## 6 Data Transmission Security and Reliability  
Let us consider parameter generation, secret sharing, and secret recovery for threshold \((k,\ n)\)-DT-RRNS. Parameter generation. A compact sequence of coprime numbers is selected \(p_{1}<p_{2}<\ldots<p_{k}<p_{k+1}<\ldots<p_{n}\); where \(p_{n}<p_{1}+p_{1}^{\theta}\) and \(\theta\in(0,1)\); secret \(s\in[0,P)\), where \(P=\prod_{i=1}^{k}p_{i}\) is the dynamic range of the RRNS. Secret Sharing. Shares \(s_{i}\) of a secret \(s\) are calculated as \(\forall\ i=\overline{1,n}:s_{i}=\text{smod}p_{i}\). Secret Recovery. Any authorized set of participants with numbers \(I\) can uniquely recover the secret, where \(|I|=m\geq k\). \(s\) is calculated using CRT \(s=\left|\sum_{j=1}^{m}s_{i_{j}}P_{j,m}\left|P_{j,m}^{-1}\right|_{p_{j}}\right| _{M}\), where \(M=\prod_{j=1}^{m}p_{i_{j}}\) and \(P_{j,m}=M/p_{i_{j}}\). Let us consider the main properties of the DT-RRNS. The following notations are introduced. \begin{tabular}{l l} \(I\) & Authorized set is subset \(of\)\(\{1,2,\ldots,n\}\), cardinality is equal to \(k\) \\ \(\widetilde{I}\) & Unauthorized set is subset \(\{1,2,\ldots,n\}\), cardinality is less to \(k\) \\ \(\widetilde{I}_{max}\) & Unauthorized set \\  & \(\{n-k+2,\ n-k+3,\ldots,n\}\) cardinality is equal to \(k-1\) \\ \(\widetilde{P}=\prod\limits_{i\in I}p_{i}\) & Dynamic range for an unauthorized set \(\widetilde{I}\) \\ \(\widetilde{p}_{max}=\prod\limits_{i\in I_{max}}p_{i}\) & Dynamic range for an unauthorized set \(\widetilde{I}_{max}\) \\ \(f\left(\widetilde{I}\right)\) & Cardinality of the set of possible secrets \(\widetilde{s}\)  for a given \(\widetilde{p}\) \\ \(f\left(\widetilde{I}_{max}\right)\) & Cardinality of the set of possible secrets \(\widetilde{s}\)  for a given \(\widetilde{p}_{max}\) \\ \(\widetilde{s}=s\) mod \(\widetilde{p}\) & Projection of the secret \(s\) modulo \(\widetilde{P}\) \\ \(f_{k}\) & Approximate value of \(f\left(\widetilde{I}\right)\) \\ \(S\) & Universal set of all subsets of possible secrets recovered from all \\  & available shares \\ \(Q_{i}\) & The probability of intercepting \(i\) nodes \\ \(R\) & Proximity of the cardinality of possible secrets \(f\left(\widetilde{I}_{max}\right)\) to \(p_{1}\) \\ \end{tabular}   
The following statement shows a lower bound for the moduli selection. **Statement 1. **_In DT-RRNS, for any unauthorized subset of participants with numbers \(\widetilde{I}\), \(P>\widetilde{P}=\prod_{i\in\widetilde{I}}p_{i}\) and \(p_{1}>2^{k-1}\)._  
**Proof. ** Using the fact that in threshold SSS the maximum unauthorized subset is the subset numbered \(n,n-1,\ldots,n-k+2\) and considering the definition of compact sequences, we obtain:  
\[\widetilde{P}<\prod_{i=0}^{k-2}p_{n-i}<\left(p_{1}+p_{1}^{\theta}\right)^{k-1} =p_{1}^{k-1}\Big{(}1+p_{1}^{\theta-1}\Big{)}^{k-1}<p_{1}^{k-1}2^{k-1}. \tag{9}\]  
On the other hand, \(P>p_{1}^{k}\). From here  
\[\frac{P}{\widetilde{P}}>\frac{p_{1}^{k}}{p_{1}^{k-1}2^{k-1}}=\frac{p_{1}}{2^{k- 1}} \tag{10}\]  
To comply with the condition \(P>\widetilde{P}\) it is necessary to fulfill the inequality \(P/\widetilde{P}>1\). This inequality will necessarily hold if the inequality \(\frac{p_{1}}{2^{k-1}}>1\), or which is equivalent, \(p_{1}>2^{k-1}\). The statement is proven. \(\square\)  
In other words, the DT-RRNS is applicable when choosing a module \(p_{1}\) at the parameter generation stage such that \(p_{1}>2^{k-1}\). **Statement 2. **_For DT-RRNS, when combining the shares of an unauthorized subset of participants of \(\widetilde{I}\), the cardinality of the enumeration set \(f\left(\widetilde{I}\right)\) is determined by the expression:_  
\[\left\lfloor\frac{P}{\widetilde{P}}\right\rfloor\leq f\left(\widetilde{I} \right)\leq\left\lfloor\frac{P}{\widetilde{P}}\right\rfloor+1, \tag{11}\]_where \(\widetilde{P}=\prod_{i\in\widetilde{I}}p_{i}\)._  
**Proof.