![1. Encryption algorithm
Without loss of generality, we assume the plain image is of MN pixels. The detailed
encryption process is described as follows:
Step 1: Iterate Eq. (1) for N0 times to avoid the harmful effect of transitional procedure,
where N0 is a constant. To solve the equation, fourth-order Runge-Kutta method isemployed,
as given by Eq. (2)
( ),
( ) ,
,
dx
y x
dt
dy
x z y
dt
dz
xy z
dt
(1)
1 1 2 3 4
1 1 2 3 4
1 1 2 3 4
( / 6)( 2 2 ),
( / 6)( 2 2 ),
( / 6)( 2 2 ),
n n
n n
n n
x x h K K K K
y y h L L L L
z z h M M M M
(2)
where
1 1
1 1 1
1 1 1
( )
( ) with 1
j n n
j n n n
j n n n
K y x
L x z y j
M x y z
緒
1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
[( / 2) ( / 2)]
( / 2)[ ( / 2)] ( / 2) with 2,3
( / 2)( / 2) ( / 2)
j n j n j
j n j n j n j
j n j n j n j
K y hL x hK
L x hK z hM y hL j
M x hK y hL z hM
緒
1 1 1 1
1 1 1 1 1 1
1 1 1 1 1 1
[( ) ( )]
( )[ ( )] ( ) with 4
( )( ) ( )
j n j n j
j n j n j n j
j n j n j n j
K y hL x hK
L x hK z hM y hL j
M x hK y hL z hM
緒
and the step h is chosen as 0.0005.
Step 2: The Lorenz system isiterated continuously. For each iteration, we can get three state
valuesand one is selected as quantification of diffusion keystream accordingto
1
1
1
, % 3 0
, % 3 1
, % 3 2
n n
n n n
n n
x for p
r y for p
z for p
緒
緒
緒
(3)
where pn-1 is the previously operated plain pixel and % denotes modulo operation. One may
set initial value p0 as a constant.
Step 3: The keystream element isquantified byusingthe following formula
14
mod[ (( ( ) ( ( ))) 10 ), ]n n nk round abs r floor abs r L , (4)](https://image.slidesharecdn.com/cdfe6ca5-373d-41bb-953a-78f07e2ca76e-151108093756-lva1-app6892/85/con-dif2-1-320.jpg)
![where abs(x) returns the absolute value of x, floor(x) returns the value of x to the nearest
integers less than or equal to x, round(x) rounds x to the nearest integers, and L is the color
level (for a 256 grey-scale image, L=256). In our scheme, all the state variables are declared as
64-bit double-precision type. According to the IEEE floating-point standard, the
computational precision of the 64-bit double-precision number is about 10-15. Therefore, the
fractional part of a state variable is multiplied by 1014 so as to ensure both the randomness
and accuracyof the quantified keystream.
Step 4: Buffer keystream element kn into a vector k={k1, k2, , kMN} as the diffusion
operation is performed after permutation operation.
Step 5: Let sn and tn denote the remaining two state variables of the Lorenz system. Swap
current pixel with the pixel at position (m, n), where
14
14
mod[ ( 10 ), ],
mod[ ( 10 ), ].
n
n
m floor s M
n floor t N
器
器
(5)
Step 6: Return toStep 1 until all the pixelsin the plain image are swapped from left to right,
top to bottom.
Step 7: Modify the pixel values sequentially from left to right, top to bottom, during which
the influence of each individual pixel is spread out over all itssubsequent pixelsin the image.
This is done by usingEq. (6).
1{[ ] mod } ,n n n n nc k p k L c (6)
where pn, cn and cn-1 are the currently operated pixel, output cipher pixel and previous
ciphered pixel, respectively, and performsbit-wise exclusiveOR operation. The initial value
c0 may also be set as a constant.
In general, 3-4 rounds of such permutation-diffusion operations are needed to achieve a
satisfactorylevel of security. To accelerate the diffusion process, the shuffled image is diffused
in order from bottom to top, right to left in every other round. With such a mechanism, the
proposed scheme requires only two encryption rounds to achieve a satisfactory level of
security.
