Partial Encryption of Color Image Using ... - Semantic Scholar

International Journal of Signal Processing, Image Processing and Pattern Recognition Vol.8, No.10 (2015), pp.171-190 http://dx.doi.org/10.14257/ijsip.2015.8.10.20

Partial Encryption of Color Image Using Quaternion Discrete Cosine Transform Jing Fan1,2, Jinwei Wang 1,2, Xingming Sun 1,2 and Ting Li 1,2 1

(Jiangsu Engineering Center of Network Monitoring, Nanjing University of Information Science & Technology, Nanjing 210044, PR China) 2 (School of Computer & Software, Nanjing University of Information Science & Technology, Nanjing 210044,PR China) Email:[email protected], [email protected],[email protected] Abstract In this paper, the quaternion discrete Cosine transform (QDCT) is applied to the color image encryption technology, and partial encryption technique of color image based on QDCT is proposed. Obtained the QDCT spectrum of color image by QDCT, the real and imaginary parts of QDCT coefficients are quantized and encoded with the color image quantization coefficient table and QDCT coefficients encoding method, and part of low and high frequency coefficients are encrypted in the encoding process. Simulation results and security performance analysis shows that the proposed scheme can obtain the effect of secure image encryption, reduce the amount of encryption data and transmission data, and well distribute the pixels of the original image when only part of the data are encrypted. What’s more, the proposed scheme can decrypt and recover the original image with high quality. Due to the large key space, the proposed scheme can resist exhaustive attacks. Keywords: quaternion discrete Cosine transform; color image; partial encryption; encode

1. Introduction With the rapid popularization of Internet technology, multimedia technology has become a popular medium of communication, convenient for information communication to the public. Color images which can fully describe the image detail, have the characteristics of high fidelity and color expressive performance, being one of the most widely used multimedia data types, even more, the transmission safety of color images in the current open Internet environment becomes an important research content in the field of information security. As color images have characteristics of multichannel and large amount of data, the traditional data encryption approach can't solve such problems as the large number of encryption data, high computational complexity and system resource consumption, while the color image partial encryption technique can not only improve the encryption efficiency, reduce the system encryption burden, but also can satisfy the need of different security applications, and has important research value and significance [1-5]. Partial encryption technique for color images can be roughly divided into three categories from the perspective of multichannel processing pattern: the two-dimensional (2D) plane, block-tree mapped and three-dimensional (3D) space encryption technique. The 2D plane partial encryption technique encrypt part of multichannel and multidimensional color image in the 2D space, namely the 2D gray image encryption technique are directly applied to multidimensional color images, that is color image channels will be encrypted separately, with encryption data of each color channel composing of the ultimate encryption data finally. Block-tree mapped partial encryption technique combine the multiple components of color image together into block-tree

ISSN: 2005-4254 IJSIP Copyright ⓒ 2015 SERSC

International Journal of Signal Processing, Image Processing and Pattern Recognition Vol.8, No.10 (2015)

