## 4 Proposed Secure MP-FrFT-OFDM Cryptosystem
### Proposed DCBS Generator  
The design of the proposed DCBS generator is based on the secure fractional number sequence generated from the proposed MCC-MF sine map and the biometric fingerprint minutiae generated from the FS83 s-FA Module [62]. We assumed that the FS83 s-FA Module generated a sequence \"\(\text{T}\text{T}^{\prime}\in[1,256]\) with a length of 2072 bytes. The block diagram of the proposed DCBS generator is shown in Figure 4. As shown in Figure 4, the proposed DCBS generator consists of 128 secret keys, an initial condition generation, and the proposed MCC-MF-sine map. The secret key is used for initial condition generation for the proposed MCC-MF-sine map with the fractional secure parameters (\(\
u_{1}\)_to_\(\
u_{4}\)) in order to generate 2072 bytes and 512 \(\times\) 512 bytes. The output of the MCC-MF-sine map is used as an input for the DCBS generator to produce two vectors \(\overline{a}\) and \(\overline{b}\) of sizes (1 \(\times\) 256) and 256 \(\times\) 256 bytes for the encryption and authentication process.  
1. The secret key (SK) of 128 bits represented by 32 hexadecimal digits \"C2250 EA6637F5A FAAF0654 9 CCD16220A\" is used to combined the biometric signature with the fractional number sequence generated from the proposed MCC-MF sine map.
2. The secret key is divided into eight sections to generate the initial conditions and the different control parameters of the proposed MCC-MF sine map. All secret parameters and the initial condition are \(10^{-15}\) decimal precision.
3. The first eight hexadecimal digits (\(k_{s}^{1}\) ) and the last eight hexadecimal digits number eight (\(k_{s}^{8}\)) are used to generate the fractional initial condition of the proposed MCC-MF sine map as: \[x_{0}=\left(\frac{hex2 dec\left(k_{s}^{1}\right)}{hex2 dec\left(k_{s}^{8}\right) }\text{ mod }1\right)\] (16)
4. The first (\(k_{s}^{1}\) ) and the second (\(k_{s}^{2}\)) eight hexadecimal digits are used to generate the first fractional secret control parameter (\(r_{1}^{f}\)) as: \[r_{1}^{f}=\left(hex2 dec\left(k_{s}^{1}\right)+hex2 dec\left(k_{s}^{2}\right) \text{ }*\text{ }10^{-6}\right)\text{ mod }20\] (17)  
Figure 4: Proposed DCBS generator block diagram based on MCC-MF sine map.  
5. The next three fractional secret control parameters are given as: \[r_{2}^{f} =\left(hex2 dec\left(k_{\rm s}^{2}\right)+hex2 dec\left(k_{\rm s}^{4} \right)\ *\ 10^{-6}\right)\ \mathrm{mod}\ 20\] (18) \[r_{3}^{f} =\left(hex2 dec\left(k_{\rm s}^{5}\right)+hex2 dec\left(k_{\rm s}^{6} \right)\ *\ 10^{-6}\right)\ \mathrm{mod}\ 20\] (19) \[r_{4}^{f} =\left(hex2 dec\left(k_{\rm s}^{2}\right)+hex2 dec\left(k_{\rm s}^{2} \right)\ *\ 10^{-8}\right)\ \mathrm{mod}\ 20\] (20)
6. The proposed MCC-MF sine map given in Equation (15) is iterated \(t=512\times 512\times 8\) times by using the generated \(x_{0},r_{1}^{f},r_{2}^{f},r_{3}^{f},r_{4}^{f}\) secret parameters and the fractional secure parameters ( \(v_{1}\)_to_\(v_{4}\)).
7. Ignore the first 1000 bits to prove the chaos property of the generated chaotic sequence. In addition, select the last 2072 bytes of the generated chaotic sequence.
8. Concatenate the chaotic sequence output (2072 bytes) with the 2072 bytes of the biometric signature to generate the dynamic chaotic biometric signature (DCBS).
9. Finally, randomly select \(256\times 256\) bytes from the iterated chaotic sequence \(512\times 512\) bytes for the diffusion process by Xoring with the original image and select the two different 256 vectors (\(\overline{a}\) and \(\overline{b}\) ), which are used as the secret multi-parameters for the confusion process in the MPFrFT OFDM transform.