## 1 Introduction
Apart from visual perception metric, some reported extra metrics including histograms and pixel correlation/key security results; for example, [PERSON] et al. [27], [PERSON] and [PERSON] [29], and [PERSON] and [PERSON] [30], which represent the most comprehensive work for security analysis. To the best of our knowledge, entropy analysis and image sensitivity analysis have never been presented by these HDR encryption algorithms in the extant literature. Without thorough security analysis, it remains unclear whether the image cipher actually does have a high security level able to resist malicious attacks, such as entropy attacks and differential attack [23; 24]. This paper proposes a novel constructive data hiding and chaotic sequence-based bit-level image encryption algorithm. Our scheme offers six significant features. First, we conceal secret messages when converting an HDR image from the OpenEXR format into the RGBE one. From the steganalysis point of view, our scheme is able to resist the steganalytic attack because we construct a stego image rather than generate it from the existing cover image. Second, we adaptively embed a secret message according to the distribution of the exponent channel shared by the Red, Green, and Blue channels. The adaptivity enables our scheme to conceal more secret messages in pixels with low luminance, where their exponent value is smaller than the median of the whole HDR image. Third, we exploit adaptive message embedding to satisfy a user's request. A user can request a large embedding rate in exchange for endurable image quality. Alternatively, an end user can exchange a limited embedding rate for a high quality stego image. Fourth, we introduce a chaotic sequence-based bit-level image encryption algorithm which not only protects the image contents, but also secures the hidden messages, thereby reducing the probability of any inappropriate use of the stego HDR RGBE image produced. We adopt a 2D Sine Logistic modulation map (2D-SLMM) [33] to generate pseudo-random sequences. The map holds two large positive Lyapunov exponent (LE) values and two large Kolmogorov entropy values, thereby providing better hyperchaotic behavior, more complexity, and unpredictability. Fifth, we apply the random permutation technique to our scheme in a bit-level permutation mechanism. As evidence from the experimental results, the bit-level permutation significantly increases the security of HDR image encryption. The final feature of our scheme is that we present six metrics to thoroughly and comprehensively evaluate the security of the cipher image. The main contributions of our work can be summarized as follows:  
* We present a novel adaptive data hiding algorithm for an HDR image. Secret messages are concealed during the HDR format conversion, signifying that the cover and stego image are in different image formats, avoiding the vulnerability caused by conventional data hiding utilizing the cover image modification approach. * Our algorithm adaptively embeds secret messages. More secret messages are conveyed in those pixels containing low luminance appearing to be dark pixels, while fewer secret messages are injected into pixels with high luminance which appear to be brighter; this coincides with the phenomena that human eyes are less sensitive to the dark pixels and more sensitive to noticing subtle changes encountered in the brighter area. * Our data hiding scheme provides the ability to satisfy the request for the embedding rate proposed by the end user. In addition, image distortion due to the message concealment can be predicted prior to real message embedding. Thanks to the optimal base mechanism we introduce, which not only satisfies users' demand, but also derives the minimal mean squared error to reduce the image distortion as small as possible. To the best of our knowledge, no algorithm in the HDR data hiding literature can comply with the embedding rate requested by the end user.