## 1 Introduction
This technique, also recognized as \(\pm\) embedding, adjusts the pixel value by randomly adding or subtracting one to ensure the message bit aligns with the cover pixel's LSB when they do not match. Such a strategy is designed to minimize pixel value alterations, thereby reducing the image distortion. After [PERSON]'s foundational LSB matching approach, various methods [11; 12; 13; 14] have emerged that enhance the technique's utility and application. In 2006, [PERSON] and [PERSON] [15] introduced a DH method using the EMD (exploiting modification direction) technique, which encodes secret data according to a \((2n+1)\) arithmetic scheme into \(n\) cover pixels for transmission. This method involves selecting a pixel from the \(n\) cover pixels and subtly modifying its value--either by incrementing or decrementing by one--to embed the secret information, while the EMD technique is lauded for its data hiding efficiency, it's crucial to note that, unlike RDH methods, the original cover image cannot be restored after data extraction with the EMD approach, highlighting a significant distinction.The data hiding method based on Hamming coding [16], a coding method known as an error correction code, is known to be an effective method of hiding data in images. Meanwhile, dual RDH [13; 14; 17; 18; 19; 20; 21; 22; 23], inspired by traditional secret image sharing methods [24], represents an advanced technique requiring two cover images for the DH process. In this method, secret data are concealed within both images. These altered images--now known as stego images--must then be combined to retrieve the embedded data. Over the past decade, this method has undergone intensive research and development, highlighting its promise for secure data transmission. Since [PERSON] et al. (2007) [17] introduced an RDH method that leverages the EMD technique, research into this and related methods [13; 14; 22; 23] has significantly expanded. Additionally, some researchers have investigated dual RDH using the LSB matching technique [13; 14; 22; 23] as an alternative approach. Unlike EMD, LSB matching aims at aligning the LSBs of pixel values in two cover images for data embedding, offering a distinct balance between data capacity and image fidelity. [PERSON] et al. (2013) [21] proposed a DH method, in which only one pixel is changed by replacing the major and minor diagonals with horizontal and vertical lines. Similarly, [PERSON] and [PERSON] (2013) [25] introduced a method for hiding data by grouping two contiguous pixels from each of the two cover images, achieving reversibility by combining pairs of pixels in a dual stego image in a specific direction. In 2015, [PERSON] et al. [14] presented a novel RDH technique using two images and LSB matching, marking a significant advance in DH technology. This method introduces a new approach by combining two images to enhance security and data capacity. Following this innovation, [PERSON] et al. [13] in 2017 published an improved version of the two-image-based reversible encryption technique, incorporating enhanced LSB matching. In this refined approach, duplicates of the cover image are created, and an advanced LSB matching method is applied to embed the secret message in both images, demonstrating a creative application of existing technologies for improved results. In 2022, [PERSON] et al. [23] made a significant contribution to the field by introducing an innovative reversible data encryption technique. Leveraging a novel LSB sorting technique combined with EMD, this method significantly expands and refines the strategies previously available in this field. A thorough analysis of this modified LSB matching technique has revealed theoretical similarities with the EMD method, enriching our understanding of its fundamental principles. In this study, we introduce a groundbreaking RDH technique that combines the exploiting modification direction (EMD) method with an optimal least significant Bit (OLSB) replacement strategy, which is applied to two identical grayscale cover images.