structure form using space mapping, then encode the block-tree structure data, meanwhile the encryption is proceeded during the encoding process, and one of the most typical representative technique was color image partial encryption technique based on CSPIHT. 3D space encryption technique is the scheme that preprocess and encrypt color image whose multiple components are put into 3D data structure space, between which and 2D plane encryption technique the biggest difference and the improvement are being able to encrypt multiple channels of color image at the same time, eliminating tedious independent channel encryption, improve the encryption efficiency, what’s more ,representative technique of the 3D space encryption is partial encryption technique based on quaternion. Quaternion representation method for color image process the pixels as a whole vector in multidimensional space considering the color association, that can really keep the correlation between color components in mathematical way comparing with the block-tree mapped encryption technique, and theoretically, it can bring about better encryption effect and has resistance to attack. Current encryption techniques for color images are mostly multichannel independent encryption method, as multiple color channels are represented as a number of gray images, which are encrypted independently with gray image encryption means, the gray image encryption data compose the final encrypted data in the final. However, such an approach ignoring the correlation between color channels of color image, has high computational complexity and wastes computation space, and to some extent, the encryption efficiency decreased greatly. Quaternions (or hypercomplex) have been widely used in geometry and physics since born, and quaternions are introduced to the field of color image processing in recent years [6-24]. Gray image encryption method mostly proceed in airspace which has little operating space and low security factor, so the improved encryption method is performed in the frequency domain, and orthogonal transformation commonly used are discrete Cosine transform (DCT), discrete Fourier transform (DFT) and discrete wavelet transform(DWT), etc. that have been expanded in quaternion field. Ell was the first person who gave the definition of quaternion (hypercomplex) Fourier transform (QFT) in his doctoral thesis[6], Bahri and Ashino introduced the correlation theorem of QFT form II[7], then Sangwine applied QFT to color images, and gave the definition of quaternion discrete Fourier transform (QDFT)[8]. Ouyang and Wen et al, applied QFT to color image watermarking, and proposed a robust blind color image watermarking algorithm [9]. GAI and Wang et al. combined DQFT and double random phase encryption technology based on optical image encryption theory, proposing a new double random phase encryption technology for color images [10-11]. Meanwhile GAI et al. put QDFT into information hiding technique of color image and put forward a new technique that imbed secret data in color image[12]. YAO et al. proceed DQFT to host image and watermarking image at the same time and then present a blind double color image watermarking algorithm based on DQFT [13]. XU et al. raised fractional quaternion Fourier transform and defined its convolution and correlative computation [14]. Chan et al. proposed quaternion wavelet transform (QWT) and successfully applied it to multidimensional signal processing [15]. Bahri and Ashino et al. also gave the definition of two-dimensional QFT with fixed core, based on which intended QWT is defined [16]. Shao et al. put forward quaternion rotation transform and also put it in use of color image encryption [17]. GONG et al. proposed color and multispectral remote sensing image water area extraction algorithm based on quaternion using the advantage of quaternion in processing color and multispectral remote sensing image [18-19]. Feng et al. extended DCT to quaternion discrete Cosine transform (QDCT), meanwhile applied it to color image template matching [23]. GAI el al. firstly used QDCT in color image digital watermarking technique and raised a blind color image watermarking algorithm [24]. However, materials that QDCT applied in color image encryption are not found referring to a number of references at home and abroad. Different from current trend multichannel independent color image encryption technique, color image processing technique based on quaternion using quaternion representation

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method for color image, regards color image pixels as vector and dispose them integrally in multidimensional space, namely processing multiple channels of color images as a whole, considering the color association, and saving the computation space, thus attracted much attention in the field of color image processing. This paper proposes a partial color image encryption algorithm based on QDCT, for the first time introducing QDCT to color image encryption technique. Utilizing quaternions’ characteristics that have a real part and three imaginary parts, we convert multiple channels of the color image to a whole discrete quaternion matrix through quaternions, extending the color images from 3D space to four-dimensional space, finally obtained the quaternion spectrum of color image by QDCT and encrypted image by aid of special encryption method. The simulation results show that this method is safe and effective to encrypt color image with high efficiency, low encryption data volume, and can meet the demand of different encryption application modes. In the four-dimensional space constructed with quaternions, dealing with color images by QDCT orthogonal transform makes processing color image information become more intuitive and sharp, and provides a new thought for research of partial color image encryption.

2. QDCT Definition for Color Image In this section, we will firstly introduce the definition and algebra of quaternions briefly, next show the quaternion representation method for color image, then present the definition and calculation of QDCT. At last, quantization tables of QDCT coefficients for color image are proposed and described in detail through a series of deducing and computation. The quantization table has great significance for the following proposed encryption technique. 2.1 Quaternion Definition and Algebra Hamilton firstly introduced the concept of quaternion in 1843[25], the quaternion q can be represented in a hypercomplex form as (1) q  a  bi  c  j  d  k Where a, b, c, d  R, and i, j, k are imaginary numbers which obey the following rules 2 i 2  j  k 2  1

i  j   j  i  k , j  k  k  j  i, k  i  i  k  j

(2)

From which we can see that the multiplication of quaternions is not commutative. Quaternions are the extension of real numbers and complex numbers, being regarded as the combination of real part and imaginary part, when a is the real part of the quaternion q , b ·i + c ·j + d ·k is the imaginary part of quaternion q , in addition, we called quaternion q pure quaternion when a = 0. The modulus of quaternion q is defined as (3) q  a 2  b2  c 2  d 2 2.2 Quaternion Representation Method for Color Image Color images are composed of multiple color components, for example, color image has R, G and B three components in RGB color model, but has H, S and I three components in HIS color model, and that every component can represent a gray image independently. RGB color model is the main describing and discussing object in this paper. Pei was the first person that applied quaternion to color image, as well proposed quaternion model of color image[22], which consider the three color components R, G, B


