# Synthesis of Sources of Common Randomness Based on Keystream Generators with Shared Secret Keys
## 4 Information-Theoretic Analysis of the Source of Common Randomness Based on KSG(Kg)
\(\Box\)  
Remark 3: To assist the reader, we emphasize that Theorem 1 provides a lower bound on the uncertainty (equivocation) that an eavesdropper retains about the secret key even after observing all public communication. This result is crucial, as it ensures that the secret key \(K_{G}\) maintains a provable level of secrecy regardless of public leakage. The bound directly quantifies how the entropy of the key is preserved, depending on the properties of the system's injected randomness and the structure of the reconciliation masks. Remark 4: Knowing the lower bound of the equivocation of the secret key \(K_{G}\) allows us to formulate conditions under which this bound never falls below a predetermined threshold \(\Delta_{K_{G}}\). According to (20), if the following condition is satisfied:  
\[|K_{G}|+\sum_{i=1}^{n_{p}}h_{b}(\varepsilon_{i})-n_{p}\geq\Delta_{K_{G}},\ 0 \leq\Delta_{K_{G}}\leq|K_{G}|, \tag{31}\]  
then, based on the observations \(S=\{S_{i}\}\), \(|S|=n_{p}\), from the public channel during the execution of the given IR protocol, Eve cannot reduce the initial uncertainty of the key \(K_{G}\) below \(\Delta_{K_{G}}\), where \(0\leq\)\(\Delta_{K_{G}}\)\(\leq|K_{G}|\). Definition 1: The admissible region \(D\big{(}\varepsilon,\ \Delta_{K_{G}},\ K_{G},\ t,\ K_{S}\big{)}\) of the system operation shown in Figure 1 is defined by the following constraints:  
\[0<\Delta_{K_{G}}<K_{G} \tag{32}\]  
\[H\Big{(}K_{G}\mid S_{1},\ S_{2},\ldots,\ S_{n_{p}}\Big{)}\geq\Delta_{K_{G}} \tag{33}\]  
\[|K|=|K_{S}|+|K_{G}| \tag{34}\]  
\[|K|+t\leq n_{IR0}\cdot\mu_{IR} \tag{35}\]  
\[IE=n_{IR0}\cdot h_{b}(\varepsilon)+|K_{G}|\geq|K| \tag{36}\]  
\[\sum_{i=1}^{n_{p}}H(K_{G}\mid S_{i})=H\Big{(}K_{G}\mid S_{1},\ S_{2},\ \ldots,\ S_{n_{p}}\Big{)}-\big{(}n_{p}-1\big{)}H(K_{G}). \tag{28}\]0  
where \(t\) is the security margin, \(n_{IR0}\) is the length of initial Alice and Bob strings before starting IR protocol, while \(\mu_{IR}\) is so-called data remaining efficiency, defined as the ratio of the length \(n_{IR}\) of reconciliated sequences after completion of the IR protocol and the length of initial sequences, i.e.,  
\[\sum_{i=1}^{n_{p}}H(K_{G}\mid S_{i})=H\Big{(}K_{G}\mid S_{1},\ S_{2},\ \ldots,\ S_{n_{p}}\Big{)}-\big{(}n_{p}-1\big{)}H(K_{G}). \tag{28}\]1  
The \(1E\) denotes the so-called innovative or fresh entropy [38]. For the considered system, IE is actually the amount of entropy introduced into the system by the local source of randomness. Condition (32) ensures the selection of the lower bound of the equivocation of the keys \(K_{G}\), as given by (33). Condition (34) guarantees the autonomy of the system in maintaining this lower bound. This is achieved through regular refreshing of the keys\(K_{G}\) after each block of newly generated sequences \(\{Y_{i}\}\) of length \(n_{IR0}\), during which the system simultaneously distills keys \(K_{S}\) for independent use. Condition (35) establishes a relation between the length of the distilled keys \(|K|\), the security margin \(t\), which reflects the compression rate of the PA block, and the length of the reconciled sequences resulting from the IR phase [1; 35]. Condition (36) imposes the constraint on the length of the distilled keys and the amount of innovative entropy introduced in each system cycle. Through condition (37), the KDR is controlled. Typical values for \(t\) range from 30 to 40, while \(\delta\) is on the order of \(10^{-5}\). The admissible region \(D\) can also be defined using quantities normalized by the length \(n_{IR0}\), which allows clearer interpretation and visualization. Let us introduce the following notation for the normalized quantities of interest  
\[\sum_{i=1}^{n_{p}}H(K_{G}\mid S_{i})=H\Big{(}K_{G}\mid S_{1},\ S_{2},\ \ldots,\ S_{n_{p}}\Big{)}-\big{(}n_{p}-1\big{)}H(K_{G}). \tag{28}\]2  
These can now be interpreted as the corresponding rates.