# A Class of Perfectly Secret Autonomous Low-Bit-Rate Voice Communication Systems
## 2 System Architecture
The cryptographic part of the system consists of the KSG(K) generator of the binary pseudorandom sequence \(C_{K}\), which is added modulo 2 with the binary sequence that comes from the analyzer. \(CR(S,\ \hat{S})\) is denoted as the source of common randomness, formed from the input speech signal \(S\) on the transmitter side and the speech signal \(\hat{S}\) synthesized on the receiver side. Figure 2: Extension of the generic scheme shown in Figure 1 with local MELPe synthesis on the transmitter side. \(CR(\hat{S}_{l},\ \hat{S})\) is denoted as the source of common randomness, formed from the locally synthesized speech signal \(\hat{S}_{l}\) on the transmitter side and the speech signal \(\hat{S}\) synthesized on the receiver side. **Lemma 1**: _Let the initial strings \(X\) and \(Y\), owned by [PERSON] and [PERSON] at the beginning of the SKD protocol, be binary iid random sequences of length \(N\) at Hamming distance \(\varepsilon\), \(\varepsilon\in[0,\ 0.5]\). Then, the expected number of parity bits that [PERSON] exchanges with [PERSON] over the public channel is provided by_  
\[N_{ADparity}=\left\lfloor\frac{N}{2}\right\rfloor+\sum_{i=1}^{s-1}\left\lfloor \frac{N}{2^{i+1}}\cdot\frac{\varepsilon^{2^{i}}+\left(1-\varepsilon\right)^{ 2^{i}}}{\prod\limits_{j=0}^{i-1}\left(\varepsilon^{2^{j}}+\left(1-\varepsilon \right)^{2^{j}}\right)}\right\rfloor, \tag{1}\]  
_where \(s\) is the number of iterations of the BP algorithm._  
The proof follows directly from the fact that the total amount of parity bits exchanged in the BP algorithm is equal to the sum of exchanges for each iteration. On the other hand, for each iteration, this value is equal to half the length of the sequences at the beginning of the iteration. This length is also equal to the length of the sequences at the end of the previous iteration. In [32], Theorem 2.2.3, p. 17, an expression for the compression rate of the BP algorithm in each iteration is provided. Based on this formula, we obtain the length of the sequences after \(i\) iterations i.e., \(\frac{N}{2^{i+1}}\cdot\frac{\varepsilon^{2^{i}}+\left(1-\varepsilon\right)^{ 2^{i}}}{\prod\limits_{j=0}^{i-1}\left(\varepsilon^{2^{j}}+\left(1-\varepsilon \right)^{2^{j}}\right)}\). Summarizing these values over all iterations, we obtain statement (1). Note that the smallest integer operator \(\left\lfloor\cdot\right\rfloor\) is applied due to the very nature of parity checking of 2-bit blocks. This completes the proof of the Lemma. \(\square\)  
The number of bits of significance for an eavesdropper may be slightly less than \(N_{ADparity}\) since some parity equations may be linearly dependent. However, as the \(\left\lfloor\frac{N}{2}\right\rfloor\) parity bit in the first iteration of the BP algorithm is mutually linearly independent, it always holds  
\[N_{ADparity}\geq\left\lfloor\frac{N}{2}\right\rfloor. \tag{2}\]  
This means that the uncertainty of the speech signal at the input to the system is at least halved, as viewed from the eavesdropper's side. In Figure 3, an example of the dependence of \(N_{ADparity}\) as a function of \(\varepsilon\) is shown for \(N=8640\) and \(s=1,\ 2,\ 3,\ 4,\ 5\). Figure 3: An example of the dependence of the number of parity bits exchanged over the public channel in AD phase, as a function of the initial Hamming distance of [PERSON] and [PERSON]’s sequence. SKD algorithm parameters: initial sequence length \(N=8640\), number of iterations of the BP algorithm \(s=1,\ 2,\ 3,\ 4,\ 5\). **Remark 1**. : _Since our goal is a perfectly secret system, we can conclude that the first two candidate sources of common randomness cannot be used during protected communication, but only in the open operating mode, which, as a rule, precedes the protected one. Namely, in the typical use of such systems, legitimate parties establish a communication link through usual open communication. After checking the connection quality and mutual consent of the parties, the switch is made to encrypted communication.