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as three imaginary parts of pure quaternion q whose real part qr = 0 , it can be described as below (4) q( x, y)  r ( x, y)  i  g ( x, y)  j  b( x, y)  k Where r(x, y), g(x, y), b(x, y) are gray values of color components R, G, B at position (x, y) , so we can absolutely represent a color image with a quaternion matrix. 2.3 QDCT Definition A color image I in size of X × Y can be described in form of discrete matrix

f (x, y), in

which x and y are row and line positions of pixel matrix, and x  [0, X－1], y  [0, Y－ 1] , then f (x, y) can be expressed as a pure quaternion whose real part is zero in such a form (5) f q ( x, y)  f r ( x, y)  i  f g ( x, y)  j  f b ( x, y)  k Where f r (x, y) , f g (x, y) , f b (x, y) are the matrix of the three color components R, G, B of f (x, y) . The forward QDCT(FQDCT) of color image f (x, y) is defined as[23] X 1 Y 1

FQDCT q( p, s)   ( p) (s)     q f q( x, y)  N ( p, s, x, y) x 0 y 0

(6)

Where μq is a pure quaternion, and satisfied with μq 2 = 1 , i.e. the modulus of μq is 1, and it can also be expressed as[21] (7)  q  i  i   j  j   k  k So different μq result in different calculation results of FQDCT. In experiment, we assume as μq = (i + j + k) / 3 , therefore μi = μj = μk = 1 / 3 . The definitions of N(p, s, x, y), α(p), α(s) are shown as below   (2 x  1) p    (2 y  1) s  N ( p, s, x, y )  cos   cos    2X 2Y     1  1 , p0 , s0 (8)    X  Y  ( p)    ( s)    2,  2, p0 s0    X  Y Considering Cq (p, s) as the QDCT of f (x, y) , Cq (p, s) can be expressed as (9) C q( p, s)  FQDCT q( f q( x, y)) And Cq (p, s) is a quaternion matrix in the same size with f (x, y) , it can be regarded as the QDCT spectrum of f (x, y) , so Cq (p, s) can be more represented as (10) C q ( p, s)  C r ( p, s)  Ci ( p, s)  i  C j ( p, s)  j  C k ( p, s)  k Where Cr (p, s) is the real part coefficient of spectrum, Ci (p, s), Cj (p, s) , Ck (p, s) are imaginary part coefficients of spectrum. The inverse QDCT(IQDCT) of f (x, y) is defined as X 1 Y 1

IQDCT q( x, y)      ( p) ( s)  qC q( p, s)  N ( p, s, x, y) x 0 y 0

Where μq , FQDCT.

N(p, s, x, y),

(11)

α(p), α(s) have the same definitions with those in

2.4 QDCT Coefficients Quantization Table for Color Image As QDCT is the extension of DCT in quaternion domain, we speculate the relationship between QDCT coefficient quantization table and the acquiescent standard JPEG

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luminance quantization table, according to computation relationship between QDCT and DCT coefficients, and then deduce the specific quantization tables of QDCT coefficients. In this section ,we will give the quantization tables of QDCT coefficients and their derivation. Taking Eq.5-7 and 10 as substitution of corresponding components in Eq.9, the QDCT spectrum Cr (p, s), Ci (p, s), Cj (p, s) , Ck (p, s) can be described as below X 1 Y 1

C r ( p, s)   ( p) (s)      i f r ( x, y)   j f g ( x, y)   k f b( x, y)   N ( p, s, x, y) x 0 y 0

X 1 Y 1

C i( p, s)   ( p) (s)  x0 y0   j f b( x, y)   k f g ( x, y)   N ( p, s, x, y) X 1 Y 1

C j ( p, s)   ( p) (s) x0 y0  k f r ( x, y)   i f b( x, y) N ( p, s, x, y)

(12)

X 1 Y 1

C k ( p, s)   ( p) (s)     i f g ( x, y)   j f r ( x, y)   N ( p, s, x, y) x 0 y 0

Under the condition μq = (i + j + k) / 3 we take, s) can be further represented as

Cr (p, s), Ci (p, s), Cj (p, s) , Ck (p,

X 1 Y 1

1 C r ( p, s)   3  ( p) (s)     f r ( x, y)  f g ( x, y)  f b( x, y)   N ( p, s, x, y) x 0 y 0

C i ( p, s )  C j ( p, s ) 

X 1 Y 1 1  ( p) ( s)     f 3 x 0 y 0

b

( x, y )  f g ( x, y )   N ( p, s, x, y )

