.

Contribution

It presents the LCRP, studies its convergence/divergence at critical indices, and proposes a novel, efficient, and secure encryption method based on these properties.

Findings

LCRP converges for chirp functions but diverges for classical Riesz potentials on grating functions.

The limit of Riesz potential at non-boundary points matches the characteristic function, but differs at boundaries.

The proposed encryption system outperforms existing methods in efficiency and demonstrates robustness against various attacks.

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.