ORIGINAL ARTICLE

Deciphering an RGB color image cryptosystem based on Choquet fuzzy integral

Yushu Zhang1 • Wenying Wen2 • Yongfei Wu3 • Rui Zhang4 • Jun-xin Chen5 •

Xing He1

Received: 20 May 2015 / Accepted: 12 August 2015 The Natural Computing Applications Forum 2015

Abstract For the purpose of designing better color image cryptosystems, it is necessary to perform a mathematically rigorous security analysis for the existing ones. In this study, a Choquet fuzzy integral-based color image cryptosystem with shift–diffusion architecture is cryptanalyzed.

Firstly, it is demonstrated that the color image cryptosystem is irreversible, since the encryption algorithm cannot be decrypted correctly. Meanwhile, a simple improvement is presented to guarantee the reversibility. Moreover, a cryptanalysis of the improved scheme is proposed and results show the improved color image cryptosystem can be broken by a known plain/cipher image pair. The shift– diffusion architecture should alter the external key every time for security consideration.

Keywords Color image encryption Cryptanalysis

Choquet fuzzy integral Shift–diffusion 1 Introduction

Nowadays a growing number of RGB color images with private and confidential information are produced, stored and transmitted over network systems and smartphone equipments at every moment. Forcolor image security purpose, it is of crucial significance to exploit some encryption operations before transmission. Inparticular,chaos-based techniquesare the most widely utilized for designing color image cryptosystems [1–8]. A hybrid color image encryption scheme based on a cyclic elliptic curve and chaotic system was presented [1]. In [2], a high-dimensional chaotic map called oneway coupled-map lattices was used to encrypt color images with a stream cipher structure. Liu and Wang also designed a stream cipher algorithm based on one-time keys and robust chaotic maps [3]. In [4], the R, G, B components of a color image are encrypted at the same time using chaos such that these three components can be affected with each other.

Zhang and Xiao [5] proposed a novel scheme based on coupled logistic map, self-adaptive permutation, S-boxes transform and combined global diffusion. Liu and Wang [6] leveraged a bit-level permutation and high-dimensional chaotic map to encrypt color images. Zhou et al. [7] combined fractional Fourier transform with chaos to build up an interesting encryption scheme. The DNA sequence operation was also embedded in chaos-based color image encryption algorithm [8, 9]. The S-boxes which were constructed by chaotic maps were also employed to perform color image encryption operations in [10] and [11], respectively.

In this study, we decipher a color image cryptosystem recently presented in [12], which is based on Choquet fuzzy integral (CFI). This cryptosystem possesses the structure of shift–diffusion. Three gray-level components, which are produced through decomposing the color pixels, are randomly shifted in bit level by the pseudo-random & Yushu Zhang yushuboshi@163.com 1 School of Electronics and Information Engineering,

Southwest University, Chongqing 400715, China 2 School of Information Technology, Jiangxi University of

Finance and Economics, Nanchang 330013, China 3 College of Mathematics and Statistics, Chongqing

University, Chongqing 401331, China 4 College of Computer Science, Chongqing University,

Chongqing 400044, China 5 School of Information Science and Engineering, Northeastern

University, Shenyang 110004, Liaoning, China 123

Neural Comput & Applic

DOI 10.1007/s00521-015-2045-2 keystreams generated by CFI. Then the shifted components are coupled with the keystreams. Our study demonstrates that the original image cryptosystem is irreversible, since the encryption mode cannot be decrypted correctly. At the same time, a simple improvement is proposed to guarantee the reversibility. Moreover, the cryptanalysis of the improved scheme is presented. The remaining proceeds with the following manner. The next section gives a redescription of the original color image cipher based on

CFI. In Sect. 3, the irreversibility of the original RGB color image cipher and a corresponding simple improvement are proposed. The cryptanalysis of the improved cipher is stated in Sect. 4. The last section concludes the paper. 2 A re-description of the original RGB color image cipher based on Choquet fuzzy integral

This section firstly introduces the CFI and corresponding pseudo-random keystream generation process and then re-describes the original color image cipher based on

CFI. 2.1 CFI and corresponding pseudo-random keystream

For a measureable function h1 : A! 0;þ1ð Þ with respect to a fuzzy measure h2, the fuzzy integral is defined asR

A h1 h2. The CFI of a finite set of A ¼ x1; x2; . . .; xnf g is computed as

Eh2 h 1 ¼

Xn i¼1 h1 xið Þ h1 xi1ð Þ h2 Aið Þ; ð1Þ or

Eh2 h 1 ¼

Xn i¼1 h1 xið Þ h2 Aið Þ h2 Aiþ1ð Þ ; ð2Þ where h x1ð Þ h x2ð Þ . . .h xnð Þ and h x0ð Þ ¼ 0. The fuzzy density h2 and the input value h1 need to be known for the calculation of CFI.

The generation of pseudo-random keystream requires three initial inputs h11; h 1 2; h 1 3 and three membership grades h21; h 2 2; h 2 3 , which are controlled by a series of algebraic transformations with a 128-bit external secret key

K ¼ k1; k2; . . .; k16. The initial condition of the CFI is calculated as ti ¼ k4i3 þ 4i 3ð Þ k4i2 þ 4i 2ð Þ þ k4i1 þ 4i 1ð Þ k4i þ 4i

Xj¼16 j¼1 kj 28 j1ð Þ 2128 ! mod 1; ð3Þ where i ¼ 1; 2; 3; 4. The values of ti are rearranged in ascending order and are set to be equal to ðh11; h12; h13Þ and the dynamic parameter b, respectively. The ðh21; h22; h23Þ are generated by the following formulas: h21 ¼ 0:5 h22 ¼ 2þ h23 þ 1 h23 1 h23 ¼ 2þ b 8 >>< > : : ð4Þ

Finally, the pseudo-random keystream is produced by y ¼ ARS Int E mod 1ð Þ 1014 ;D þ