Page 442 - Mathematical Techniques of Fractional Order Systems

Applications of Continuous-time Fractional Order Chapter | 14 427

α
D y 5 Ay 1 fðy; zÞ 1 u; ð14:53cÞ
T T
where x 5 ðx 1 ; x 2 Þ and y 5 ðy 1 ; y 2 Þ are the state complex variables vectors
and z is the real variable. A hybrid synchronization scheme was proposed as
antisynchronization between the real variables and projective synchronization
between the imaginary ones. This whole system was used in a switched mod-
ulated digital communication scheme. The switching is gained from chang-
ing the derivative order. The first master system is the real part and the
second master is the imaginary part of x. The modulated signal can be
retrieved once the slave systems are in sync with the master. This happens in
less than 4 seconds according to the simulations in Luo and Wang (2013).

14.5.2 Substitution Boxes

S-Box is a key component in block cipher as it assures the security of the
entire system. It is the only nonlinear component in many encryption algo-
rithms such as IDEA, DES, and AES and most of the attacks on block cipher
algorithms are on the S-Box. Two recent papers studied the construction of
¨
new 8 bit S-Boxes based on FOCS (Khan and Shah, 2015; Ozkaynak et al.,
2017). The first one used the fractional order Ro ¨ssler system as the basis of
their construction:
D x 52 y 2 x; ð14:54aÞ
α 1
D y 5 x 1 δy; ð14:54bÞ
α 2
D z 5 μ 1 zðx 2 λÞ; ð14:54cÞ
α 3
where δ, μ, and λ are system parameters. The steps for constructing the S-
Box are (Khan and Shah, 2015):
1. Solve the system in Eq. (14.54a c) to generate a time series S of length n.
2. Generate the vector U such that U 5 floorðS 3 256Þ.
3. Generate another vector V such that V 5 modðU; 256Þ.
4. Construct the S-Box by choosing the first 256 distinct values in V.
The other paper used the fractional order chaotic Chen system to generate
¨
the time series but used it in a different algorithm (Ozkaynak et al., 2017).
¨
The steps for generating the S-box are (Ozkaynak et al., 2017):
1. Generate the time series of the FOC Chen system.
2. Store the three digits after the comma in a vector S.
3. Normalize the digits in S to the range ½0; 255Š by applying
U 5 modðS; 256Þ.
4. The first distinct 256 values are used to construct the table.