## 1 Introduction  
The Internet's public accessibility facilitates the easy exchange of data. Yet, this openness also increases the risk of surveillance, theft, and corruption of transmitted data. To mitigate these risks, data hiding (DH) technology offers a robust solution to enhance the security of data in transit. DH techniques protect data by embedding it within another file or message, often within a 'cover image', to create what is known as a'stego image' [1,2]. This technique allows the cover image to conceal the secret data, effectively shielding it from unauthorized viewers, and it is then transmitted to the intended recipient [3,4]. The alterations to the cover image are typically imperceptible, making the hidden data nearly undetectable without specialized analysis tools. This characteristic significantly enhances the confidentiality of messages hidden within stego images, positioning DH as an excellent strategy for secure communication via the Internet. DH techniques are categorized into the following two main types: reversible and irreversible. Reversible data hiding (RDH) [5] technology is vital for ensuring information security and maintaining data integrity. The primary advantage of RDH is its ability to perfectly restore the original cover image to its original state after the extraction of the embedded data, thus preserving the quality of the cover image without any loss of the original data. This feature not only enhances information security, but also maintainsthe high quality of the original data and images, offering a robust level of security. RDH is especially beneficial in fields requiring high precision and protection, including legal documents, medical images, and copyrighted content, due to its unique capability of the complete recovery of the original data and strict security requirements. This makes RDH an invaluable tool in areas where data integrity and security are of the utmost importance. Building on the foundation of RDH's significance, various methods [5; 6; 7; 8] have been developed to optimize its application while addressing specific challenges. The difference expansion (DE) method, introduced by [PERSON] [5], and the histogram shift (HS) method, developed by [PERSON] [6], illustrate RDH's versatility; moreover, while DE offers substantial data capacity by modifying the pixel value differences, it may introduce visual distortions in sensitive applications. Conversely, HS [6; 7] aims to minimize such distortions by adjusting the image's histogram for data embedding, thus preserving the visual integrity of the cover image. The strategic application of RDH highlights its indispensable role in ensuring both the secure transmission of information and the impeccable recovery of the original data. The predictive coding (PE) method, proposed by [PERSON] [8], leverages the correlation between neighboring pixels to accurately predict their values, embedding data within the prediction's error margins. By capitalizing on the inherent redundancy in images, [PERSON]'s method discreetly embeds data, minimizes noticeable alterations, and preserves the original image's quality more effectively than the DE method [5]. The field of DH technology has seen significant advancements with the introduction of various techniques, notably, the least significant bit (LSB) substitution method. Pioneered by [PERSON] and [PERSON] in 2004 [9], this technique embeds hidden data into an image by altering its LSB. In this method, the \(k\) secret bits are used to replace the \(k\) least significant bits of each cover pixel, thus embedding the hidden data into the image. Building on this foundational approach, numerous enhancements have been proposed, driving the continuous evolution of DH technology. Initially introduced by [PERSON] in 2006, the LSB matching method [10; 11; 12; 13; 14] embeds messages by altering the least significant bit (LSB) of the cover pixel.