## 2 Related Preliminary Basics  
### Multiple Parameters FrFT  
The MPFrFT was presented with its applications and its advantages in signal processing, image encryption and communications in [60]. The \(a^{th}\)-order continuous FRT of \(x(t)\) is given by:  
\[X_{a}=\int_{-\infty}^{+\infty}x(t)K_{a}(u,t)dt \tag{1}\]  
\[K_{a}(u,t)=\sqrt{1-jcot\alpha}\cdot e^{j\pi(j^{2}cot\alpha-tu\text{csc}\left(a \right)+u^{2}cot\alpha)} \tag{2}\]  
where \(K_{a}(u,t)\) is the transform kernel and \(\alpha=a\pi/2\). The matrix \(F\) is \(N\times NDFT\) can be defined as:  
\[F_{k,n}=\frac{1}{\sqrt{N}}e^{-j\frac{2\pi}{N}kn},0\leq k,n\leq N-1 \tag{3}\]  
The DFT matrix \(F\) has only four different eigenvalues \(\{1,-j,-1,j\}\). Consider \(S\) as a nearly tri-diagonal \(N\times N\) matrix whose nonzero entries are \(S_{n,n}=2\cos 2\pi n/N,0\leq n\geq N\) and \(S_{n,n+1}\)=\(S_{n+1,n=1}\), \(0\leq n\geq N-2\), and \(S_{n-1,0}\)=\(S_{0,n-1=1}\). The matrices S and F will have the same eigenvectors if they commute with the matrix \(\text{F}\) (\(S\cdot F=F\cdot S\)) but will not have the same eigenvectors \(\lambda_{k}=e\frac{-j\pi k}{2}\). Based on the four different eigenvectors \(\{1,-j,-1,j\}\), the ath-order FrFT matrix of size \(N\times N\) denoted by \(F^{a}\) is defined by [61]:  
\[F^{a}=V\Lambda^{a}V^{\text{T}}=\begin{cases}\sum_{k=0}^{N-1}\lambda_{k}^{a}v_{ k}v_{k}^{\text{T}},&\text{for $N$ odd}\\ \sum_{k=0}^{N-2}\lambda_{k}^{a}v_{k}v_{k}^{\text{T}}+\lambda_{N}^{a}v_{N}v_{N}^ {\text{T}},&\text{for $N$ even}\end{cases} \tag{4}\]  
where \((\cdot)^{\text{T}}\) denotes the matrix transpose operation, \(V=[v_{0}v_{1}v_{2}\cdots v_{N-2}v_{N-1}]\) for \(N\) odd and \(V=[v_{0}v_{1}v_{2}\cdots v_{N-2}v_{N}]\) for \(N\) even, \(v_{k}\) is the normalized \(k^{th}\)-order discrete Hermite-Gaussian-like eigenvector of S, and \(\Lambda^{a}\) is a diagonal matrix whose entries are \(\lambda_{k}^{a}\) with fractional order \(a\). The MPFrFT can be defined as an extension of the FrFT with multiple parameters by replacing the order \(a\) with the vector of fraction orders \(\overline{a}\) of length \(1\times N\) which are independent fraction orders; then, the MPFrFT denoted by \(F^{\overline{a}}\) is defined as:  
\[F^{\overline{a}}=V\Lambda^{\overline{a}}V^{T} \tag{5}\]  
where \(\Lambda^{a}\) is given by:  
\[\Lambda^{a}=\begin{cases}diag\Big{(}\lambda_{0}^{a(0)},\lambda_{1}^{a(1)}, \cdots,\lambda_{N-1}^{a(N-1)},&for\ N\ odd\\ diag\Big{(}\lambda_{0}^{a(0)},\lambda_{1}^{a(1)},\cdots,\lambda_{N-2}^{a(N-2)},\lambda_{N}^{a(N)},&for\ N\ even\end{cases} \tag{6}\]  
In addition, this model of 1D MPFrFT can be modeled as 2D MPFrFT by using two vectors of fraction orders \(\overline{a}\) and \(\overline{b}\) with lengths of \(1\times N\) and \(1\times M\). The two vectors of fraction orders \(\overline{a}\) and \(\overline{b}\) are independent fraction orders. The 2D MPFrFT can be performed by applying one 1D MPFrFT along rows followed by applying another 1D MPFrFT along columns. The 2D MPFrFT denoted by \(F^{(\overline{a}\overline{b})}\) is defined as [61]:  
\[F^{(\overline{a}\overline{b})}=V\Lambda^{(\overline{a}\overline{b})}V^{T} \tag{7}\]  
Then, the 2D MPFrFT of a 2D input P of size \(M\times N\) can be defined in a row-column scheme as:  
\[F^{(\overline{a}\overline{b})}[P]=F^{(\overline{a})}PF^{(\overline{b})} \tag{8}\]  
The properties of the MPFrFT are given in [62]. The main advantage of the 2D MPFrFT is that the two vectors of fraction orders \(\overline{a}\) and \(\overline{b}\) with lengths of \(1\times N\) and \(1\times M\) can be used as an additional secret key for secure applications.