## 5 Secret Sharing Scheme with Residue Number System  
In this section, we introduce the basic concepts of security of two well-known SSS based on RRNS: Asmuth-Bloom and Mignotte. Let each participant have a unique number or identifier. The entire set of these numbers we call the universal set of numbers and denote \(U\) (in the simplest case \(U=\{1,2,\ldots,n\}\), where \(n\) is the number of participants in the scheme). The set of authorized (qualified) coalitions is called the authorized subsets of \(U\) denoted by \(I\). Participants of the qualified subsets can recover the secret from their shares when they act together to pool their knowledge. An unauthorized subset is a subset \(\widetilde{I}\) of participants of any coalition that does not have the right to recover the secret. In the Asmuth-Bloom scheme, \(p_{0}\) is a secret key, and \(s\in[0,p_{0})\). The moduli \(p_{1}<p_{2}<\ldots<p_{k}<p_{k+1}<\ldots<p_{n}\) have to be chosen, so that \(\prod_{i=1}^{k}p_{i}>p_{0}\prod_{i=0}^{k-2}p_{n-i}\). The last inequality is usually called the Asmuth-Bloom condition. At the stage of sharing the secret, a random number \(r_{n}\) is generated such that \(s^{\prime}=s+r_{n}p_{0}<\prod_{i=1}^{k}p_{i}\). Secrets \(s^{\prime}\) is divided so that \(s_{i}=s^{\prime}\text{mod}\,p_{i}\) is a share for participant \(i\), where \(i=1,2,\ldots,n\). Any set of authorized participants with numbers from \(I\) can recover the secret; wherein \(n\geq|I|=m\geq k\). Using the CRT, \(s^{\prime}\) is recovered based on its RRNS representation \(\big{(}s_{i_{1}},s_{i_{2}},\ldots,s_{i_{m}}\big{)}\) with moduli \(p_{i_{1}},p_{i_{2}},\ldots,p_{i_{m}}\), where \(i_{j}\in I\), for all \(j=1,2,\ldots,m\). \(s\) is recovered as the remainder of the division of \(s^{\prime}\) on \(p_{0}\): \(s=s^{\prime}\) mod \(p_{0}\). Let us consider an unauthorized coalition of participants with numbers from \(\widetilde{I}\). Then \(\left|\widetilde{I}\right|\leq k-1\), let \(P=\prod_{i=1}^{k}p_{i}\) and \(\widetilde{P}=\prod_{i\in\widetilde{I}}p_{i}\). In this case, \(\widetilde{s}=s^{\prime}\text{mod}\,\,\widetilde{P}\). According to the Asmuth-Bloom conditions \(P/\widetilde{P}>p_{0}\) and \(\left(\widetilde{P},p_{0}\right)=1\). Thus, as shown in [34], an unauthorized coalition obtained fewer than \(k\) shares does not receive any useful information about the secret. In the Mignotte \((k,\,n)\) threshold scheme, moduli \(p_{1}<p_{2}<\ldots<p_{k}<p_{k+1}<\ldots<p_{n}\) are chosen to satisfy the inequality:  
\[\alpha=\prod_{i=0}^{k-2}p_{n-i}<\prod_{i=1}^{k}p_{i}=\beta. \tag{6}\]  
To achieve security, secret \(s\) has to be in the interval \((\alpha,\beta)\). Any set \(I\) of authorized participants can recover the secret, wherein \(|I|=m\geq k\). \(s\) is recovered by CRT using \(\big{(}s_{i_{1}},s_{i_{2}},\ldots,s_{i_{m}}\big{)}\) and moduli \(p_{i_{1}},p_{i_{2}},\ldots,p_{i_{m}}\), where \(i_{j}\in I\), for all \(j=1,2,\ldots,m\). To ensure security, Mignotte sequences with a large value \((\beta-\alpha)/\beta\) should be used [35]. This scheme is not computationally secure but has practical applications due to reduced redundancy compared with Asmuth-Bloom. Let us consider the concept of entropy, which plays an important role in SSS security theory. We denote the entropy of the secret as \(H(s)=\log_{2}s\). In this case, the entropy is maximum. Knowing the subset of the shares \(s^{*}=\big{(}s_{i_{1}},s_{i_{2}},\ldots,s_{i_{m}}\big{)}\), we denote entropy as \(H(s^{*},I)=\min(\log_{2}s,\sum_{i\in I}\log_{2}p_{i})\), where \(\forall j:i_{j}\in I\) and \(|I|=m\). If \(I\) is the set of authorized participants, then \(H(s^{*},I)=H(s)=\log_{2}s\). The important characteristic of the SSS is the uncertainty of the secret that is defined by  
\[\Delta(s,s^{*},I)=H(s)-H(s^{*},I). \tag{7}\]  
SSS is computationally secure if \(\Delta\big{(}s,s^{*},\overline{I}\big{)}=H(s)\) for all \(\overline{I}\), where \(\overline{I}\) is the set of unauthorized participants.