X 1 Y 1 1  ( p) ( s)     f r ( x, y )  3 x 0 y 0

(13)

f b( x, y )  N ( p, s, x, y )

X 1 Y 1

1 C k ( p, s)  3  ( p) (s)     f g ( x, y)  f r ( x, y)   N ( p, s, x, y) x 0 y 0

What’s more, they have numerical relationship of Ci (p, s) + Cj (p, s) + Ck (p, s) = 0. The definition of DCT of a gray image f ′(x, y) in size of X  Y can be described as X 1 Y 1

C( p, s)   ( p) (s)   f ( x, y)  N ( p, s, x, y) x 0 y 0

(14)

Where N(p, s, x, y), α(p), α(s) have the same definitions with those in FQDCT. Analyzing the relationship among Eq.13, we have found that f r (x, y), f g (x, y) , f b ( x, y) and f ′(x, y) are all gray matrix in size of X×Y , and there are 3 times numerical relationship between Cr (p, s) which is the real part of QDCT spectrum of f (x, y) and C′(p, s) which is the DCT spectrum of f ′(x, y), meanwhile there are also 2 / 3 times numerical relationship between Ci (p, s), Cj (p, s) , Ck (p, s) and C′(p, s). Assuming that the JPEG standard luminance quantization table is TJPEG , which is a integer matrix in size of 8×8, we define QDCT coefficient quantization table TQDCT as real part table Treal and imaginary part table Timag two parts. Treal and Timag are both integer matrix in size of 8×8 , in addition , there are numerical relationship between TJPEG , Treal and Timag as T real  3 T JPEG  1.732T JPEG 2 (15) T imag  T JPEG  1.155T JPEG 3 According to the matrix elements of quantization table are integers, we conduct the round computation with Treal and Timag , generating the final specific quantization table as below


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 28  21   24  24 T real   31   41  85  125  18  14   16  16 T imag   21   28 57  83

19 21 23

17 24 28

28 33 42

42 45 69

69 100 99

88 104 120

29 38 61 111

38 64 95 135

50 97 111 151

88 118 140 178

151 189 180 210

139 178 196 208

159

165

170

194

173

178

13 14 15

12 16 18

18 22 28

28 30 46

46 67 66

59 69 80

20 25 40 74

25 43 64 90

33 65 74 100

59 79 94 119

100 126 120 140

92 119 131 139

106

110

113

129

116

119

106 95  97  107 133 159 175  171

(16)

    72  89  106 117  114 70 64 65

The proposed QDCT coefficient quantization table for color image is universal and completely compression in 100%, while quality factor which was usually mentioned during quantization procedure will directly affect image compression rate and compressed image quality, and commonly when quality factor increases, compression rate decreased and the compressed image quality increases. Assuming Q is the quality factor, the scaling factor α and Q obey the rules as 

50 / Q (100  Q) / 50

 

0  Q  50 50  Q  100

(17)

The unit of Q is % , and the quantization table used in the experiment is or T = α ·Timag .

T = α ·Treal

3. Color Image Partial Encryption Technique based on QDCT In this section, the encryption and decryption procedure are briefly introduced, then we proposed the method of QDCT coefficient coding for color image which is the expansion and based on JPEG compression and coding, finally the encryption technique based on QDCT coefficient coding for color image is proposed and described in detail. 3.1 Encryption and Decryption Procedure In this paper ,we extended JPEG compression encoding to QDCT, as the color image are compressed and encoded in QDCT domain, meanwhile the partial encryption is proceeded, realizing the simultaneity of encoding and encryption. The encryption and decryption procedure are shown in Figure 1(a), in which the encrypted data and encryption position are given separately, and correlative encoding and encryption method will be introduced in detail in section 3.2 and 3.3. The encryption procedure is divided into such steps as below: 1) The original color image I are 8  8 blocked and the pixel matrix is migrated; 2) FQDCT, obtaining the QDCT spectrum Cr, Ci, Cj, Ck ; 3) Quantization; 4) Scanning and sorting in Zigzag; 5) Entropy encoding and encryption; 6) Getting image encryption data D. The decryption algorithm is the converse procedure of encryption algorithm as Fig.1(b) has shown, therefore the decryption procedure is divided into such steps as below: 1) Entropy decoding and decryption with D; 2) Inverse scanning in Zigzag;

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3) 4) 5)

Inverse quantization; IQDCT and inverse pixel matrix migration; Obtaining the decrypted image I ′.

