# Synthesis of Sources of Common Randomness Based on Keystream Generators with Shared Secret Keys
## 3 System Architecture
In the SKD block, a suitably chosen IR algorithm is first applied, whose primary function is to eliminate differences between [PERSON]'s and [PERSON]'s sequences by exchanging various types of parity information over the available public authenticated channel [1; 4; 30]. As a result of this phase, [PERSON] and [PERSON] share identical sequences \(W_{i}\) with high probability (see Figure 1). In the final phase of the SKD protocol, PA is performed by [PERSON] and [PERSON] agreeing over the public channel on the choice of a compression function \(G\), to distill from \(W_{i}\) a shorter string \(K_{i}\) about which [PERSON] has only a negligible amount of information [31; 32]. As a rule, it is achieved by applying a suitable hash function from the class of universal hash functions [1; 33; 34]. Universality means that for any two different inputs, the chance they collide (hash to the same output) is at most \(\frac{1}{\text{output space}}\). From Theorem 3 and Corollary 4 in [35], it follows that if the hash output length is at most the Renyi entropy of order 2 of the input minus a small security parameter, then the output is nearly indistinguishable from uniform. Moreover, if [PERSON] holds some side information about the reconciled sequence (the input to the hash function) after the IR phase, then her information about the hash output becomes negligibly small, provided her conditional Renyi entropy of order 2 is greater than the hash output length minus a small security parameter. In the splitter block, from the distilled key \(K_{i}\), the key \(K_{G,i}\) is selected to refresh the secret key of \(KSG(K_{G,i-1})\), resulting in the updated \(KSG(K_{G,i})\), while the remaining part, \(K_{Si}\), is reserved for subsequent use. From the system's end-use perspective, the generation rate of usable secret keys is equal to the generation rate of \(K_{Si}\). The refresh rate of \(KSG(K_{G,i})\), is chosen such that [PERSON]'s equivocation  
\[X\sim B(\varepsilon),\ \varepsilon=Pr\{X=1\},\ 0<\varepsilon\leq 1, \tag{2}\]0  
never falls below a predetermined margin \(\Delta_{K_{G}}\) for every operation cycle \(i\), where \(S\) denotes the total information that [PERSON] obtains by observing the public channel during the execution of the SKD protocol for the \(i\)-th block cycle. Summary of Assumptions Made in the Design of Proposed System  
Finally, let us summarize the assumptions made in the design of the proposed autonomous SKD systems from \[X\sim B(\varepsilon),\ \varepsilon=Pr\{X=1\},\ 0<\varepsilon\leq 1, \tag{2}\]1 The proposed architecture operates under the following assumptions:  
1. Pre-shared secret key \(K_{G}\) is available to [PERSON] and [PERSON] for initializing the keystream generator \(KSG(K_{G})\) and is refreshed securely in each cycle. 2. Both parties have trusted local sources of true randomness generating Bernoulli-distributed sequences with a tunable parameter \(\varepsilon\), enabling precise control over the BER in the system, thus emulating a synthetic Maurer satellite scenario. 3. The keystream generator \(KSG(K_{G})\) satisfies the avalanche criterion, ensuring that without knowledge of \(K_{G}\), the keystream appears statistically indistinguishable from a Bernoulli(0.5) sequence to an adversary. 4. An authenticated public channel is available for IR and PA, assumed to be observable but not modifiable by an adversary. 5. [PERSON] is assumed to have unlimited computational power (information-theoretic security model) and complete access to the public channel, but no access to \(K_{G}\) or the synchronized keystream. 6. The system assumes the use of efficient IR protocols capable of reconciling [PERSON]'s and [PERSON]'s sequences to enable secret key rates approaching the secrecy capacity. 7. PA is performed using universal hash functions, assuming sufficiently high conditional Renyi-2 entropy (or Shannon equivocation) of the reconciled sequences with respect to [PERSON]'s view to ensure negligible leakage. 8.