## 3 Preliminaries About Weighted Moore-Penrose Inverse  
As usual, let \(C\) be the set of complex numbers, \(c^{m\times n}\), be the set of \(m\times n\) complex matrices of rank \(C_{r}^{m\times n}=\{C^{m\times n}\mid rank(X)=r\}\). An \(m\times n\) matrix: the \(m\) rows are horizontal and the \(n\) columns are vertical. For any matrix \(A\in C^{m\times n}\) (\(A\)) and positive-definite Hermitian matrices \(M\) and \(N\) of the orders \(m\) and \(n\) respectively, consider the following equations in \(X\)[23]:  
\[\text{AXA} = \text{A} \tag{1}\] \[\text{XAX} = \text{X}\] (2) \[\left(\text{MAX}\right)^{*} = \text{MAX}\] (3) \[\left(\text{NXA}\right)^{*} = \text{NXA} \tag{4}\]  
Character \(*\) denotes conjugate transpose. The conjugate transpose, also known as the Hermitian transpose, of \(m\times n\) complex matrix \(A\) is an \(n\times m\) matrix obtained by transposing \(A\) and applying complex conjugate on each entry. Moreover, the system of matrix Equations (1)-(4) has a unique solution, and the matrix \(X\) which satisfies Equations (1)-(4) is called the \"weighted Moore-Penrose inverse of \(A\), and it is denoted \(X=A_{M,N}^{\dagger}\)\" [23]. Especially in the case \(M=I_{m}\) and \(N=I_{n}\), this becomes \(A_{M,N}^{\dagger}=A^{\dagger}\), i.e., the \"weighted Moore-Penrose inverse is reduced to the Moore-Penrose inverse\" [24; 25]. The concept of using weighted MP inverses in cryptography originated from our earlier publications [26; 27], in which we examined their fundamental characteristics. **Theorem 1**. : _([PERSON] and [PERSON] [28]). Let \(A\in C^{m\times n}\) and \(A_{k}\) is a submatrix that consists of the first \(k\) columns of the matrix \(A\). For \(k=2,\ldots,n\) the matrix \(A_{k}\) is represented with_  
\[A_{k}=[A_{k-1}|a_{k}] \tag{5}\]  
_and \(N_{k}\in C^{k\times k}\) is a submatrix of the matrix \(N\). Then the matrix \(N_{k}\) is given as_  
\[N_{k}= \begin{bmatrix}N_{k-1}&I_{k}\\ I_{k}^{*}&n_{kk}\end{bmatrix} . \tag{6}\]_Let the matrices_ \(X_{k-1}\) _and_ \(X_{k}\) _and the vectors_ \(d_{k}\) _and_ \(c_{k}\) _be defined as_  
\[X_{k-1}=\left(A_{k-1}\right)^{\dagger}_{MN_{k-1}}X_{k}=\left(A_{k}\right)^{ \dagger}_{MN_{k}} \tag{7}\]  
\[d_{k}=X_{k-1}a_{k} \tag{8}\]  
\[c_{k}=a_{k}-A_{k-1}d_{k}=(I-A_{k-1}X_{k-1})a_{k}. \tag{9}\]  
_Then_  
\[X_{k}= \begin{bmatrix}X_{k-1}-\left(d_{k}+(I-X_{k-1}A_{k-1})N_{k-1}^{-1}l_{k} \right)b_{k}^{*}\\ b_{k}^{*}\end{bmatrix} , \tag{10}\]  
_where_  
\[b_{k}^{*}=\left\{ \begin{array}{cc}\left(c_{k}^{*}Mc_{k}\right)^{-1}c_{k}^{* }M&c_{k}\
eq 0\\ \delta_{k}^{-1}\big{(}d_{k}^{*}N_{k-1}-l_{k}^{*}\big{)}X_{k-1}&c_{k}=0\end{array} \right. \tag{11}\]  
_and_  
\[\delta_{k}=n_{kk}+d_{k}^{*}N_{k-1}d_{k}-l_{k}^{*}d_{k}-d_{k}^{*}l_{k}-l_{k}^{* }(I-X_{k-1}A_{k-1})N_{k-1}^{-1}l_{k}. \tag{12}\]  
Since the number of combinations of the possible positive definite Hermitian \(8\times 8\) matrix we mentioned is huge, it is clear that this type of matrix could be used in developing an encryption algorithm and would be extremely strong. For the calculation of Algorithm 1 from [28], auxiliary Algorithm 2 is needed. ```
0: Let \(A\in C^{n\times n}\) and \(M\) and \(N\) are positive definite matrices with dimensions \(m\times m\) and \(n\times n\), respectively. 1:\(A_{1}=a_{1}\)
2:if\(a_{1}=0\)then
3:\(X_{1}^{\dagger}=\frac{1}{a_{1}^{*}Min_{1}}a_{1}^{*}M\)
4:else
5:\(X_{1}=0\)
6:endif
7:for\(k=2\) to \(n\)do
8:\(d_{k}=X_{k-1}a_{k}\)
9:\(c_{k}=a_{k}-\hat{A}_{k-1}d_{k}\)
10:if\(c_{k}\
eq 0\)then
11:\(b_{k}^{*}=\big{(}c_{k}^{*}Mc_{k}\big{)}^{-1}c_{k}^{*}M\), go to Step 16
12:else
13:\(\delta_{k}=n_{kk}+d_{k}^{*}N_{k-1}d_{k}-\big{(}d_{k}^{*}l_{k}+l_{k}^{*}d_{k} \big{)}-l_{k}^{*}\big{(}I-X_{k-1}\hat{A}_{k-1}\big{)}N_{k-1}^{-1}l_{k}\)
14:\(b_{k}^{*}=\delta_{k}^{-1}\big{(}d_{k}^{*}N_{k-1}-l_{k}^{*}X_{k-1}\big{)}\)
15:endif
16:\(X_{k}= \begin{bmatrix}X_{k-1}-\left(d_{k}+(I-X_{k-1}\hat{A}_{k-1})N_{k-1}^{-1}l_ {i}\right)b_{k}^{*}\\ b_{k}^{*}\end{bmatrix}