## 3 Development of an Imitation-Resistant Zero-Knowledge Authentication Protocol Based on an MPPC
In [71], it is suggested to use the interval-index characteristic for error correction. The Chinese Remainder Theorem (CRT) is proposed to compute this positional characteristic, which requires \(k+r\) multiplication operations, \(k+r-1\) addition operations modulo two and one comparison operation. In [72], it is proposed to use the interval estimation of modular code combinations. However, this algorithm has a disadvantage. To obtain the interval estimation of the modular code combination, it is necessary to additionally calculate the rank of a number in the set of informational bases. In this case, when implementing the algorithm [72], it is necessary to perform \(k+r\) modular multiplication operations and \(k+r\) addition operations modulo two. The paper [73] describes a method of comparing two combinations in a modular code without performing a reverse conversion to a positional code. However, this method only allows for detecting an error in the combination. In the developed algorithm, orthogonal bases with broken orthogonality on redundant bases are used to detect and correct errors in MPPC. Let \(k\) informational bases be defined. Let us perform a CRTP-based reverse conversion of the combination \(U(x)=(U_{1}(x),\,\ldots,\,U_{k}(x))\) without redundant residues. For this purpose, we use \(k\) orthogonal bases, which are defined as  
\[B_{i}(x)=\Big{|}P_{i}^{-1}(x)\Big{|}_{m_{i}(x)}P_{i}(x), \tag{25}\]  
where \(P_{i}(x)=\frac{M_{k}(x)}{m_{i}(x)}\); \(\Big{|}P_{i}^{-1}(x)\Big{|}_{m_{i}(x)}\) is the weight of the \(i\)-th orthogonal basis; and \(i=1,\,\ldots,\,k\). The result is a polynomial \(U(x)\), for which \(\text{deg}U(x)<\text{deg}M_{k}(x)\). Let us represent the orthogonal bases using additional \(r\) redundant bases. Then, we have  
\[ \begin{array}{l}B_{1}^{\prime\prime}(x)=(1,\,0,\,0,\,\ldots,\,0,\,\psi_{k+ 1}^{1}(x),\,\psi_{k+2}^{1}(x),\,\ldots,\,\psi_{k+r}^{1}(x)),\\ B_{2}^{\prime\prime}(x)=(0,\,1,\,0,\,\ldots,\,0,\,\psi_{k+1}^{2}(x),\,\psi_{k+ 2}^{2}(x),\,\ldots,\,\psi_{k+r}^{2}(x)),\\ \vdots\\ B_{k}^{\prime\prime}(x)=(0,\,0,\,0,\,\ldots,\,1,\,\psi_{k+1}^{k}(x),\,\psi_{k +2}^{k}(x),\,\ldots,\,\psi_{k+r}^{k}(x)),\end{array}  \tag{26}\]  
where \(\psi_{j}^{i}(x)\equiv B_{i}(x)\text{mod}m_{j}(x)\); and \(j=k+1,\,\ldots,\,k+r\). Let us use the orthogonal bases (26) and implement CRTP-based reverse conversion using informational bases \(m_{1}(x),\,\ldots,\,m_{k}(x)\). Then, we obtain  
\[U(x)=\sum_{i=1}^{k}U_{i}(x)B_{i}^{\prime\prime}(x)\text{mod}M_{k}(x). \tag{27}\]  
Then, pairwise products can be represented as  
\[\left\{ \begin{array}{l}U_{1}(x)B_{1}^{\prime\prime}(x)\text{mod}M_{k}(x)=(U _{1}(x),0,0,\ldots,0,\gamma_{k+1}^{1}(x),\ldots,\gamma_{k+r}^{1}(x)),\\ U_{2}(x)B_{2}^{\prime\prime}(x)\text{mod}M_{k}(x)=(0,\,U_{2}(x),0,\ldots,0, \gamma_{k+1}^{2}(x),\ldots,\gamma_{k+r}^{2}(x)),\\ \vdots\\ U_{k}(x)B_{k}^{\prime\prime}(x)\text{mod}M_{k}(x)=(0,\,0,\,0,\,\ldots,\,U_{ k}(x),\gamma_{k+1}^{k}(x),\ldots,\gamma_{k+r}^{k}(x)),\end{array} \right. \tag{28}\]  
where \(\gamma_{j}^{i}(x)=U_{i}(x)\psi_{i}^{i}(x)\text{mod}m_{j}(x)\); and \(i=1,\,\ldots,\,k;j=k+1,\,\ldots,k+\,r\). Let us substitute expressions (28) into equality (27). We obtain  
\[ \begin{array}{l}U(x)=(U_{1}(x),\,0,\,0,\ldots,0,\gamma_{k+1}^{1}(x),\ldots, \gamma_{k+r}^{1}(x))\\ +(0,U_{2}(x),\,0,\ldots,0,\gamma_{k+1}^{2}(x),\ldots,\gamma_{k+r}^{2}(x))+ \ldots\\ +(0,\,0,\,0,\ldots,U_{k}(x),\gamma_{k+1}^{2}(x),\ldots,\gamma_{k+r}^{2}(x)). \end{array}  \tag{29}\]  
Since the equality  
\[U(x)=(U_{1}(x),\,U_{2}(x),\,\ldots,\,U_{k}(x))=(U_{1}(x),\,U_{2}(x),\,\ldots, \,U_{k+r}(x)) \tag{30}\]is true, we can obtain the positional characteristic called the error syndrome from expression (29). In this case, the error syndrome for the redundant combination \(U(x)\) can be represented in the form of the following equations:  
\[\left\{