# Synthesis of Sources of Common Randomness Based on Keystream Generators with Shared Secret Keys
## 5 Optimal SKD System Based on CR KSG and Winnow IR Protocol
Figure 6: Variation of normalized values \(\hat{n}_{p}\), \(\hat{n}_{b}\), and \(\hat{n}_{H}\), as functions of \(p_{IR0}\) after four iterations of the Winnow protocol. Figure 7: Variation of the final BER \(p_{IRend}\) between [PERSON]’s and [PERSON]’s sequences after the first four iterations of the Winnow protocol. In Figure 8 (top), the correspondence between the parameter \(p_{IRend}=\delta=10^{-4}\) and \(p_{IR0}(\delta)=0.0995\) is shown, which determines the upper bound of the admissible region \(\widetilde{D}^{W}\) along the \(p_{IR0}\) axis. Taking into account relation (7), the corresponding parameter is \(\epsilon(\delta)=0.0525\). Based on Theorem 3, it follows that the maximum value of \(\left|\widetilde{K}_{S}\right|_{max}\), regardless of the choice of \(\tilde{\Delta}_{K_{C}max}\), is achieved by finding the maximum of the function  
\[f(p_{IR0})=\Psi_{1l}{}^{Win}(p_{IR0}\,\mu_{IR})-\Psi_{L}{}^{Win}(p_{IR0},\mu_{ IR})\,\ p_{IR0}\in\widetilde{D}^{W}. \tag{81}\]  
Figure 8: Correspondence between the parameters \(p_{IRend}=\delta=10^{-4}\) and \(p_{IR0}(\delta)=0.0995\) (**top**). Admissible region for the key distillation rate \(CR\ KSG_{W}\) system, with Winnow protocol parameters \(N=8\), \(maxiter=4\), and an upper bound on the KDR of \(\delta\leq 10^{-4}\) (**middle**). The function \(f(p_{IR0})\) and its maximum \(f(p_{IR0 max})\), achieved at \(p_{IR0 max}=0.0905\), corresponding to \(\epsilon_{max}=0.0525\) (**bottom**). Due to the highly complex analytical expressions for \(\Psi_{U}{}^{Win}(p_{IR0}\,,\mu_{IR})\) and \(\Psi_{L}{}^{Win}(p_{IR0},\mu_{IR})\), this maximum can be obtained numerically. Figure 8 (bottom) shows the function \(f(p_{IR0})\) and its maximum \(f(p_{IR0 max})\), which is achieved at \(p_{IR0 max}=0.0905\), corresponding to \(\varepsilon_{max}=0.0525\). Figure 8 (middle) shows points \(A\) and \(B\) on \(\Psi_{L}{}^{Win}\) and \(\Psi_{U}{}^{Win}\), respectively, for the \(p_{IR0 max}\) value. The coordinates of these points are  
\[A(0.0905,\ C_{1})\,\ B(0.0905,\ C_{2}-\bar{t}), \tag{82}\]  
where \(C_{1}\) and \(C_{2}\) denote  
\[m_{Hi}=[m_{Hi1},\ m_{Hi2},\ \ldots,m_{Hi7}]\,\ \sum_{j=1}^{7}m_{Hij}=4\,\ i=1,2,3. \tag{78}\]0  
The value \(p_{IRend}(0.0905)=2.5822\cdot 10^{-5}\), indicates an extremely low probability of KDR. **Remark 10**. : _From Equation (60) and the definition of normalized values given in (39), we obtain_  
\[m_{Hi}=[m_{Hi1},\ m_{Hi2},\ \ldots,m_{Hi7}]\,\ \sum_{j=1}^{7}m_{Hij}=4\,\ i=1,2,3. \tag{78}\]1  
_and based on Equations (59) and (39), we obtain_  
\[m_{Hi}=[m_{Hi1},\ m_{Hi2},\ \ldots,m_{Hi7}]\,\ \sum_{j=1}^{7}m_{Hij}=4\,\ i=1,2,3. \tag{78}\]2  
\[m_{Hi}=[m_{Hi1},\ m_{Hi2},\ \ldots,m_{Hi7}]\,\ \sum_{j=1}^{7}m_{Hij}=4\,\ i=1,2,3. \tag{78}\]3  
_where the constants \(C_{1}\) and \(C_{2}\) are given in Equation (83). For sufficiently large \(n_{IR0}\) and sufficiently small \(r\), from (84) we conclude that the asymptotic value of the maximal effective secrecy capacity is equal to \(\left|\widetilde{K}_{s}\right|_{max}=C_{2}-C_{1}=0.2272\), or \(22.72\%\)._  
The effective secrecy capacity (84) can also be expressed in terms of \(K_{G}\), by substituting (85) into (84), which gives  
\[m_{Hi}=[m_{Hi1},\ m_{Hi2},\ \ldots,m_{Hi7}]\,\ \sum_{j=1}^{7}m_{Hij}=4\,\ i=1,2,3. \tag{78}\]4  
The synthesis of the optimal \(CR\)\(KSG_{W}\) system begins with the selection of the parameter \(r\) and \(K_{G}\), which according to (87) uniquely determine the secrecy capacity \(\left|\widetilde{K}_{S}\right|\); see \[m_{Hi}=[m_{Hi1},\ m_{Hi2},\ \ldots,m_{Hi7}]\,\ \sum_{j=1}^{7}m_{Hij}=4\,\ i=1,2,3. \tag{78}\]5  
The selection of the secret key length \(K_{G}\) can be made based on various practical considerations. For example, if a standard KSG is used, such as AES, DES, or 3 DES, which typically comes with predefined key lengths [41], then one of the available options is chosen. This value uniquely determines the base block length \(n_{IR0}\), according to (85). Figure 10 shows the functions \(n_{IR0}(r)\), for several typical values of \(|K_{G}|\). **Example 1**.