# Synthesis of Sources of Common Randomness Based on Keystream Generators with Shared Secret Keys
## 3 System Architecture
[PERSON]'s optimal strategy for selecting the sequence \(X_{E}\) is  
\[Y=X\ \oplus\ C_{K} \tag{1}\]3  
which is structurally identical to [PERSON]'s and [PERSON]'s sequences \(X_{A}\) and \(X_{B}\), but generated with her own encryption key \(C_{KE}\). Then, [PERSON]'s equivalent binary noise \(\tilde{E}_{E}\) relative to the satellite signal \(C_{K}\) is given by  
\[Y=X\ \oplus\ C_{K} \tag{1}\]4  
\[Y=X\ \oplus\ C_{K} \tag{1}\]5  
which has a Bernoulli distribution with parameter \(\varepsilon_{\tilde{E}}\) given by  
\[Y=X\ \oplus\ C_{K} \tag{1}\]6  
\[Y=X\ \oplus\ C_{K} \tag{1}\]7 Equivalent Maurer satellite scenario representation within the proposed SKD system. The satellite represents the shared-key keystream generator \(KSG(K_{G})\), which broadcasts the keystream \(C_{K}\). Independent binary additive noises \(E_{A}\), \(E_{B}\), and \(\tilde{E}_{E}\) (locally generated binary Bernoulli sequences) are added on [PERSON]’s, [PERSON]’s, and [PERSON]’s channels, respectively, illustrating how the system enables controlled correlation between legitimate parties while preserving secrecy against an adversary without the secret key \(K_{G}\). **Remark 2**. : _In the analyses presented, see (8)-(14), we have, among other considerations, employed the well-known result that the modulo-two sums of a binary sequence distributed as \(B(0.5)\) with an independent sequence distributed as \(B(\varepsilon)\), for any \(\varepsilon\in[0,1]\), results in a sequence that remains distributed as \(B(0.5)\). This result is a classical property in information theory, particularly within the analysis of the Binary Symmetric Channel (BSC), where it is well known that if the channel input is uniformly distributed (i.e., each bit is \(B(0.5)\)) and the channel noise is an independent \(B(\varepsilon)\) random variable, the channel output remains uniformly distributed regardless of the parameter \(\varepsilon\). This property ensures that an adversary observing the output gains no information about the input, which underlies the statement that the \"most secure change\" of a sequence is achieved when each bit changes with probability 0.5. Although this property is often considered folklore, it is systematically presented in standard references such as [26; 27; 28]. Additionally, we would like to point out that this result aligns with the random-in/random-out property of non-expanding ciphers, as discussed by [PERSON] [29], where it is shown that the cascade of a binary symmetric source (BSS), corresponding to a \(B(0.5)\) source, with a non-expanding cipher (such as additive stream ciphers or block ciphers) yields an output that is also a BSS, regardless of the secret key distribution. This implies that the ciphertext sequence is statistically independent of the secret key, guaranteeing that no information about the key can be obtained from the ciphertext alone, no matter how much ciphertext is observed by an adversary. This further supports the security principle that the highest level of uncertainty--and thus security--is achieved when bit changes occur with probability 0.5._  
It is well known that, in the case of the satellite scenario (see [1]), the secrecy capacity, thus the maximum length of a secret key that can be distilled in the presence of an eavesdropper, is  
\[Y=X\ \oplus\ C_{K} \tag{1}\]8  
where \(I(X_{A};X_{B})\) denotes the mutual information between \(X_{A}\) and \(X_{B}\), while \(I(X_{A};X_{B}\ |X_{E})\) denotes this mutual information conditioned by \(X_{E}\). Taking into account (8) and (10), we conclude that [PERSON]'s sequence \(X_{E}\) is independent of [PERSON] and [PERSON] sequence \(X_{A}\) and \(X_{B}\), which reduces (15) to  
\[Y=X\ \oplus\ C_{K} \tag{1}\]9  
The advantage of the SKD strategy, which consists of applying PA to the result of the IR protocol, lies in achieving all secret key rates below the secrecy capacity \(C_{S}^{max}\), as well as its explicit practical implementation [1].