(a)Encryption algorithm

(b)Decryption algorithm

Figure 1. Flow Diagram of Encryption and Decryption Algorithm

(a)JPEG encoder

(b)JPEG decoder

Figure 2. Flow Diagram of JPEG Encoder and Decoder 3.2 QDCT Coefficient Coding 3.2.1 JPEG Compression and Coding: As is known, JPEG lossy compression algorithm which is based on DCT remove redundant information visual and data itself taking


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advantage of human visual system. Figure 2(a) shows the JPEG encoder process, in which DCT based coding is the core part including frequency domain orthogonal transformation DCT, quantizer and entropy encoder, furthermore, quantizer needs the JPEG standard quantization table and entropy encoder needs the entropy encoding table(usually adopting the Huffman encoding table). Figure 2(b) shows the JPEG decoding process which is the inverse process of encoding.

(a)QDCT encoder

(b)QDCT decoder

Figure 3. Flow Diagram of QDCT Encoder 3.2.2 QDCT Coefficient Coding: The proposed QDCT coefficient coding is the extension of JPEG coding, and Figure 3(a) shows the QDCT encoder process, in which the entropy encoder as Figure 3(b) has shown is the extension of JPEG entropy encoder in quaternion domain. The main computation steps of QDCT encoder are listed as below: 1) Proceeding the FQDCT with original image; 2) Quantizing the QDCT coefficient matrix, in addition, quantizing the real part coefficients Cr with Treal which is defined in section 2.3, and quantizing the imaginary part coefficients Ci , Cj , Ck with Timag ; 3) Scanning the quantized QDCT coefficients real part and imaginary part either in Zigzag, converting the two-dimensional matrix to one-dimensional vector; 4) Encoding the DC(Direct Current) coefficients of both real part coefficients and imaginary part coefficients with DPCM(Differential Pulse Code Modulation), presenting them in form of intermediate 2-tuple (s, a) , in which s is the size, a is the amplitude; 5) Encoding the AC(Alternating Current) coefficients of both real part coefficients and imaginary part coefficients with RLE(Run-Length Encoding), presenting them in form of intermediate 3-tuple (r, s, a), in which r is the run-length, s is the size and a is the amplitude; 6) VLC-VLI coding, in which VLC is variable length coding, usually adopting the Huffman coding and VLI is variable length integer coding; proceeding VLC coding with s and VLI coding with a in intermediate 2-tuple of DC coefficients separately; proceeding VLC coding with (r, s) and VLI coding with a in intermediate 3-tuple of AC coefficients; 7) The code stream entropy coded is the compressed image data. QDCT coefficient decoding is also the inverse process of encoding, as Fig.4 has shown, its calculation steps are not discussed here.

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Figure 4. Flow Diagram of QDCT Decoder 3.3 Encryption Technique based on QDCT Encoding 3.3.1 DC Coefficient Encryption: The encryption algorithm process introduced in section 3.1 demonstrates that the encryption algorithm is proceeding during the entropy encoding, specifically DC intermediate 2-tuple and AC intermediate 3-tuple of QDCT entropy encoder. We choose to encrypt DC coefficients before intermediate 2-tuple coding and after Zigzag scanning either. The DC coefficient encryption can be divided into such steps as below: 1) Reconstructing DC coefficients of all the sub-matrix that blocked in size of 8×8 2)

to DC coefficient matrix as {p(i, j)}, i  [0, X/8－1], j  [0, Y/8－1]; Proceeding Logistic chaos scrambling with DC coefficient matrix, generating the initial value x0 (key 1) with pseudo-random number generator, we set the precision of x0 32 places after the decimal point. The equation of Logistic chaos mapping is described as xn+1 = μ xn(1－xn), where μ  (0, 4) is the bifurcation parameter which is near to 4, and

3)

4)

5)

6)

xn  (0, 1) is the state variable of system;

Encrypting DC coefficient value p when it’s nonzero and in the range of [－m, m], and m is the degeneration degree parameter which represents the quality of reconstructed image, in addition, m > 0; Mapping p to index q according to Eq.18, and q  (0, 2m] [26]: p, p0 (18) q   p0 m  p , Generating the random integer d (key 2) which is in the range of (0, 2m) with pseudo-random number generator, and mapping q to index q′ according to Eq.19, q′  (0, 2m]: q  2m  d  q  d, (19) q   q  d  2m, q  2m  d Inverse mapping index q′ to corresponding DC coefficient value p′ according to Eq.20, and p′  [－m, 0) ∪ (0, m]:

q  m  q, (20) p   m  q, q  m So p′ is the encrypted DC coefficient value, then we continue to proceed the following intermediate 2-tuple coding and VLC-VLI encoding with encrypted DC coefficients. The degeneration degree parameter m affect encryption result directly


