## 1 Introduction
Unlike natural images, encrypted images are not informative and, hence, are not easily decodable. Hence, it is much more difficult to identify the presence of any information if the secret information is embedded in an encrypted image. Moreover, encrypting the cover image protects the image from disclosing itself to unauthorized parties, concealing its meaning and, hence, safeguarding its privacy. Thus, the use of RDH in the encrypted domain has become important, attracting researchers to identify new approaches and applications. RDH in the encrypted domain is a challenging task, as the pixel correlation on the cover image no longer exists. Hence, the space for hiding the secret data needs to be vacated, either before encrypting the cover image [22; 23] or after the encryption [24; 25]. The data hider can embed the secret data into these vacated spaces but RDH requires the receiver to recover the original cover and the hidden secret without any loss. This, in turn, implies that the quantity of data that can be hidden is related to the recovery of the original cover. For example, the embedding capacity of the RDH scheme described in [26] improves when the cover is processed with a block side-length less than 32, but complete reversibility is guaranteed only at a block side-length of 32 and higher. Thus, a well-designed RDH needs to maintain a positive trade-off between the quantity of the secret and the quality of the recovered cover. To the best of our knowledge, most of the existing RDH schemes for encrypted images alter the distribution of the pixel intensities of the encrypted image while hiding the secret message [27; 28; 29]. Entropy is a metric that measures the randomness of the pixels in an image. The randomness of pixels indicates how different each pixel is from another. Generally, for an encrypted image, the entropy will yield a value near 8 (cover image, being an 8-bit grayscale image) and histogram analysis will generate a flat or uniform histogram. These are indicators of a good encryption algorithm. Conversely, entropy and histogram analysis are good metrics by which to evaluate the quality of the encrypted image. For example, [30] used histogram analysis as one of the metrics to demonstrate that the encrypted image and watermarked image were very different and, hence, that nothing could be inferred about the cover image from the watermarked image. The entropy of the encrypted image and the hidden data image described in [31] are slightly different, demonstrating the efficiency of the scheme. In this context, we propose an RDH scheme for encrypted images which is very different from most such schemes described in the existing literature. The proposed RDH scheme is designed in such a way that the pixel intensities of the encrypted image are never modified during the process of data hiding. To implement such an RDH scheme, we utilized a well-known image-scrambling algorithm known as the Fibonacci transform. Image-scrambling is treated as a kind of encryption that provides security of the image data by making the contents undecipherable and unreadable. It destroys the high correlation that exists between neighboring pixels. The Fibonacci transform performs scrambling by modifying the pixel locations through simple matrix transformations. Thus, such an algorithm can be tuned for data hiding, meeting our requirement of retaining the pixel intensities of the encrypted image. Many studies are available which address securing data by encryption [32] through the use of pell sequences [33]. Image-scrambling involves matrix transformations that can also be used in information-covering. The Fibonacci transform is an image-scrambling technique that uses a \(2\times 2\) transform matrix with four elements that follows four consecutive numbers from the Fibonacci series [34]. Suppose \((x,y)\) is the original coordinate and a transformation to another coordinate \((x^{\prime},y^{\prime})\) is given by,  
\[\left[ \begin{array}{c}x^{\prime}\\ y^{\prime}\end{array} \right]=\left[ \begin{array}{cc}1&1\\ 0&1\end{array}