Page 443 - Mathematical Techniques of Fractional Order Systems

428 Mathematical Techniques of Fractional Order Systems

There are numerous criterion in literature for the design of a strong S-
Box such as: nonlinearity, strict avalanche criterion (SAC), bit independent
criterion (BIC), linear and differential approximation property (LAP, DAP).
When compared with previous chaotic-based S-Box designs, the construc-
¨
tions in Ozkaynak et al. (2017) and Khan and Shah (2015) had higher
degrees of nonlinearity.
14.5.3 Image Encryption

FOCS are used as pseudo-random sequence generators in image encryption
systems as the fractional orders increase the key space. True color images
are stored as 8-bit unsigned integers so the transition from double precision
generated from the FOCS simulation to the ½0; 255Š integer range is an
important part of the encryption algorithm.
Dasgupta et al. (2014) used four different FOCS to generate the key.
Each system has three variables ½x 1 ; x 2 ; x 3 Š and these are designated by addi-
tional subscripts for each system: a for FO Lorenz, b for FO Chen, c for FO
Liu, and d for FO Volta. The fractional order used in all these systems is
q 5 0:95. The number of pixels in the image is computed and then an equal
number of iterations is made for each system variable. The 8-bit unsigned
integers are calculated from the system variables as (Dasgupta et al., 2014):
14
X i 5 ½ðjx i j 2 bx i cÞ 3 10 Šmodð256Þ: ð14:55Þ
14 4 3 6 336
The key space is equal to ð10 Þ 5 10 . The key image is generated
according to the following Xor operations (Dasgupta et al., 2014):
X R 5 X 1a "X 3b "X 2c "X 3d ; ð14:56aÞ

X G 5 X 2a "X 1b "X 1c "X 2d ; ð14:56bÞ

X B 5 X 3a "X 2b "X 3c "X 1d ; ð14:56cÞ
where X R , X G and X B are the red, green, and blue components of the key
image, respectively. Then the original image pixels are Xored with the key
image for encryption. For decryption, the encrypted image is then Xored
with the key image. The most time-consuming step of this algorithm is the
simulation of FOCS as it takes much longer as the number of pixels increase.
So, the authors recommended using C language for this time-consuming step
as well as using the short memory principle to approximate the generated
sequence. Fig. 14.10 shows an example encrypted image based on the pro-
posed algorithm in Dasgupta et al. (2014). The image is the standard colored
lena image of size 256 pixels.
There are numerous algorithms in literature that depend, in some steps,
on FOCS. For example, the one discussed by Radwan et al. (2012) used the
fractional order Lorenz system to generate the pseudo-random sequence. One
interesting part of this algorithm is the inclusion of a least significant bit