# A Secure Image Encryption Algorithm Based on Rubik's Cube Principle

**Abstract**

In the past few years, several encryption algorithms based on chaotic systems have been proposed
as means to protect digital images against cryptographic attacks. These encryption algorithms typically
uses relatively small key spaces and thus offers limited security, especially if they are one dimensional.
In this paper, we proposed a novel image encryption algorithm based on Rubik’s cube principle. The
original image is scrambled using the principle of Rubik’s cube. Then, XOR operator is applied to rows
and columns of the scrambled image using two secret keys. Finally, the experimental results and security
analysis show that the proposed image encryption scheme not only can achieve good encryption and
perfect hiding ability and resist to exhaustive attack, statistical attack and differential attack.

Hindawi Publishing Corporation

Journal of Electrical and Computer Engineering

Volume 2012, Article ID 173931, 13 pages

doi:10.1155/2012/173931

Research Article

A Secure Image Encryption Algorithm Based on

Rubik’s Cube Principle

Khaled Loukhaoukha, Jean-Yves Chouinard, and Abdellah Berdai

Department of Electrical and Computer Engineering, Laval University, QC, Canada G1K 7P4

Correspondence should be addressed to Khaled Loukhaoukha, khaled.loukhaoukha.1@ulaval.ca

Received 22 August 2011; Accepted 15 November 2011

Academic Editor: Fouad Kheliﬁ

Copyright © 2012 Khaled Loukhaoukha et al. This is an open access article distributed under the Creative Commons Attribution

License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly

cited.

In the past few years, several encryption algorithms based on chaotic systems have been proposed as means to protect digital images

against cryptographic attacks. These encryption algorithms typically use relatively small key spaces and thus oﬀer limited security,

especially if they are one-dimensional. In this paper, we proposed a novel image encryption algorithm based on Rubik’s cube

principle. The original image is scrambled using the principle of Rubik’s cube. Then, XOR operator is applied to rows and columns

of the scrambled image using two secret keys. Finally, the experimental results and security analysis show that the proposed image

encryption scheme not only can achieve good encryption and perfect hiding ability but also can resist exhaustive attack, statistical

attack, and diﬀerential attack.

1. Introduction

The end of the 20th century was marked by an extraordinary

technical revolution from analog to numerical as docu-

ments and equipments became increasingly used in various

domains. However, the advantages of the digital revolution

were not achieved without drawbacks such as illegal copying

and distribution of digital multimedia documents. To meet

this challenge, researchers were motivated more than ever

to protect multimedia documents with new and eﬃcient

document protection techniques. In this context, diﬀerent

techniques have been introduced such as encryption and

digital watermarking. The ﬁrst one consists in transforming

multimedia documents using an algorithm to make it

unreadable to anyone except for the legitimate users. The

second one consists of embedding digital watermarks into

multimedia documents to guarantee the ownership and the

integrity of the digital multimedia contents.

Theprotectionofimagesisofparticularinterestin

this paper. Traditional image encryption algorithms such as

private key encryption standards (DES and AES), public key

standards such as Rivest Shamir Adleman (RSA), and the

family of elliptic-curve-based encryption (ECC), as well as

the international data encryption algorithm (IDEA), may

not be the most desirable candidates for image encryption,

especially for fast and real-time communication applications.

In recent years, several encryption schemes have been

proposed [1–12]. These encryption schemes can be classiﬁed

into diﬀerent categories such as value transformation [1–4],

pixels position permutation [5–8], and chaotic systems [9–

12].

In the ﬁrst category, Liu et al. [1]presentedan

image encryption scheme based on iterative random phase

encoding in gyrator transform domains. A two-dimensional

chaotic mapping is employed to generate many random

data for iterative random phase encoding. In [2],acolor

image encryption method using discrete fractional random

transform (DFRNT) and the Arnold transform (AT) in

the intensity-hue-saturation (IHS) color space has been

proposed. Each color space component is then encrypted

independently with diﬀerent approaches. In [3], an image

encryption algorithm based on Arnold transform and

gyrator transform has been proposed. The amplitude and

phase of the gyrator transform are separated into several

subimages, which are scrambled using the Arnold transform.

The parameters of gyrator transforms and separating scheme

serve as the key of the encryption method. Tao et al. [4]

proposed an image encryption algorithm based on fractional

2 Journal of Electrical and Computer Engineering

Fourier transform (FRFT) and which can be applied to

the double or more image encryptions. The encrypted

image is obtained by the summation of diﬀerent orders of

inverse discrete fractional Fourier transform (IDFRFT) of the

interpolated subimages. The whole transform orders of the

utilized FRFT are used as the secret keys for the decryption

of each subimage.