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which can decide the application mode of encrypted image whether is degeneration mode or security mode, what’s more, the value of m is related to quality factor Q, therefore different original image and Q result in different m . In the experiment , we set m = max {|p(i, j)|}, and give four chaos initial values x0r , x0i , x0j , x0k , four degeneration degree parameters mr , mi , mj , mk , and four random integers dr , di , dj , dk considering of Cr and Ci , Cj , Ck , avoiding of relevance between real part and imaginary part of QDCT coefficients, that can acquire better encryption effect, in addition, Logistic chaos initial values and random integers are part of the key. As every step of DC coefficient encryption procedure is inverse mapping, we can decrypt the DC coefficients when proceeding the encryption procedure inversely with correct key. 3.3.2 AC Coefficient Encryption：We choose to encrypt the AC coefficients after the intermediate 3-tuple coding and before the VLC-VLI encoding either, and the encryption procedure can be divided into such steps as: 1) Encrypting nonzero AC coefficients; 2) Getting s and a from intermediate 3-tuple coding of AC coefficients; 3) Conducting VLI coding with a and obtaining the result b ; 4) Generating the random binary number c (key 3) of M (M ≥ s) places with pseudo-random number generator; 5) Selecting the higher s places of c , and mapping b to b′ according to Eq.21: (21) b  b  c b′ is the VLI coding value of encrypted a . In the experiment, considering of Cr and Ci , Cj , Ck , we set four random binary numbers cr , ci , cj , ck which are part of the key. As VLI coding is implemented during encryption process, next we only have to proceed the VLC coding. The AC coefficient decoding procedure is the inverse process of encoding as well, therefore, data can be correctly decrypted with correct key.

4. Experiment Results and Discussion The experiment is conducted under MATLAB simulation platform, we tested a mount of RGB color image in size of 512×512, and due to space limitations, we choose classic images Lenna, Baboon, Lake and Pepper to show experiment results as Figure 5 shows. We carry out a series of experiments, mainly discussing about encryption and decryption results, peak signal to noise ratio, statistical analysis, encryption performance, key space analysis and some data comparison analysis with other algorithm in references.

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Figure 5. Original Images 4.1 Encryption and Decryption Results Analysis We choose the quality factor Q under condition of Q = 50 and Q = 80 to test the encryption and decryption results with different color images. Fig.6-7 show the encrypted image, correctly decrypted image and wrongly decrypted image of Lenna and Baboon. The results demonstrate that the encrypted images of test image are unrecognizable, realized secure encryption effect, in addition, encrypted images appear snowflake-like distribution which has the same hue with original image, for example, the encryption image of Lenna shows a red hue. The original images can be recovered in high quality accurately with correct key, and the decrypted images under condition of Q = 50 and Q = 80 have no difference with original images when observed with naked eye. Due to the quantization table is the original quantization table with no expansion nor shrink when Q = 50, we set the image quality of decrypted image under condition of Q = 50 as the unified standard of measuring the performance of decryption algorithm proposed in this article. The recovered images with wrong random key are unrecognizable as well, reflecting the sensitivity of key, what’s more, the wrongly decrypted images present a complementary hue with encrypted image, as the wrongly decrypted image of Lenna shows a green hue.


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(a)Encrypted images

(b)Correctly decrypted images

(c)Wrongly decrypted images

Figure 6. Encrypted and Decrypted Images of Lenna; Lines from First to Second are Q=50, Q=80

(a)Encrypted images

(b)Correctly decrypted images (c)Wrongly decrypted images

Figure 7. Encrypted and Decrypted Images of Baboon; Lines from First to Second are Q=50, Q=80 4.2 Peak Signal to Noise Ratio In the experiment, we select the peak signal to noise ratio (PSNR) as the comparison test indicator of encrypted image and decrypted image with original image, for PSNR provides an objective standard to measure image distortion and noise level with the evaluation result shown as unit dB. The definition and calculation of PSNR is:

182



PSNR  10  log10 X 1Y 1

2552 MSE

  ( I  P )2

MSE 

(22)

x 0 y 0

X Y

Where MSE is the mean square error, X and Y are the size of input image, I and P are the images being tested and compared. The image distortion degree is lower when the PSNR is higher, otherwise it’s higher, and the PSNR tends to infinity when the image is without distortion. Normally the image distortion is unrecognizable even none with naked eye observing when the PSNR is higher than 28 dB, however, when the PSNR is lower than 11 dB, the image has a serious distortion.

