## 4 Method for Determining Optimal Weights
The results obtained are reflected in the figures (Figures 1 and 2), where is the absolute error equal to \(0.1\cdot 10^{-2^{16}}\), and the tables (Tables 1-3). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Number of** & **Moduli** & **Modulo Size, Bit** & **Optimal Value \(minmax(C(\mathrm{B}_{i}))\)** & **Set of Optimal Weights** & **Absolute Error** \\ \hline \multirow{10}{*}{10} & \multirow{3}{*}{32} & \multirow{3}{*}{3319015825.455541} & (1812568780, 1744305376, 2879177275, 27045322353, & \\  & & & 740106217, 2132  
Figure 1: The results of the study: (**a**) stage A and (**b**) stage B.  
Figure 2: The results of the study: (**a**) stage A and (**b**) stage B.  
Namely, sets of moduli and their numbers, as well as optimal weights, were taken. With these values, the positional characteristics of the number were calculated as \(X\). The same was undertaken under the conditions of the weights obtained by Formula (5). Based on which three-time characteristics were obtained:  
1. Positional characteristics are obtained based on optimal weights;
2. Positional characteristics are obtained based on the calculated weights with the measurement of the time of their calculation;
3. Positional characteristics are obtained based on the calculated weights without measuring the time of their calculation. Based on the experiment, the following tables were obtained. Let us consider the results obtained. The results obtained are presented both as tables (Tables 1 and 2). Based on the data obtained, we can say that the method we have developed for finding optimal weights is effective. Efficiency was confirmed in both cases. When weights are calculated and when weights are stored on disk. In both cases, the use of optimal weights makes it possible to reduce the computational complexity of determining the positional characteristics based on ACF. The result is explained by the reduction in the size of \(C(B)\), which reduces the computational cost. Analyzing the results of Table 2, we can say that the result obtained in Table 1 is adequate. With this study, we confirmed that the resulting sets are stable, both with an increase in the size of the moduli and with an increase in their number. Thus, it is possible to use both the method presented in the work and the obtained weight table in the operation of real systems. The benefit obtained in our study is within 1%. Despite the small productivity gains, this is an important result. When using RNS methods for the safety and reliability of systems, this increase is noticeable. For example, RNS is used in homomorphic encryption [26; 27; 28]. Homomorphic encryption arithmetic has high computational complexity. RNS is used to speed it up due to some of its properties. Thus, an increase of even 1% will allow the use of this type of encryption with greater efficiency. We examine the data indicated in the table as follows. Since, based on the data, the performance advantage of our method is obvious, let us compare our method with the experiment closest in performance. Namely, with the classical calculation of the core without calculating the weights. Additionally, we obtain the following graphs (Figure 1). To do this, we divide the values without calculation weight by the optimal weight and obtain how many times our method is more effective than the classical one. Here, we can observe the following picture. In general, throughout the study, the proposed method had an advantage. However, we can observe the following trend. Depending on the size of the moduli, the performance ratio of both methods tends to unity (Figure 1a) exponentially. This is explained by the fact that with an increase in the size of the moduli, the computational complexity of the calculations also increases. The essence of our method is to manage this complexity by introducing optimal weights. However, due to the fact that for huge values of moduli, the computational complexity is colossal, the influence of optimal weights on it is reduced. However, there is still relevance here.