## 2 Related Works
### BPCX Analysis  
The traditional stream encryption is not secure enough: it is easily attacked by key repetition, and it has limited key space, which affects the security. New encryption techniques, such as the PSO-based image encryption scheme using a modular integrated logistic exponential map, aim to enhance security through optimization and chaos theory, providing robust and complex encryption processes. Designs of two-dimensional hyperchaotic systems and the use of fractional-order 3D Lorenz chaotic systems and 2D discrete polynomial hyper-chaotic maps offer high-performance multi-image encryption. Additionally, methods like image encryption algorithms based on plane-level image filtering and discrete logarithmic transform have introduced novel approaches to increasing encryption complexity and security. These advancements aim to address the limitations of traditional stream ciphers and provide more secure and efficient encryption methods for modern needs. It bears emphasis that the BPCX encryption algorithm commonly employed during the encryption phase exhibits three salient characteristics:  
1. The maximum number of 0-bit and 1-bit values within an image block remains constant before and after encryption.
2. The arrangement of pixels within a block does not alter the average value of the block.
3. The arrangement of pixels within a block does not effectively cover the low frequency information in the image, resulting only in a transformation to a high-definition and fuzzy image.  
In response to the first two properties, [PERSON] et al. [26] proposed a KPA based on bit-block inversion and mean equivalent division (MED), to estimate the block permutation key for BPCX image encryption. The experimental results indicated that even for the smallest 2 \(\times\) 2 blocks the average estimated accuracy of the block arrangement sequence was over 40%. When the block size exceeded 3 \(\times\) 3, the accuracy of the block permutation key estimation surpassed 50%. This suggests that utilizing the BPCX encryption algorithm may pose certain security concerns. Based on the data, it is evident that using stream encryption alone is insecure. In recent years, new image encryption technologies have developed rapidly, such as PSO-based image encryption using modular integrated logistic exponential maps [27]; the MILE map addresses issues of low performance and the narrow chaos range in existing chaotic maps; the encryption efficiency is enhanced by optimizing the key, using the particle swarm optimization (PSO) algorithm. The novel 2D Rosenbrock map, based on the Rosenbrock function with excellent oscillating characteristics, is a two-dimensional hyperchaotic system that works through the optimization benchmark function [28]. Its application in multimedia encryption demonstrates outstanding chaotic capability, due to its regenerative and oscillating properties. Additionally, a newly designed fractional-order 3D Lorenz chaotic system and a 2D sine-constrained polynomial hyper-chaotic map (2D-SCPM) have been exploited for high-performance multi-image encryption [29]. Effective encryption of multiple images has been achieved, enhancing image encryption efficiency.