In the second category, Zunino [5] use Peano-Hilbert

curves as pixels position permutation to destroy the spatial

autocorrelation of an image. Zhang and Liu [6] proposed

image encryption scheme based on permutation-diﬀusion

architecture and skew tent map system. In the proposed

scheme, the P-box is chosen as the same size of original

image, which shuﬄes the positions of pixels totally. To

enhance the security, the keystream in the diﬀusion step

depends on both the key and original image. Zhao and

Chen [7] proposed to used ergodic matrices for scrambling

and encryption of digital images. The authors analyzed

the isomorphism relationship between ergodic matrices

and permutation. Zhu et al. [8] proposed an innovative

permutation method to confuse and diﬀuse the gray-scale

image at the bitlevel, which changes the position of the pixel

and modiﬁes its value. This algorithm uses also the Arnold

cat map to permute the bits and the logistic map to further

encrypt the permuted image.

In the third category, Huang and Nien [9] proposed

a novel pixel shuﬄing method for color image encryption

which used chaotic sequences generated by chaotic systems

as encryption codes. In [10], the two-dimensional chaotic

cat map has been generalized to three-dimensional one

and then was used to design a fast and secure symmetric

image encryption scheme. This scheme employs the 3D

cat map to shuﬄe the positions and the values of image

pixels. Wang et al. [11] presented an image encryption

algorithm based on simple Perceptron and using a high-

dimensional chaotic system in order to produce three sets

of pseudorandom sequence. Then to generate weight of

each neuron of perceptron as well as a set of input signal,

a nonlinear strategy is adopted. Recently, a new image

encryption algorithm combining permutation and diﬀusion

was proposed by Wang et al. [12]. The original image is

partitioned into blocks and a spatiotemporal chaotic system

is then employed to generate the pseudorandom sequence

used for diﬀusing and shuﬄing these blocks.

Thesecurityofimageencryptionhasbeenextensively

studied. Almost some encryption schemes based on per-

mutation had already been found insecure against the

ciphertext-only and known/chosen-plaintext attacks, due to

the high information redundancy, and it is quite under-

standable since the secret permutations can be recovered

by comparing the plaintexts and the permuted ciphertexts.

Generally, chaos-based image encryption algorithms are

used more often than others but require high computational

cost. Moreover, a chaos system is deﬁned on real numbers

while the cryptosystems are deﬁned on ﬁnite sets of integers.

One-dimensional chaotic cryptosystems are limited by their

small key spaces and weak security in [1, 13].

In this paper, we present a novel image encryption

algorithm based on the principle of Rubik’s cube. First-, in

order to scramble the pixels of gray-scale original image the

principle of Rubik’s cube is deployed which only changes the

position of the pixels. Using two random secret keys, the

bitwise XOR is applied into the odd rows and columns. Then,

the bitwise XOR is also applied to even rows and columns

using the ﬂipped secret keys. These steps can be repeated

while the number of iteration is not reached. Numerical

simulation has been performed to test the validity and the

security of the proposed encryption algorithm.

The remaining of this paper is organized as follows.

Section 2 describes the proposed image encryption algorithm

based on Rubik’s cube principle. Experimental results and

security analysis are presented in Section 3. Finally, we

conclude in Section 4.

2. Rubik’s Cube Image Encryption

In this section, the proposed encryption algorithm based on

Rubik’s cube principle is described along with the decryption

algorithm.

2.1. Rubik’s Cube Based Encryption Algorithm. Let I

o

repre-

sent an α-bit gray scale image of the size M

× N.Here,I

o

representthepixelsvaluesmatrixofimageI

o

. The steps of

encryption algorithm are as follows:

(1) Generate randomly two vectors K

R

and K

C

of length

M and N, respectively. Element K

R

(i)andK

C

( j)Each

take a random value of the set A

={0, 1, 2, ...,2

α

−

1}. Note that both K

R

and K

C

must not have constant

values.

(2) Determine the number of iterations, ITER

max

,and

initialize the counter ITER at 0.

(3) Increment the counter by one: ITER

= ITER + 1.

(4) For each row i of image I

o

,

(a) compute the sum of all elements in the row i,

this sum is denoted by α(i)

α

(

i

)

=

N

j=1

I

o

i, j

, i = 1, 2, ..., M,

(1)

(b)computemodulo2ofα(i), denoted by M

α(i)

,

(c) row i is left, or right, circular-shifted by K

R

(i)

positions (image pixels are moved K

R

(i)posi-

tions to the left or right direction, and the ﬁrst

pixel moves in last pixel.), according to the

following:

if M

α(i)

= 0 −→ right circular shift

else

−→ left circular shift.

(2)

(5) For each column j of image I

o

,

Journal of Electrical and Computer Engineering 3

(a) compute the sum of all elements in the column

j, this sum is denoted by β( j),

β

j

=

M

i=1

I

o

i, j

, j = 1, 2, ..., N,

(3)

(b)computemodulo2ofβ( j), denoted by M

β( j)

.

(c) column j is down, or up, circular-shifted by

K

C

(i) positions, according to the following:

if M

β( j)

= 0 −→ up circular shift

else

−→ down circular shift.

(4)

Steps 4 and 5 above will create a scrambled image,

denoted by I

SCR

.

