CommentaryCommentary on “A block chaotic image encryption scheme based on self-adaptive modelling” [Applied Soft Computing 22 (2014) 351–357]
Introduction
In [1], Ye and Zhou designed and then applied essentially the same round function on a plain image for a number of times, denoted as the round number r, in order to achieve image encryption. They analysed the security of their scheme against differential plaintext analysis based on two statistical measures i.e., the unified average changing intensity and number of pixels change rate. We term the scheme proposed by Ye and Zhou as YEZ's scheme throughout the paper.
This paper presents r-round differentials with probability 1 on the YEZ's scheme, for any number of rounds r. Subsequently, a distinguishing attack and a chosen-ciphertext attack are presented against YEZ's scheme by exploiting the discovered r-round differentials. To our knowledge, our attacks are the first to show that image encryption schemes should consider the security against distinguishing attacks, as commonly expected by the cryptographic community. The results indicate that YEZ's scheme cannot provide adequate security against differential attack even if one increases the number of rounds.
Section snippets
The specification of YEZ's scheme
A plain image P is defined as being of size m × n pixels where each pixel is of 8 bits, and where m is the number of rows and n is the number of columns. The m × n image P, in vectorized form is denoted as P(i), for i = 1, 2, …, m × n.
Preliminaries
Some basic functions used in its construction are described as follows:
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The floor operation b = ⌊a⌋: Integer b is the largest integer not greater than a.
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The ceiling operation b = ⌈a⌉: Integer b as the smallest integer not less than a.
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The modulus operation c = mod(a,
The r-round differentials with probability 1
Three patterns of r-round differentials with probability 1 are found based on the following observations. Observation 1 Given the same secret keys , x0, y0, z0 and rn, if the sum si of all pixels in XR,i is the same for two different images, then the same rkL,i will be generated. Observation 2 Given the same secret keys and r2, the same rkR = rkR,i will be generated as it does not depend on the image for any of the rounds i. Observation 3 To result in the same sum of all pixels for two different images, the number of pixel differences
Attacks based on r-round differentials
The r-round differentials with probability 1 can be exploited to effectively distinguish images encrypted using YEZ's scheme from a random encrypted image. The differentials can also be exploited under the chosen-ciphertext attack model [4] to recover any plaintext given any new ciphertext.
Conclusion
Several r-round differentials were presented against YEZ's scheme that occur with probability 1, for any number of rounds r. By exploiting such differentials, encrypted images can be distinguished and furthermore the plaintext can be recovered when given a ciphertext generated using this scheme. Lessons learnt from the results include: (1) using modular addition on its own is not a good design principle to provide nonlinearity; (2) statistical measures do not guarantee the security of image
Acknowledgment
W.S. Yap's work was fully supported by the UTARRF. R.C.-W. Phan's research was supported by the Ministry of Higher Education's FRGS under the project ProvAdverse.
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