2 Decryption algorithm
In general, the decryption procedure is similar to that of the encryption process except that
some steps are followed in a reversed order. However, there are still some slight differences
between the two processes as the permutation table and the diffusion keystream are
generated from Lorenz system simultaneously. Moreover, as the proposed cryptosystem is a
symmetric key cipher, the same secret key (x0, y0, z0) and initial conditions (p0, c0) should be
used for decryption. The detailed decryption process is described as follows:
Steps 1 to 3 are the same as those of the encryption algorithm, except pn-1 denotes the
previouslydeciphered pixel.
Step 4: Buffer (sn , tn) into a M-by-N-by-2 permutation matrix Mp as the decryption is done
in reverse order of encryption.](https://image.slidesharecdn.com/cdfe6ca5-373d-41bb-953a-78f07e2ca76e-151108093756-lva1-app6892/85/con-dif2-2-320.jpg)
![Step5: Remove the effect of diffusion from the cipher image toobtain an intermediate image,
i.e., the shuffled image. The detailed operations are the same as those described in Step 7 in
encryption, except that the inverse of Eq. (6) is applied, as given by
LkLcckp nnnnn mod][ 1
. (7)
Step6: Remove the effect of permutation from the shuffled image torecover the plain image.
This isdone byswappingthe pixelsof the shuffled image accordingto the permutation matrix
Mp in reverse order of Step 6 in encryption, i.e., from, bottom to top, right to left. Obviously,
matrix Mp should also be used reversely.
Asboth decipher and encipher procedureshave similar structures, they have essentiallythe
same algorithmiccomplexityand time consumption.
3. Conclusions
This paper has suggested a chaos-based image cipher with improved confusion-diffusion
strategies. To address the security and efficiency problems encountered by many existing
image ciphers, the new scheme utilizes a single chaotic system, Lorenz system, for both
permutation and diffusion. In the permutation stage, we introduce a novel shuffling
mechanism, which shuffles each pixel in the plain image by swapping it with another pixel
chosen bytwoof the three state variablesof Lorenzsystem. The remainingvariable isused for
quantification of pseudorandom keystream for image diffusion. Results of permutation
performance analysis have indicated that the proposed permutation method outperforms
existing methods with respect to either effectiveness or efficiency. Moreover, the selection of
state variablesis controlled byplain pixel. As a result, the quantified keystream is related not
onlyto the keybut also tothe plain image, which enhancesthe securityagainst known/chosen
plaintext attack.](https://image.slidesharecdn.com/cdfe6ca5-373d-41bb-953a-78f07e2ca76e-151108093756-lva1-app6892/85/con-dif2-3-320.jpg)
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- 1. 1. Encryption algorithm Without loss of generality, we assume the plain image is of MN pixels. The detailed encryption process is described as follows: Step 1: Iterate Eq. (1) for N0 times to avoid the harmful effect of transitional procedure, where N0 is a constant. To solve the equation, fourth-order Runge-Kutta method isemployed, as given by Eq. (2) ( ), ( ) , , dx y x dt dy x z y dt dz xy z dt (1) 1 1 2 3 4 1 1 2 3 4 1 1 2 3 4 ( / 6)( 2 2 ), ( / 6)( 2 2 ), ( / 6)( 2 2 ), n n n n n n x x h K K K K y y h L L L L z z h M M M M (2) where 1 1 1 1 1 1 1 1 ( ) ( ) with 1 j n n j n n n j n n n K y x L x z y j M x y z 緒 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 [( / 2) ( / 2)] ( / 2)[ ( / 2)] ( / 2) with 2,3 ( / 2)( / 2) ( / 2) j n j n j j n j n j n j j n j n j n j K y hL x hK L x hK z hM y hL j M x hK y hL z hM 緒 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 [( ) ( )] ( )[ ( )] ( ) with 4 ( )( ) ( ) j n j n j j n j n j n j j n j n j n j K y hL x hK L x hK z hM y hL j M x hK y hL z hM 緒 and the step h is chosen as 0.0005. Step 2: The Lorenz system isiterated continuously. For each iteration, we can get three state valuesand one is selected as quantification of diffusion keystream accordingto 1 1 1 , % 3 0 , % 3 1 , % 3 2 n n n n n n n x for p r y for p z for p 緒 緒 緒 (3) where pn-1 is the previously operated plain pixel and % denotes modulo operation. One may set initial value p0 as a constant. Step 3: The keystream element isquantified byusingthe following formula 14 mod[ (( ( ) ( ( ))) 10 ), ]n n nk round abs r floor abs r L , (4)