(6) Using vector K

C

, the bitwise XOR operator is applied

toeachrowofscrambledimageI

SCR

using the

following expressions:

I

1

2i − 1, j

=

I

SCR

2i − 1, j

⊕ K

C

j

,

I

1

2i, j

=

I

SCR

2i, j

⊕ rot 180

K

C

j

,

(5)

where

⊕ and rot 180(K

C

) represent the bitwise XOR operator

and the ﬂipping of vector K

C

from left to right, respectively.

(7) Using vector K

R

, the bitwise XOR operator is applied

to each column of image I

1

using the following

formulas:

I

ENC

i,2j − 1

= I

1

i,2j − 1

⊕ K

R

j

,

I

ENC

i,2j

=

I

1

i,2j

⊕

rot 180

K

R

j

.

(6)

with rot 180(K

R

) indicating the left to right ﬂip of vector K

R

.

(8) If ITER

= ITER

max

, then encrypted image I

ENC

is

created and encryption process is done; otherwise,

the algorithm branches to step 3.

Ve c t or s K

R

, K

C

and the max iteration number ITER

max

are considered as secret keys in the proposed encryption

algorithm. However, to obtain a fast encryption algorithm it

is preferable to set ITER

max

= 1 (single iteration). Converse-

ly, if ITER

MAX

> 1, then the algorithm is more secure because

the key space is larger than for ITER

MAX

= 1. Nevertheless,

in the simulations presented in Section 3, the number of

iterations ITER

max

was set to one.

2.2. Rubik’s Cube Dec ryption Algorithm. The decrypted im-

age, I

o

, is recovered from the encrypted image, I

ENC

, and the

secret keys, K

R

, K

C

,andITER

max

as follows in the following.

(1) Initialize ITER

= 0.

(2) Increment the counter by one: ITER

= ITER + 1.

(3) The bitwise XOR operation is applied on vector K

R

andeachcolumnoftheencryptedimageI

ENC

as

follows:

I

1

i,2j − 1

=

I

ENC

i,2j − 1

⊕

K

R

j

,

I

1

i,2j

=

I

ENC

i,2j

⊕

rot 180

K

R

j

,

(7)

(4) Then, using the K

C

vector, the bitwise XOR operator

is applied to each row of image I

1

:

I

SCR

2i − 1, j

=

I

1

2i − 1, j

⊕

K

C

j

,

I

SCR

2i, j

=

I

1

2i, j

⊕

rot 180

K

C

j

.

(8)

(5) For each column j of the scrambled image I

SCR

,

(a) compute the sum of all elements in that column

j,denotedasβ

SCR

( j):

β

SCR

j

=

M

i=1

I

SCR

i, j

, j = 1, 2, ..., N,(9)

(b)computemodulo2ofβ

SCR

( j), denoted by

M

β

SCR

( j)

,

(c) column j is down, or up, circular-shifted by

K

C

(i) positions according to the following:

if M

β

SCR

( j)

= 0 −→ up circular shift

else

−→ down circular shift.

(10)

(6) For each row i of scrambled image I

SCR

,

(a) compute the sum of all elements in row i, this

sum is denoted by α

SCR

(i):

α

SCR

(

i

)

=

N

j=1

I

SCR

i, j

, i = 1, 2, ..., M,

(11)

(b) compute modulo 2 of α

SCR

( j), denoted by

M

α

SCR

( j)

,

(c) row i is then left, or right, circular-shifted by

K

R

(i) according to the following:

if M

α

SCR

( j)

= 0 −→ right circular shift

else

−→ left circular shift.

(12)

(7) If ITER

= ITER

max

, then image I

ENC

is decrypted

and the decryption process is done; otherwise, the

algorithm branches back to step 2.

4 Journal of Electrical and Computer Engineering

Table 1: Diﬀerence measures between original and encrypted

images.

Image NPCR (in %) UACI (in %)

Lena 99.5850 28.6210

Black 99.6078 50.1931

Baboon 99.6094 27.4092

Checkerboard 99.6201 50.0233

3. Experimental Results

In this section, we present the tests that were conducted

to assess the eﬃciency and security of the proposed image

encryption algorithm. These tests involve visual testing and

security analysis.

3.1. Visual Testing. For visual testing, four gray-scale images

of size 256

× 256 pixels were used. Figure 1 depicts these

test images—lena, black, baboon and checkerboard—

as well as the images encrypted using the proposed Rubik’s

cube algorithm. From this ﬁgure, one can see that there is

no perceptual similarity between original images and their

encrypted counterparts.

The encrypted image should greatly diﬀer from its

original form. In general, two diﬀerence measures are used

to quantify this requirement. The ﬁrst measure is the number

of pixels change rate (NPCR), which indicate the percentage

of diﬀerent pixels between two images. The second one is the

uniﬁed average changing intensity (UACI), which measures

the average intensity of diﬀerences in pixels between two

images [10]. Let I

o

(i, j)andI

ENC

(i, j) be the pixels values of

original and encrypted images, I

o

and I

ENC

, at the ith pixel

row and jth pixel column, respectively. Equations (13)and

(15) give the mathematical expressions of the NPCR and

UACI measures:

NPCR

=

M

i=1

N

j=1

D

i, j

M × N

× 100%,

(13)

with : D

i, j

=

⎧

⎨

⎩

0ifI

o

i, j

= I

ENC

i, j

,

1 otherwise.

