Cracking a hierarchical chaotic image encryption algorithm based on permutation
Introduction
With the increasing transmission speeds of wired/wireless networks and popularization of image capturing devices and cloud storage services, image data are transmitted over open networks more and more frequently. This makes security of image data become more and more important. The public concern of it becomes serious as news about the illegal online leak of personal photos of some celebrities was released. As a chaotic system owns some similar properties as that of modern encryption schemes, it has been intensively studied as an alternative approach for designing secure and efficient encryption schemes [1], [2], [3]. The main idea and principle of applying chaos theory to protecting images can be traced back to 1986 [4], which demonstrates the stretching effect of a chaotic map on a painting of Henri Poincaré, a founder of modern chaos theory.
The simplest and most efficient method for protecting multimedia data is permuting the positions of their spatial pixels [5] or frequency coefficients [6]. In the literature, some synonyms of permutation, transposition, shuffle, scramble [6], swap and shift, are used. Security scrutiny on some specific permutation-only encryption algorithms against known/chosen-plaintext attacks were previously developed [7], [8]. In [9], a ciphertext-only attack on a specially simple permutation-only encryption algorithm was proposed utilizing correlation redundancy remaining in the cipher-image. No matter how the permutation relationships are generated and what the permutation object is, any permutation-only encryption algorithm can always be represented by a permutation relationship matrix, whose entry stores the corresponding permuted location in the cipher-text [10]. The security of permutation-only encryption algorithm relies on its real permutation domain, in which any element in the permutation object can be permuted independently. As for a permutation domain of size with T possible value levels, it is estimated that the required number of known/chosen-plaintexts for an efficient plaintext attack is , where denotes the ceiling function. An upper bound of the attack complexity is also derived therein to be , where n is the number of known/chosen plain-images [10]. In [11], the computational complexity of the attack is further reduced to by replacing the set intersection operations of quadratic complexity with linear element access operations. Even so, all kinds of permutation operations are still being used in multimedia protection today [12], [13], [14], [15].
In [16], a typical example of permutation-only image encryption algorithms, called HCIE (Hierarchical Chaotic Image Encryption), was proposed. Although security performance of general permutation-only image encryption algorithms against plaintext attack has been quantitatively analyzed, specific security performance of HCIE is still not evaluated. The core of HCIE is a permutation function composed of rotation operations of four directions, originates from an intellectual toy, Rubik׳s Cube [17]. In [16], the authors claimed about the security property of HCIE as follows: “By way of collecting some original images and their encryption results or collecting some specified images and their corresponding encryption results, it is still difficult for the cryptanalysts to decrypt an encrypted image correctly because the permutation relationship is different for each image.” In this paper, we will demonstrate that the claim on the robustness of HCIE against known/chosen-plaintext attack is groundless. Further more, we find that the hierarchical encryption structure suggested in HCIE does not provide any higher security against known/chosen-plaintext attack, but actually make the overall security performance even weaker. In addition, we find that the capability of HCIE against ciphertext-only attack was much over-estimated.
The rest of this paper is organized as follows. The algorithm HCIE is briefly introduced in Section 2. Detailed cryptanalysis on HCIE is provided in Section 3, with some experimental results. The last section concludes the paper.
Section snippets
The hierarchical chaotic image encryption algorithm (HCIE)
HCIE is a two-level hierarchical permutation-only image encryption algorithm, in which all involved permutation relationships are defined by pseudo-random combinations of four rotation mappings with pseudo-random parameters. For an image, , the four mapping operations are described as follows, where holds for each mapping. Definition 1 The mapping is defined to rotate the i-th row of f, in the left (when b=0) or right (when b=1) direction, by p pixels. Definition 2 The
The ciphertext-only attack on HCIE
In [16], it was claimed that the complexity of brute-force attacks to HCIE is , since there are secret chaotic bits in that are unknown to the attackers. However, this statement is not true due to the following fact: the Lb bits are uniquely determined by the secret key, i.e., the initial condition and the control parameter μ, which have only secret bits. This means that there are only different chaotic bit sequences.
Now, let us
Conclusion
Specific security performance of a typical permutation-only encryption algorithm, called HCIE, against ciphertext-only attack and known/chosen-plaintext attacks has been studied in detail. It is found that the capability of HCIE against the former attack was over-estimated much and hierarchical permutation-only image encryption algorithms such as HCIE are less secure than normal permutation-only ones without using hierarchical encryption structures. This work effectively demonstrates that the
Acknowledgments
This research was supported by the Distinguished Young Scholar Program of the Hunan Provincial Natural Science Foundation of China (No. 2015JJ1013), and Scientific Research Fund of Hunan Provincial Education Department (15A186). Some parts of Section 3 were completed with the help of Dr. Shujun Li, from Surrey University, UK.
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