[Paper Review] Coupling Chaotic System Based on Unit Transform and Its Applications in Image Encryption

Guozhen Hu, Baobin Li|arXiv (Cornell University)|Sep 18, 2019

Chaos-based Image/Signal Encryption45 references4 citations

TL;DR

This paper proposes a novel unit transform-based coupling chaotic system (UT-CCS) that combines any two 1D chaotic maps to generate a new chaotic map with enhanced dynamical properties. A chaos-based pseudo-random number generator (CBPRNG) is designed with proven uniformity and randomness, enabling a secure image encryption scheme that achieves strong confusion and diffusion, outperforming existing methods in correlation reduction and robustness to data loss.

ABSTRACT

Chaotic maps are very important for establishing chaos-based image encryption systems. This paper introduces a coupling chaotic system based on a certain unit transform, which can combine any two 1D chaotic maps to generate a new one with excellent performance. The chaotic behavior analysis has verified this coupling system's effectiveness and progress. In particular, we give a specific strategy about selecting an appropriate unit transform function to enhance chaos of generated maps. Besides, a new chaos based pseudo-random number generator, shorted as CBPRNG, is designed to improve the distribution of chaotic sequences. We give a mathematical illustration on the uniformity of CBPRNG, and test the randomness of it. Moreover, based on CBPRNG, an image encryption algorithm is introduced. Simulation results and security analysis indicate that the proposed image encryption scheme is competitive with some advanced existing methods.

Motivation & Objective

• To address the limitations of simple 1D chaotic maps in image encryption, such as narrow chaotic parameter ranges, finite precision sensitivity, and non-uniform sequence distributions.
• To develop a general framework that combines any two 1D chaotic maps into a new, higher-performance chaotic system using unit transform functions (UTFs).
• To design a chaos-based pseudo-random number generator (CBPRNG) with mathematically proven uniformity and improved randomness for cryptographic use.
• To construct a secure image encryption algorithm based on CBPRNG that ensures strong confusion and diffusion, with robustness to data loss.
• To validate the proposed system through comprehensive security analysis, including correlation, randomness, and robustness tests.

Proposed method

• Introduces a general framework called UT-CCS (Unit Transform-based Coupled Chaotic System) that combines two 1D chaotic maps via a unit transform function (UTF) to generate a new 2D chaotic map with improved dynamics.
• Proposes a theorem to guide the selection of optimal UTFs that enhance the chaotic behavior of the resulting map, ensuring broader chaotic parameter ranges and better ergodicity.
• Designs a chaos-based pseudo-random number generator (CBPRNG) that maps chaotic sequences through a transformation to achieve uniform distribution, proven mathematically using statistical moments.
• Employs CBPRNG in a two-phase image encryption scheme: confusion using one CBPRNG output and diffusion using another, ensuring high randomness and strong diffusion properties.
• Applies standard image security metrics—correlation analysis, entropy, NPCR, UACI, and robustness to data loss—using benchmark images like Lena, Cameraman, and Peppers.
• Uses the Pearson correlation coefficient to quantitatively evaluate pixel correlation reduction, with values near zero post-encryption indicating effective decorrelation.

Experimental results

Research questions

• RQ1 Can a general framework be developed to combine any two 1D chaotic maps into a new chaotic system with superior dynamical performance using unit transform functions?
• RQ2 What criteria or mathematical principles can guide the selection of unit transform functions to maximize chaos enhancement in the resulting coupled system?
• RQ3 Can a chaos-based pseudo-random number generator be constructed with provable uniformity and high randomness for use in image encryption?
• RQ4 How effective is the proposed encryption scheme in breaking pixel correlation and achieving strong confusion and diffusion?
• RQ5 To what extent does the proposed scheme maintain image recovery quality under data loss conditions, indicating robustness in real-world transmission?

Key findings

• The proposed UT-CCS framework successfully combines any two 1D chaotic maps into a new 2D system with broader chaotic parameter ranges and improved ergodicity, as validated by chaotic behavior analysis.
• The CBPRNG achieves uniform distribution of chaotic sequences, proven mathematically via moment analysis, and passes standard randomness tests, ensuring high-quality pseudo-random output.
• After encryption, the average correlation coefficients across all directions (horizontal, vertical, diagonal) for the Lena image drop to 0.001135, 0.002457, and 0.003213, respectively, indicating effective decorrelation of adjacent pixels.
• The proposed scheme achieves NPCR and UACI values close to ideal levels (NPCR ≈ 99.6%, UACI ≈ 33.4%), demonstrating strong sensitivity to plaintext and key changes.
• Even with 50% data loss in the encrypted image, the decrypted image retains high visual quality, confirming the scheme’s robustness to partial data loss in transmission.
• Compared to existing methods, the proposed algorithm achieves competitive or superior performance in correlation reduction and randomness metrics, as shown in Table 14.