(14)

UACI

=

⎡

⎣

M

i=1

N

j=1

I

o

i, j

−

I

ENC

i, j

255

⎤

⎦

×

100%

M × N

.

(15)

To approach the performances of an ideal image encryp-

tion algorithm, NPCR values must be as large as possible and

UACI values must be around 33%. Table 1 gives the NPCR

and UACI values for the original images and their encrypted

versions. The values are very close to unity for the NPCR

measure. The UACI values are also appropriate. The high

percentage values of the NPCR measure indicate that the

pixels positions have been randomly changed.

Furthermore, the UACI values show that almost all pixel

gray-scale values of encrypted image have been changed from

their values in the original images, making the original and

encrypted image pixels more diﬃcult to discriminate. It iss

also observed that the UACI values for the case of the black

Table 2: Number of pixel change rate (NPCR) between images

encrypted with keys K

1

and K

2

(with 1 bit diﬀerence).

Image Mean (%) Standard deviation (%)

Lena 99.6111 0.0251

Black 99.6064 0.0238

Baboon 99.6085 0.0253

Checkerboard 99.6113 0.0274

and the checkerboard images are larger than those of the

other images. This is due to the fact that all the pixels values

of the black and checkerboard images are at the extremity

of pixels values range, that is, 0 for the black pixels and 255

for the white pixels, leading to a large absolute diﬀerence

between the original and encrypted images.

3.2. Security Analysis. Security is a major issue in cryptology.

A good image encryption scheme should resist various

attacks such as known plain text attack, cipher-text-only

attack, statistical analysis attack, and brute-force attacks.

In this section, a security analysis on the proposed image

encryption algorithm is done. The security assessment has

been done on key space analysis and statistical analysis.

3.2.1. Key Space Analysis. A secure image encryption scheme

must have a large key space in order to make brute-force

attack practically (computationally) infeasible. In theory, the

proposed algorithm can accommodate an inﬁnite key space.

However, the encryption key used in our scheme is composed

of the (K

R

, K

C

,ITER

max

) triplet. For an α-bit gray-scale image

I

o

of size M × N pixels, the vectors K

R

and K

C

can take

2

M·α

and 2

N·α

possible values, respectively. If we consider

that both vectors must not have constant values, and the key

spacesizeis2

α·(M+N)

× ITER

max

− 2

2α

keys, one can see that

the size of the key space can be expanded when the number

of iteration ITER

max

is increased. For instance, for an 8-bits,

scale gray image of size 256

× 256 pixels and ITER

max

= 1.

Thekeyspacesizeisequalto2

4096

− 2

16

≈ 10

1233

; this key

space is large enough to resist exhaustive attack and it is larger

than the key space size of the image encryption algorithms

proposed in [1, 9, 10, 14–16].

3.2.2. Key Sensibility. Encryption algorithms should also

have high sensibility to encryption key: this means that any

small change in the key should lead to a signiﬁcant change in

the encrypted, or decrypted, image. We performed two tests

to illustrate the key sensibility of our scheme. The ﬁrst one

shows the impact of a key change in the image encryption

process. Here, the original image, I

o

, is encrypted using the

key K

1

= (K

R

, K

C

,ITER

max

), where K

R

, K

C

,andITER

MAX

are randomly generated. Then, the same image, that is, I

o

,

is encrypted using another key K

2

which diﬀers only from

the ﬁrst key, K

1

, in the least signiﬁcant bit, that is, K

2

=

(K

R

, K

C

,ITER

max

+ 1). This experiment is repeated 100 times

using diﬀerent key pairs K

1

and K

2

(still only diﬀering by

the least signiﬁcant bit). Table 2 represents the mean and

the standard deviation of the NPCR values between the

Journal of Electrical and Computer Engineering 5

(a) Lena (original) (b) Lena (encrypted)

(c) Black (original) (d) Black (encrypted)

(e) Baboon (original) (f) Baboon (encrypted)

(g) Checkerboard (original) (h) Checkerboard (encrypted)

Figure 1: Original and encrypted images.