- 2. where abs(x) returns the absolute value of x, floor(x) returns the value of x to the nearest integers less than or equal to x, round(x) rounds x to the nearest integers, and L is the color level (for a 256 grey-scale image, L=256). In our scheme, all the state variables are declared as 64-bit double-precision type. According to the IEEE floating-point standard, the computational precision of the 64-bit double-precision number is about 10-15. Therefore, the fractional part of a state variable is multiplied by 1014 so as to ensure both the randomness and accuracyof the quantified keystream. Step 4: Buffer keystream element kn into a vector k={k1, k2, , kMN} as the diffusion operation is performed after permutation operation. Step 5: Let sn and tn denote the remaining two state variables of the Lorenz system. Swap current pixel with the pixel at position (m, n), where 14 14 mod[ ( 10 ), ], mod[ ( 10 ), ]. n n m floor s M n floor t N 器 器 (5) Step 6: Return toStep 1 until all the pixelsin the plain image are swapped from left to right, top to bottom. Step 7: Modify the pixel values sequentially from left to right, top to bottom, during which the influence of each individual pixel is spread out over all itssubsequent pixelsin the image. This is done by usingEq. (6). 1{[ ] mod } ,n n n n nc k p k L c (6) where pn, cn and cn-1 are the currently operated pixel, output cipher pixel and previous ciphered pixel, respectively, and performsbit-wise exclusiveOR operation. The initial value c0 may also be set as a constant. In general, 3-4 rounds of such permutation-diffusion operations are needed to achieve a satisfactorylevel of security. To accelerate the diffusion process, the shuffled image is diffused in order from bottom to top, right to left in every other round. With such a mechanism, the proposed scheme requires only two encryption rounds to achieve a satisfactory level of security. 2 Decryption algorithm In general, the decryption procedure is similar to that of the encryption process except that some steps are followed in a reversed order. However, there are still some slight differences between the two processes as the permutation table and the diffusion keystream are generated from Lorenz system simultaneously. Moreover, as the proposed cryptosystem is a symmetric key cipher, the same secret key (x0, y0, z0) and initial conditions (p0, c0) should be used for decryption. The detailed decryption process is described as follows: Steps 1 to 3 are the same as those of the encryption algorithm, except pn-1 denotes the previouslydeciphered pixel. Step 4: Buffer (sn , tn) into a M-by-N-by-2 permutation matrix Mp as the decryption is done in reverse order of encryption.
- 3. Step5: Remove the effect of diffusion from the cipher image toobtain an intermediate image, i.e., the shuffled image. The detailed operations are the same as those described in Step 7 in encryption, except that the inverse of Eq. (6) is applied, as given by LkLcckp nnnnn mod][ 1 . (7) Step6: Remove the effect of permutation from the shuffled image torecover the plain image. This isdone byswappingthe pixelsof the shuffled image accordingto the permutation matrix Mp in reverse order of Step 6 in encryption, i.e., from, bottom to top, right to left. Obviously, matrix Mp should also be used reversely. Asboth decipher and encipher procedureshave similar structures, they have essentiallythe same algorithmiccomplexityand time consumption. 3. Conclusions This paper has suggested a chaos-based image cipher with improved confusion-diffusion strategies. To address the security and efficiency problems encountered by many existing image ciphers, the new scheme utilizes a single chaotic system, Lorenz system, for both permutation and diffusion. In the permutation stage, we introduce a novel shuffling mechanism, which shuffles each pixel in the plain image by swapping it with another pixel chosen bytwoof the three state variablesof Lorenzsystem. The remainingvariable isused for quantification of pseudorandom keystream for image diffusion. Results of permutation performance analysis have indicated that the proposed permutation method outperforms existing methods with respect to either effectiveness or efficiency. Moreover, the selection of state variablesis controlled byplain pixel. As a result, the quantified keystream is related not onlyto the keybut also tothe plain image, which enhancesthe securityagainst known/chosen plaintext attack.