## 6 Data Transmission Security and Reliability
Let us now compare the DT-RRNS with the Mignotte scheme. The basic design requirement of the Mignotte scheme is the inclusion of a secret \(s\) into the interval \(\left(\alpha=\prod_{i=0}^{k-2}p_{n-i},\beta=\prod_{i=1}^{k}p_{i}\right)\). The statements proved earlier regarding the size of the set of enumerations of the DT-RRNS allow us to deviate from this rule in favor of increasing the dynamic range of the secret representation. Based on the assumption of a uniform distribution of the secret in the interval \(\left[0,P\right)\), compactness of the set \(p_{0}<p_{1}<\ldots<p_{n}\) and a sufficiently large number \(p_{0}\), it is easy to show that the probability of a secret falling into the interval \(\left[0,\alpha\right)\) approaches the probability of \"guessing\" an arbitrary secret in the Asmuth-Bloom scheme. Note that in the notation used, \(\alpha=\widetilde{P}_{max}\) and \(\beta=P\). Indeed, the secret in the Asmuth-Bloom scheme is in the range \(\left[0,p_{0}\right)\) and is determined by \(p_{0}\). With a uniform distribution of the secret on this set, the probability of choosing an arbitrary secret is \(\frac{1}{p_{0}}\). On the other hand, the probability of a number falling into the interval \(\left[0,\alpha\right)=\left[0,\widetilde{P}_{max}\right)\) is equal to \(\frac{\left|\left[0,\widetilde{P}_{max}\right)\right|}{\left|\left[0,P\right] \right|\left[\frac{p_{0}}{P_{max}}\right]}\). According to Statement 3, for a sufficiently large \(p_{0}\), the magnitude \(\frac{P}{P_{max}}\) is equivalent to \(p_{0}\). It follows that parameters that are determined by DT-RRNS can eliminate restrictions imposed on the parameters of the Mignotte scheme. Let us consider examples of generating DT-RRNS parameters. **Example 1**. : _Let \(p_{1}=1024\), \(n=10\), and \(k=5\), and let it be required that the deviation from the Asmuth-Bloom search power does not exceed 10%. Determine what should be \(\theta\) in this case. According to estimates (24) and (28), we obtain:_  
\[\frac{P}{\widetilde{P}}>\frac{p_{1}^{k}}{p_{1}^{k-1}2^{k-1}}=\frac{p_{1}}{2^{k- 1}} \tag{10}\]5  
_whereas a required deviation. In our case,\(\epsilon=0.05\)._  
_Substituting the available data into the formula, we have \(\theta<0.477\)._  
_Consequently, the numbers that provide the required cardinality of the enumeration set must be within the interval \(\left[1024,1051\right]\). Using Statement 4, we have_  
\[\frac{P}{\widetilde{P}}>\frac{p_{1}^{k}}{p_{1}^{k-1}2^{k-1}}=\frac{p_{1}}{2^{k- 1}} \tag{10}\]6  
Fewer unique divisors of \(p_{1}\), the more beneficial to use them for building a compact sequence. It increases the number of the coprime numbers in the interval from \(p_{1}\) to \(2p_{1}\). It is worth noting that the proof of the possibility of generating compact sequences is a difficult number-theoretic problem. The generation of a variety of compact sequences is the subject of further research. Now, we can limit ourselves to practical recommendations, which consist in choosing sufficiently large \(p_{1}\) and with the least number of divisors. Statements 1-4 evaluate the security of the DT-RRNS scheme. First, according to Statement 1, \(p_{1}>2^{k-1}\) determines a lower bound of \(p_{1}\). For maximum security, \(p_{1}\) must be significantly higher than \(2^{k-1}\). Secondly, an important parameter of the scheme is the value \(\theta\), defining a compact sequence. The closer \(\theta\) to zero, the better the SSS properties in terms of security, which follows from Statement 4 and inequality (24).