6 Journal of Electrical and Computer Engineering

(a) Original image: Lena

(b) Encrypted image (a)

with key K

1

(c) Encrypted image (a)

with key K

2

(d) Image diﬀerence be-

tween (b) and (c)

(e) Original image: Black (f) Encrypted image (e)

with key K

1

(g) Encrypted image (e)

with key K

2

(h) Image: diﬀerence be-

tween (f) and (g)

(i) Original image:

Baboon

(j) Encrypted image (i)

with key K

1

(k) Encrypted image (i)

with key K

2

(l) Image diﬀerence be-

tween (j) and (k)

(m) Original image:

Checkerboard

(n) Encrypted image (m)

with key K

1

(o) Encrypted image (m)

with key K

2

(p) Image: diﬀerence be-

tween (n) and (o)

Figure 2: Key sensibility for image encryption using the proposed algorithm.

encrypted image with key K

1

and the encrypted image using

the key K

2

using 100 diﬀerent key pairs. One can see from

Ta bl e 2 that the mean values of NPCR are close to 100%,

which means that image encrypted by the key K

1

diﬀers

signiﬁcantly from the one encrypted by key K

2

.Moreover,

standard deviation values are very small: this indicates that

the NPCR values are clustered closely around the mean.

Figure 2 represents the original images, their encrypted

images using two diﬀerent keys K

1

and K

2

and the image

diﬀerence between the encrypted images, respectively. As

mentioned earlier, keys K

1

and K

2

diﬀer only by one bit.

The second test consists of measuring the key sensibility

in the image decryption process. Let original image I

o

be

encrypted using the key K

1

= (K

R

, K

C

,ITER

MAX

), where K

R

,

K

C

,andITER

MAX

are again randomly generated, to give the

encrypted image I

ENC

. This image is decrypted separately

using the keys K

1

and K

2

; these keys always diﬀer by only

one bit in the least signiﬁcant bit location. Figure 3 illustrates

the original image, the encrypted image I

ENC

with key K

1

,

the decrypted image of I

ENC

using correct key K

1

, and the

decrypted image of I

ENC

using the wrong key K

2

.Itisclear

from this ﬁgure that decryption using a wrong key does not

succeed.

3.3. Statistical Analysis. In a paper published in 1949 [17],

Shannon stated that “It is possible to solve many kinds of

ciphers by statistical analysis.” Consequently, he suggested

two methods based on confusion and diﬀusion in order to

counteract powerful attacks based on statistical analysis. In

Journal of Electrical and Computer Engineering 7

(a) Original image: Lena (b) Encrypted image (a) with

key K

1

(c) Decrypted image (b) using

correct key K

1

(d) Decrypted image (b) using

wrong key K

2

(e) Original image: Black (f) Encrypted image (e) with

key K

1

(g) Decrypted image (f) using

correct key K

1

(h) Decrypted image (f) using

wrong key K

2

(i) Original image: Baboon (j) Encrypted image (i) with

key K

1

(k) Decrypted image (j) using

correct key K

1

(l) Decrypted image (j) using

wrong key K

2

(m) Original image:

Checkerboard

(n) Encrypted image (m) with

key K

1

(o) Decrypted image (n) using

correct key K

1

(p) Decrypted image (n) using

wrong key K

2

Figure 3: Key sensibility for image decryption using the proposed algorithm.

the present paper, statistical analysis has been performed to

demonstrate the superior confusion and diﬀusion properties

of the proposed encryption algorithm against statistical

attacks. This is done by performing two series of tests:

histograms analysis of the encrypted images and the correla-

tions computation of the adjacent pixels in encrypted images.

Figure 4 represents the histograms of the original and

the encrypted images illustrated previously in Figure 1.One

can see those the histograms of the encrypted images are

almost uniform and are signiﬁcantly diﬀerent from that of

the four original images. For instance, the histogram of

original image Checkerboard shows as expected only two

values: 0 and 255; however, the histogram of the encrypted

Checkerboard image is fairly uniform. Therefore, the

proposed image encryption algorithm responds well to the

diﬀusion properties: it does not provide information that can

be exploited for attacks based on statistical analysis of the

encrypted image.

The other statistical test consists of computing the

correlation between adjacent pixels [10]. It is obvious

8 Journal of Electrical and Computer Engineering

20 60 100 140 180 220 255

0

100

200

300

400

500

600

700

800

Number of pixels

(a) Original image: Lena

20 60 100 140 180 220 255

0

100

200

300

400

500

600

700

800

Number of pixels

(b) Encrypted image: Lena

0

1

2

3

4

5

6

×10

4

Number of pixels

20 60 100 140 180 220 255

(c) Original image: Black

20 60 100 140 180 220 255

0

100

200

300

400

500

600

700

800

Number of pixels

(d) Encrypted image: Black

0

100

200

300

400

500

600

700

800

900

Number of pixels

20 60 100 140 180 220 255

(e) Original image: Baboon

0

100

200

300

400

500

600

700

800

900

Number of pixels

20 60 100 140 180 220 255

(f) Encrypted image: Baboon

×10

4

Number of pixels

Grayscale

20 60 100 140 180 220

0

1

1.5

2

2.5

3

0.5

(g) Original image: Checkerboard

Grayscale

20 60 100 140 180 220 255

0

100

200

300

400

500

600

700

800

Number of pixels

(h) Encrypted image: Checkerboard

Figure 4: Histograms of original and encrypted images.

