Recently, a chaotic image encryption algorithm based on perceptron model was proposed. The present paper analyzes security of the algorithm and finds that the equivalent secret key can be reconstructed with only one pair of known-plaintext/ciphertext, which is supported by both mathematical proof and experiment results. In addition, some other security defects are also reported.
more higher frequency; 2) low efficiency of the traditional text encryption algorithms like DES with respect to protecting image data due to some intrinsic features of images such as bulk data capacity, high redundancy, strong correlation among adjacent pixels, etc; 3) some fundamental features of the chaotic dynamical systems such as ergodicity, mixing property, sensitivity to initial conditions/system parameter can be considered analogous to some ideal cryptographic properties such as confusion, diffusion, avalanche properties. According to the record of Web of Science, more than five hundreds of articles on chaos-based image encryption algorithms have been published in the past fifteen years [3]. Some other related papers consider video, audio (speech) or text as encryption objects [4,5,17].
Roughly speaking, the usage of chaos in designing digital symmetric encryption algorithms can be categorized as three classes: 1) creating position permutation relations directly or indirectly; 2) generating pseudorandom bit sequence (PRBS) controlling composition and combination of some basic encryption operations; 3) producing ciphertext directly by assigning plaintext as initial conditions or control parameters of a chaos system. As native opposite of cryptography, some cryptanalysis work was also developed and some chaos-based encryption algorithms are found to be not secure of different extents from the viewpoint of modern cryptology [1,2,8,15,18,20]. Due to owning complex dynamical properties, artificial neural network was combined with chaos to design symmetric and public encryption algorithms [6,7,13,14,21]. Unfortunately, some of them are found to be equivalent to much simper form in terms of essential structure and can be broken easily [10,11].
In [19], a chaotic image encryption algorithm based on perceptron model, a type of artificial neural network invented in 1957, was proposed, where two PRBSs, gen-erated from quantized orbit of Lorenz system under given secret key, is used to control the perceptron model to output cipher-image from the input of plain-image. However, we proved that the seeming complex image encryption algorithm is equivalent to a stream cipher of exclusive or (XOR) operation. So, it can be easily broken with only one pair of plain-image and cipherimage. In addition, some other security defects about secret key, insensitive of cipher-image with respect to plain-image and low randomness of the used PRBS are reported.
The rest of the paper is organized as follows. The next section briefly introduces the proposed image encryption algorithm. Section 3 presents detailed cryptanalysis on the encryption algorithm with experiment results and some other security defects. The last section concludes the paper.
The plaintext encrypted by the image encryption algorithm under study is a gray-scale digital image. Without loss of generality, the plaintext can be represented as a one-dimensional 8-bit integer sequences P = {p n } N -1 n=0 by scanning it in a raster order. Correspondingly, the ciphertext is denoted by
The kernel of the encryption algorithm is based on a threshold function
which was considered by the proposers as simple variant of a single layer perceptron model who inputs m variables, s 0 , s 1 , • • • , s m-1 , and outputs m ones by calculating
where w ij denotes the weight of the i-th input for the j-th neuron, θ i is the threshold of the i-th neuron, and i = 0 ∼ m -1. With these preliminary introduction, the image encryption algorithm under study can be described as follows 1 .
-The secret key: initial state (x * 0 , y * 0 , z * 0 ) and the step length h of an approximation method solving the 1 To make the presentation more concise and complete, some notations in the original paper [19] are modified, and some details about the algorithm are also amended under the condition that its security property is not influenced.
under fixed control parameters (a, b, c) = (10, 8 3 , 28). -The initialization procedures:
- in double-precision floating-point arithmetic, solve the Lorenz system (1) with the fourth-order Runge-Kutta method of step h iteratively 3001 times from (x * 0 , y * 0 , z * 0 ) and obtain the current approximation state (x 0 , y 0 , z 0 ) . 2) run the above solution approximation step seven more times from the current approximation state to get {(x j , y j , z j )} 7 j=1 , and set
and
), ŷ = max({y j } 7 j=0 ), y = min({y j } 7 j=0 ). 3) reset the current approximation state of the Lorenz system as
where r = ⌊(z 8 -⌊z 8 ⌋) • 256⌋. 4) repeat the above two steps N -1 times and get two sequences
k=0 . -The encryption procedure: for the n-th plain-byte
where
-The decryption procedure is the same as the encryption one except that the locations of p n,i and p ′ n,i in the encryption function (2) are swapped.
Breaking a chaotic image encryption algorithm based on perceptron model 3 3 Cryptanalysis
It is well-known that any detail of an encryption algorithm, except
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