## 4 Residue Number System and Secret Sharing  
\((k,\ n)\)-RRNS is determined by a system of pairwise coprime moduli \(\{p_{1},p_{2},\ldots,p_{n}\}\). Positional integer number \(s\) such that \(0\leq s<P\), where \(P=\prod_{i=1}^{k}p_{i}\), is represented as a tuple of \(n\) numbers \(s\stackrel{{ RRNS}}{{\rightarrow}}(s_{1},s_{2},\ldots,s_{n})\), where \(n=k+r\) and  
\[s_{i}=\text{smod}p_{i},\ i=1,2,\ldots,n. \tag{2}\]  
RRNS is a redundant representation of the Residue Number System (RNS). Redundancy is represented by additional moduli in the moduli set. \(k\) is the RNS dimension; \(r\) is the dimension of redundant moduli; and \(n\) is the RRNS dimension. According to the RRNS property, if the number of moduli is \(r\), then it can detect \(r\) and correct \(\lfloor r/2\rfloor\) errors. Redundancy supports reliable data processing and transmission systems with multiple error detection and correction. To detect and correct errors in RRNS, several methods are used, for instance, syndrome and projection methods [28; 29]. If we consider RRNS not only as the error detection, localization, and correction code but also as the Mignotte SSS, then we can conclude that RRNS ensures data security. RRNS has many applications because of its properties such as parallelism and modularity, among which we can mention hardware and software acceleration, information integrity control, digital signal processing, increasing the robustness of information transmission between computers, etc. Modular calculus is based on the Chinese Remainder Theorem (CRT) [28], according to which the number \(s\) can be uniquely calculated by the formula  
\[s=\left|\sum_{i=1}^{k}\left|P_{i}{}^{-1}\right|_{p_{i}}P_{i}s_{i}\right|_{p}, \tag{3}\]  
where \(P_{i}=\frac{p}{p_{i}}\), \(\left|P_{i}{}^{-1}\right|_{p_{i}}\)--multiplicative inversion \(P_{i}\) modulo \(p_{i}\), for \(i=1,2,\ldots,k\). This method is called the CRT method or the Garner method. However, it is computationally complex, since it requires division by a sufficiently large number \(P\). It is worth noting that there are many well-developed methods for an efficient implementation of calculating the remainder of the division and converting numbers back from RRNS to a WNS. It makes this system suitable for use as the basics of a SSS [15; 26; 28]. Let us consider SSSs using [PERSON]'s threshold scheme as an example [27]. The idea of this scheme is that the secret is represented as a polynomial \(k-1\) degrees. Then, to interpolate the resulting polynomial, it is necessary \(k\) points, and the polynomial can be divided into \(n\) shares. Then, the secret-sharing process is as follows. Let we need to divide the secret \(s\) on \(n\) shares. To do this, take a prime number \(p>s\). The following polynomial is constructed:  
\[F(x)=\left(a_{k-1}x^{k-1}+a_{k-2}x^{k-2}+\ldots+a_{1}x+s\right)\bmod p, \tag{4}\]  
where \(a_{k-1},a_{k-2},\ldots,a_{1}\)--random numbers that are only known when the secret is shared. The secret recovery occurs due to the calculation of the Lagrange interpolation polynomial according to the following formula:  
\[F(x)=\sum_{i}l_{i}(x)y_{i}\bmod p;l_{i}(x)=\prod{}_{i\
eq j}\frac{x-x_{i}}{x_{ i}-x_{j}}\bmod p, \tag{5}\]  
where \((x_{i},y_{i})\)--polynomial point coordinates. In addition, there is a limitation: all calculations are performed only in the final field \(p\). In this scheme, an integer polynomial is used. Despite the low redundancy and high scalability, the field space \(p\) is not used efficiently. This scheme was developed by [PERSON] in 1993 [37]. In this scheme, integer coefficients are shares. It is a \((k,n)\) threshold SSS. It distributes \(s\) among \(k\) participants randomly. The recovery of the secret is possible from \(k\) shares, while \(k-1\) shares do not allow you to recover \(s\). Let us consider the Information Dissemination Algorithm designated as \(IDA\) (Algorithm 1). This algorithm works for parameters \(n\) (total number of shares) and \(k\) (required number of shares for recovery).