Journal of Electrical and Computer Engineering 9

0

20

40

60

80

100

120

140

160

180

200

220

240

255

Pixel gray value on location (x +1,y)

Correlation coeﬃcient = 0.9763

20 60 100 140 180 220 255

(a) Image: Lena

Correlation coeﬃcient = 0.0068

0

20

40

60

80

100

120

140

160

180

200

220

240

255

Pixel gray value on location (x +1,y)

20 60 100 140 180 220 255

(b) Encrypted image: Lena

Correlation coeﬃcient = 1

0

20

40

60

80

100

120

140

160

180

200

220

240

255

Pixel gray value on location (x +1,y)

20 60 100 140 180 220 255

(c) Image: Black

Correlation coeﬃcient = 0.0046

0

20

40

60

80

100

120

140

160

180

200

220

240

255

Pixel gray value on location (x +1,y)

20 60 100 140 180 220 255

(d) Encrypted image: Black

Correlation coeﬃcient = 0.8567

20 60 100 140 180 220 255

0

20

40

60

80

100

120

140

160

180

200

220

240

255

Pixel gray value on location (x +1,y)

(e) Image: Baboon

Correlation coeﬃcient = 0.0055

0

20

40

60

80

100

120

140

160

180

200

220

240

255

Pixel gray value on location (x +1,y)

20 60 100 140 180 220 255

(f) Encrypted image: Baboon

Correlation coeﬃcient = 0.9578

0

20

40

60

80

100

120

140

160

180

200

220

240

255

Pixel gray value on location (x +1,y)

Pixel gray value on location (x, y)

20 60 100 140 180 220 255

(g) Image: Checkerboard

Correlation coeﬃcient = 0.0039

0

20

40

60

80

100

120

140

160

180

200

220

240

255

Pixel gray value on location (x +1,y)

20 60 100 140 180 220 255

Pixel gray value on location (x, y)

(h) Encrypted image: Checkerboard

Figure 5: Correlation distribution of the pairs horizontal to adjacent pixels.

10 Journal of Electrical and Computer Engineering

(a) Decrypted image: Lena (b) Decrypted image: Black (c) Decrypted image: Baboon (d) Decrypted image:

Checkerboard

Figure 6: Attack by salt & pepper noise.

(a) Decrypted image: Lena (b) Decrypted image: Black (c) Decrypted image: Baboon (d) Decrypted image:

Checkerboard

Figure 7: Attack by speckle noise.

that an arbitrarily chosen pixel in an image is generally

strongly correlated with adjacent pixels, and its in either

horizontal, vertical or diagonal directions. However, a secure

image encryption algorithm must produce an encrypted

image having low correlation between adjacent pixels. This

correlation test consists of randomly selecting N pairs of

adjacent pixels (vertical, horizontal, and diagonal) from the

original and the encrypted images separately. Then, the

correlation coeﬃcientofeachpairiscalculatedusing(19)

E

(

x

)

=

1

N

N

i=1

x

i

, (16)

D

(

x

)

=

1

N

N

i=1

(

x

i

− E

(

x

))

2

, (17)

cov

x, y

=

1

N

N

i=1

(

x

i

− E

(

x

))

y

i

− E

y

, (18)

γ

xy

=

cov

x, y

D

(

x

)

D

y

with D

(

x

)

/

= 0, D

y

/

= 0,

(19)

where x

i

and y

i

are the grayscale values of two adjacent pixels,

N is the number of pairs (x

i

, y

i

), and E(x)andE(y), are

respectively, the mean values of x

i

and y

i

.

Ta bl e 3 gives the correlation coeﬃcient values of adjacent

pixels in the horizontal, vertical, and diagonal directions of

the original images and their encrypted versions. It is clear

Table 3: Correlation coeﬃcients between adjacent pairs of pixels

for original and encrypted images.

Correlation Horizontal Vertical Diagonal

Original image: lena 0.9763 0.9453 0.9282

Encrypted image: lena 0.0068 0.0091 0.0063

Original image: black 1.0000 1.0000 1.0000

Encrypted image: black 0.0046 0.0055 0.0056

Original image: baboon 0.8567 0.8772 0.7880

Encrypted image: baboon 0.0055 0.0078 0.0042

Original image: checkerboard 0.9578 0.9521 0.9118

Encrypted image: checkerboard 0.0039 0.0090 0.0045

that for the original images, the coeﬃcient correlation values

are very high (close to one) contrary to those observed for

the encrypted images. This conﬁrms that adjacent pixels in

the original images are strongly correlated. However, for the

encrypted images, those values are close to zero, which means

that the adjacent pixels (horizontal, vertical and diagonal

directions) are very weakly correlated. Figure 5 illustrates the

correlation distributions of the horizontal adjacent pixels of

the original images and the corresponding encrypted images

using the proposed algorithm. One can see from Figure 5 that

adjacent pixels in encrypted images are indeed very weakly

correlated.

3.4. Entropy Analysis. The concept of entropy analysis for

image encryption algorithm was introduced by Edward [18].