(a)PSNR1(PSNR of encrypted images)

(b)PSNR2(PSNR of decrypted images)

Figure 8. Trend of PSNR with Changes of Q Figure 8 shows the PSNR trend of encryption and decryption image of Lenna, Baboon, Lake and Pepper with changes of Q, where the abscissa is Q at 5 intervals ranging from 5 to 95, the ordinate is PSNR value, of which PSNR1 is the PSNR of encrypted image, PSNR2 is the PSNR of decrypted image. Simulation results show that the PSNR of encrypted image of different test images are very low with different Q, the encrypted images have serious quality loss realizing the security of image information. Due to the influence of quantization degree, when Q < 30, the quantization degree is high, QDCT coefficient value and the number of non-zero coefficient change a lot influencing the reconstructed image quality, reflected as the PSNR goes up and down with the increase of Q, but when Q > 30, the quantization degree is low, having little impact on QDCT coefficients, the PSNR gradually stabilized. The PSNR of decrypted image of Lenna, Lake, Pepper reach higher than 30 when Q = 50, when Baboon needs the condition Q = 80 to obtain the same result, so different test images can decrypt the encryption image in high quality under certain conditions. The PSNR of decrypted image is in direct proportional to Q, that is, the decryption effect is better when Q is higher, this is because Q decides the degradation degree parameter m and the QDCT coefficient quantization degree, while the m directly affects the number of encrypted DC coefficient and the quantization degree affects the number of encrypted AC coefficient, therefore when Q gets larger, the quantization degree gets lower and the m gets larger, thus the number of encrypted DC and AC coefficient is more, with increasing amount of encryption data making higher encryption accuracy, that can get better decryption effect, but at the same time the execution time of the encryption and decryption algorithm will correspondingly extend.


183


(a)R channel

(b)G channel

(c)B channel

Figure 9. Histogram of Encrypted Baboon; Lines from First to Fourth are Original Image, Q=10,Q=50 ,Q=90 4.3 Statistical Analysis Histogram can reflect the distribution of different pixel values of image, due to space limitations, Figures 9-10 only show the R, G, B three-channel-histogram of original image and encrypted image when Q =10, Q =50 , Q =90 of Baboon and Lake, in which each channel is regarded as gray image, the pixel value range is 0 to 255, the abscissa is pixel value, the ordinate is the number of the pixels of according value. Results demonstrate that the three-channel-histogram of encryption images compared with the three-channel-histogram of original images, to a certain extent, the pixel distribution is more even, and with the increase of Q, the even degree increases. When Q is low, the quantization degree is high, the QDCT AC coefficients of original image are mostly set to zero, resulting in the great loss of high frequency information of image and obvious image blocked effect, so uneven jagged phenomenon of three-channel-histogram of encrypted image appeared . When Q is high, the quantization degree is low with less high frequency information of the image losing and rich detail, so the three-channel-histogram of encrypted image appear smooth. Due to the random encryption leads to the even distribution of pixel value, the histogram of encryption images also appears even.

184



(a)R channel

(b)G channel

(c)B channel

Figure 10. Histogram of Encrypted Lake; Lines from First to Fourth are Original Image, Q=10,Q=50 ,Q=90 4.4 Encryption Performance In order to test the influence of the encryption algorithm proposed on image transmission rate, we define the percentage of transmission data(PTD) to analyze the effect of compression coding in the encryption algorithm on transmission data volume, PTD is the ratio of the transmission encryption code stream bits number to the original image bits number: PTD 

Length ( D DC  D AC ) 10 ( X  Y )  Num  8

(23) Where DDC, DAC are code stream of DC coefficients and and AC coefficients encoded and encrypted, measured in unit bits, function Length calculate the bit length, X and Y is the size of original picture , Num is the number of the color image channel. Lower PTD indicates less transmission data volume and higher transmission rate of encrypted images. The number of encryption data can measure the performance of a partial encryption algorithm, we define the percentage of encryption data(PED) as the test standard of encryption data volume, and the PED is the ratio of the encrypted QDCT coefficient quantity to original QDCT coefficient quantity:


185


PED 

Numcoef _ enc 100% Numcoef

(24) Numcoef_enc is the quantity of encrypted QDCT coefficients, Numcoef is the quantity of original QDCT coefficients, due to the degradation degree parameter m in the experiment is m = max {|p(i, j)|}, so the DC coefficients are all encrypted, then the PED computational formula updates to: X Y  Num AC 0 PED  64 100% X Y

(25) Where the NumAC≠0 is non-zero AC coefficient number, the encryption data volume is less when the PED is lower. From the definition and computational formula of the PTD and PED, we can see that the value of PTD and PED are related to Q, Figure 11 shows the trend of PTD and PED of Lenna, Baboon, Lake, Pepper while Q changes at 5 intervals ranging from 5 to 95, and the abscissa is Q, the ordinate is the percentage. The PTD and PED of the four pictures are direct proportional to Q, and that when Q < 70,the increase of PTD and PED is even and slow, when Q >70,the increase is rapid, this is because Q decides the quantification degree, being inverse with quantification degree ,when the quantization degree is higher, most of the QDCT AC coefficients are set to zero, but the encryption algorithm only encrypt non-zero AC coefficients whose encryption code stream are composed of the final encryption data, so the encryption and transmission data volume will decrease accordingly. On the other hand, when the quantization degree decreases, the encryption and transmission data volume will decrease, therefore the trend of PTD and PED indicate that Q = 70 is a critical point, so various performance need to be considered and balanced when we choose Q. Under the condition of Q = 50, the PTD of four test image are all less than 20% when the PED are all less than 30%, it is clear that the encryption algorithm proposed greatly reduces the amount of transmission and encryption data.

Figure 11. Transmission and Encryption Data Volume Rate of Different Images

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4.5 Key Space Analysis The encryption algorithm proposed involves three kinds of key, that is, four Logistic chaos initial values x0r , x0i , x0j , x0k , four random integers dr , di , dj , dk , and four random binary numbers cr , ci , cj , ck , so the key number reaches twelve. Logistic chaos initial value x0 is a real number in the range of (0,1), its precision is 32 digits after the decimal point, and the key space of x0 reaches (1032)4; the range of the random integer d is (0,2m) ; while the degradation degree parameter m is an integer within three digits, the key space of d is (104)4; the digits M of the binary random number c took in the experiment of this paper is 16, the key space of c is (216)4 , of course, the value of M can expand according to demand, thus expanding the key space. So the key space of the algorithm in this paper is (1032)4 ·(104)4 ·(216)4≈2544, that is, the key space reaches to 544 digits, it can resist exhaustive attack. 4.6 Comparison Analysis In the image partial encryption algorithm proposed by Kuppusamy[27], the PSNR of encryption image value is 8.417dB, the key space is 256 digits; in the algorithm proposed by Nidhi[28], the PSNR of encryption image of eight test images are all lower than 11dB, and the key space is 43 digits; but in the algorithm we proposed, the PSNR of encrypted image of test images Baboon, Lake, Pepper are all lower than 8dB when Q = 50, the PSNR of encrypted image of Lenna is lower than 10.2dB, and the key space is 544 digits, by comparison, the encryption algorithm we proposed in this paper has better encryption effect.

5. Conclusion In this paper, we deduced the QDCT coefficient quantitation table for color image according to the relationship of QDCT and DCT, and then brought out QDCT coefficient coding method based on JPEG coding, at last, combining QDCT and compression coding, partial encryption algorithm for color image based on QDCT is proposed, realized the security encryption of color image. The coding method reduced the transmission data volume of encryption image, the transmission speed is greatly improved. The simulation results show that the algorithm can get good effect on encryption and decryption, provides a new method and new thought for quaternion applied on the field of color image partial encryption.

Acknowledgements This work was jointly supported by the National Natural Science Foundation of China (Grant No. 61272421, 61232016, 61103141, 61173141, 61300237, 61402234, 61402235), the Priority Academic Program Development of Jiangsu Higher Education Institutions, Jiangsu Government Scholarship for Overseas Studies and CICAEET.

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Authors Jing Fan, (1989-), female, graduate student, mainly engaged in digital image processing and information security.

Jinwei Wang, (1978-), male, Ph.D., Associate professor, mainly engaged in multimedia watermarking, multimedia encryption and multimedia forensics.

Xingming Sun, (1963-), male, Ph.D., Professor, main engaged in areas such as network and information safety, sensor networks, and automatic weather observation.

Ting Li, (1990-), male, graduate student, mainly engaged in digital image processing and multimedia forensics.


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