## 5 Secret Sharing Scheme with Residue Number System
For the set of authorized participants, the secret can be recovered correctly; hence, the uncertainty is equal to:  
\[\Delta\Big{(}s_{i}:i\in\widetilde{I}\Big{)}=0. \tag{8}\]  
To analyze the security of SSS based on RRNS, an additional concept of the perfect SSS was introduced in [38]. SSS is called perfect if any unauthorized subset participants cannot obtain any information about the secret. Hence, the scheme is perfect if, for all unauthorized subsets of participants with numbers \(\overline{I}\) and for any \(\varepsilon>0\), there is \(s\), such that, for \(p_{0}<p_{1}<\ldots<p_{n}\), \(p_{i}>s\) (\(i=0,1,\ldots,n\)), and \(\Delta\big{(}s,s^{*},\overline{I}\big{)}<\varepsilon\). The scheme is called ideal if the space of share has the same dimension as the secret space. An ideal SSS is perfect with the smallest possible size of each share. The question of how exactly it is necessary to choose the parameters of the SSS on the RRNS so that it has the asymptotic idealness property remains open. In [38], the authors show the asymptotic idealness of the Asmuth-Bloom scheme using \"sufficiently close\" coprime numbers for RRNS moduli. The work [39] considers so-called compact sequences of coprime numbers with an initial value \(p_{0}\) when \(p_{n}<p_{0}+p_{0}^{\theta}\) for some real number \(\theta\in(0,1)\). In the following analysis, we assume that the compact sequences of coprime numbers are used as the moduli sets. Let us now consider the concept of computationally secure SSS. Assume that at some point in time, unauthorized participants collect several shares with numbers \(\widetilde{I}\). The objective of the unauthorized participants is to recover the secret based on the available data. Let \(S\) be a universal set of all subsets of possible secrets recovered from all available shares. \(S\) can be divided into two subsets. First subset \(S_{1}\) consists of all possible secrets that cannot be used to obtain the secret. The second subset \(S_{2}\) contains all remaining possible secrets. For example, if the Mignotte scheme knows the share of the secret \(s_{j}\) for modulo \(p_{j}\), \(0\leq j\leq n\), then the secret must satisfy the condition: \(s\equiv s_{j}\text{mod}p_{j}\). Therefore, in this case, \(S_{1}=\{s:s\in S\wedge s\
ot\equiv s_{j}\text{mod}p_{j}\}\) and \(S_{2}=\{s:s\in S\wedge s\equiv s_{j}\text{mod}p_{j}\}\). Note that if the SSS is perfect, then \(S_{1}=\varnothing\) and \(S_{2}=S\). Thus, to obtain the original secret, it is necessary to use all combinations of indexes included in \(S_{2}\) and the security of the scheme depends on the cardinality of this set and the computational complexity of the complete permutation. It is necessary to generate the scheme parameters in such a way that unauthorized participants cannot, using modern computing resources, obtain the secret in a reasonable time. A scheme that meets these conditions is called a computationally secure scheme. As a measure of computationally secure, we take the cardinality of the set \(S_{2}\): \(f(\overline{I})=|S_{2}|\). For the Asmuth-Bloom scheme, considering its asymptotic idealness, and Asmuth-Bloom condition, \(f(\widetilde{I})=|S|\leq p_{0}\) for \(\widetilde{I}\). Computationally secure schemes are not always ideal but have reduced redundancy, which is important in practical applications.