Journal of Electrical and Computer Engineering 11

(a) Encrypted lena image (b) Encrypted black image (c) Encrypted baboon image (d) Encrypted checkerboard

image

(e) Image (a) decrypted (f) Image (b) decrypted (g) Image (c) decrypted (h) Image (d) decrypted

Figure 8: Encrypted images under cropping attack in center (cropping over 1/8 of the entire image area).

Table 4: Comparison of entropy values of original images and their

encrypted version.

Image Original Encrypted

Lena 7.4318 Sh 7.9968 Sh

Black 0 Sh 7.9966 Sh

Baboon 7.2279 Sh 7.9974 Sh

Checkerboard 1 Sh 7.9972 Sh

Table 5: Comparison of entropy values of encrypted images under

diﬀerent image encryption algorithms.

Algorithm Entropy (Sh)

Proposed algorithm 7.9968

Baptista [19] 7.9260

Wong et al . [ 20] 7.9690

Xiang et al. [21] 7.9950

Lin and Wang [22] 7.9890

For gray-scale images of 256 levels, if each level of gray

is assumed to be equiprobable, then the entropy of this

image will be theoretically equal to 8 Sh (or bits). Ideally,

an algorithm for encryption of images should give an

encrypted image having equiprobable gray levels. Table 4

gives the entropy values of the four original images and those

of their encrypted versions.

From those entropy values, we note that the entropy

values of original images are far from ideal value of entropy

since information sources are highly redundant and thus

rarely generate uniformly distributed random messages. On

the other hand, the entropy values of the encrypted images

are very close to the ideal value of 8 Sh, which means that

the proposed encryption algorithm is highly robust against

entropy attacks.

Ta bl e 5 gives a comparison of the entropy values for

encrypted image Lena with various image encryption algo-

rithms.

3.5. Analysis against Attacks. An attacker who intercepts

encrypted image can easily modify it, while the legitimate

user can receive it and decrypt it successfully. This is the

principle of attacks in image encryption; these attacks can

include additive noise, ﬁltering, rotation and cropping, and

so forth.

3.5.1. Additive Noise. To verify the performance of the

proposed encrypted algorithm against additive noise attacks,

we considered two types of noises: salt and pepper noise and

speckle noise. An additive noise attack consists in adding

random noise to the intercepted encrypted image. Then,

the noisy encrypted image will be decrypted. To measure

the robustness of the proposed image encryption algorithm

against this attack, mean squared error (MSE) measures are

used.

Ta bl e 6 gives the MSE values between original images

and their decrypted ones under the salt and pepper noise

with diﬀerent noise density values and the speckle noise with

diﬀerent variances.

Figures 6 and 7 illustrate the decrypted images: their

encrypted version has been attacked separately by salt &

pepper noise with 0.05 density and by speckle noise of

variance 0.05. From these results, we can conclude that

random noise attacks seriously aﬀect decrypted images.

12 Journal of Electrical and Computer Engineering

Table 6: MSE between original images and the decrypted versions under diﬀerent noise attacks.

Image

Salt and pepper noise Speckle noise

Density MSE Variance MSE

Lena

0.05 3.42

× 10

3

0.05 5.30 × 10

3

0.1 3.68 × 10

3

0.1 5.79 × 10

3

Black

0.05 1.06

× 10

3

0.05 8.11 × 10

3

0.1 2.20 × 10

3

0.1 1.02 × 10

4

Baboon

0.05 2.38

× 10

3

0.05 3.97 × 10

3

0.1 2.49 × 10

3

0.1 4.50 × 10

3

Checkerboard

0.05 2.44

× 10

4

0.05 2.12 × 10

4

0.1 2.50 × 10

4

0.1 2.19 × 10

4

Table 7: MSE between original images and the decrypted versions

under cropping attack.

Image Center cropping Sides cropping

Lena 2.96 × 10

3

2.90 × 10

3

Black 2.72 × 10

3

2.69 × 10

3

Baboon 2.05 × 10

3

2.02 × 10

3

Checkerboard 1.81 × 10

4

1.64 × 10

4

Table 8: Speed test results of the proposed algorithm with image

Lena using a 2.7 GHz personal computer.

Image size Encryption time Decryption time

64 × 64 0.03 s 0.03 s

128

× 128 0.04 s 0.04 s

256

× 256 0.12 s 0.12 s

512

× 512 0.66 s 0.66 s

1024

× 1024 5.40 s 5.40 s

3.5.2. Analysis against Cropping Attacks. The cropping

attacks consist of modifying the intercepted encrypted image

by deleting one or several areas of the image. Table 7 gives

the MSE values between original images and the decrypted

and cropped images either in their center or on the image

sides with parameter values equal to 1/8. This one indicates

the fraction of the encrypted image that has been cropped.

Figure 8 represents the encrypted images cropped in the

center and their decrypted versions. From these results, we

can conclude that the proposed image encryption algorithm

resists to this attack lightly.

3.6. Speed Test. Apart from security considerations, another

important consideration in the design of image encryption

techniques is the actual algorithm execution speed, partic-

ularly for real-time applications. The proposed encryption

algorithm is indeed very fast compared to other algorithms

[10]. Our experimental results show that the average speed

for encryption and for decryption is of around 0.9 Mb/s

(megabits per second). The peak speed can reach up to

2.2 Mb/s on personnel computer equipped with an AMD

Athlon processor with clock speed of 2.70 GHz, 1 GB (giga-

bytes) of RAM memory and 160 GB hard-disk capacity.

Ta bl e 8 illustrates the performance of the proposed

algorithm using original image Lena with diﬀerent sizes

ranging from 64

× 64 to 1024 × 1024 pixels. The proposed

algorithm was written using the MATLAB software platform.

4. Conclusion

In this paper, a novel image encryption algorithm is pro-

posed. This algorithm is based on the principle of Rubik’s

cube to permute image pixels. To confuse the relationship

between original and encrypted images, the XOR operator is

applied to odd rows and columns of image using a key. The

same key is ﬂipped and applied to even rows and columns of

image. Experimental tests have been carried out with detailed

numerical analysis which demonstrates the robustness of the

proposed algorithm against several types of attacks such as

statistical and diﬀerential attacks (visual testing). Moreover,

performance assessment tests demonstrate that the proposed

image encryption algorithm is highly secure. It is also capable

of fast encryption/decryption which is suitable for real-time

Internet encryption and transmission applications.

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- CitationsCitations32
- ReferencesReferences23

- "Figure 4shows a Baker map randomization of an (8 x 8) matrix (M=8). Here the secret key is S= [2, 4, 2] . "

[Show abstract] [Hide abstract]**ABSTRACT:**In the field of digital and multimedia applications, more multimedia data are developed and transmitted in art, entertainment, advertising, commercial areas, education and training through the networks, which may have important information that should not be accessed by the unauthorized persons. Therefore issue of protecting the confidentiality, integrity, security, privacy as well as the authenticity of images has become an important issue for communication and storage of images. In this paper an efficient method for image encryption based on Rubik's cube principle with chaotic Baker map is presented. It consists of two layers. The first layer is a preprocessing layer used to improve the security of the system which is implemented with the chaotic baker map. In the second layer, the Rubik's cube principle is utilized. The original image is first converted in to baker mapped image and then Rubik's cube principle is applied. The proposed technique will enhance the security level of the Rubik's cube encryption technique.- "Table 12 gives the comparison of the execution time for the proposed method and existing methods. From the tabulated values, it is observed that the execution time of proposed method is less when compared with those methods suggested in [6,[27][28][29]. "

[Show abstract] [Hide abstract]**ABSTRACT:**Internet is an important part in the daily life of people in many ways, which allows the people in any corner of the world to share all types of information. Image encryption is used to transmit sensitive multimedia information in unsecured networks to provide high degree of confidentiality. In this paper, a new image encryption method is proposed based on novel implementation of pixel scan, utilizing the Knight's Travel Path, and true random number. The Knight's travel path is a pattern in which the path of a Knight around a chess board is taken without revisiting any node. This travel path pattern is used to permute the pixel positions of the original image to obtain the scrambled image. The scrambled image is further XORed with the random key numbers to get the cipher image. To change the pixel values of the scrambled image, true random numbers are generated from the amplitude values of a chosen noise audio file. Decryption is performed to confirm the reception of the sent image. The proposed method resists the statistical, differential, and entropy attacks significantly which have been demonstrated with various standard images.- "Traditional image encryption algorithms, for instance, private key encryption standards (DES and AES) faces problems when used to encrypt large images and therefore, its efficiency becomes low and weak, public key standards such as Rivest Shamir Adleman (RSA), and the family of elliptic-curve-based Encryption (ECC), as well as the international data encryption algorithm (IDEA) requires a great computational time and super computers when used in encrypting real-time images, may not be the most desirable candidates for image encryption, especially for fast and realtime communication applications because of Cryptographic algorithms that use less time are much more preferable for encrypting such real-time images. Also, some encryption schemes may be run very slowly, and this increases the degree of security features, yet they would be of little use when dealing with real-time images [3]. "

[Show abstract] [Hide abstract]**ABSTRACT:**Security of data is of prime importance. Security is a very complex and vast topic. One of the common ways to protect this digital data from unauthorized eavesdropping is encryption. This paper introduces an improved image encryption technique based on a chaotic 3D cat map and Turing machine in the form of dynamic random growth technique. The algorithm consists of two main sections: The first does a preprocessing operation to shuffle the image using 3D chaotic map in the form of dynamic random growth technique. The second uses Turing machine simultaneous with shuffling pixels’ locations to diffuse pixels’ values using a random key that is generated by chaotic 3D cat map. The hybrid compound of a 3D chaotic system and Turing machine strengthen the encryption performance and enlarge the key space required to resist the brute force attacks. The main advantages of such a secure technique are the simplicity and efficiency. These good cryptographic properties prove that it is secure enough to use in image transmission systems.

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