# A Class of Perfectly Secret Autonomous Low-Bit-Rate Voice Communication Systems
## 4 LSP-Based LP Synthesizer  
In the open communication phase, the analyzer and synthesizer of the built-in MELPe vocoder are used to form the \(CR(\hat{S}_{l},\,\hat{S})\) source and distill the secret keys used to initially fill the FIFO memories. For the purposes of protected communication, it is necessary to form a \(CR(\hat{S}_{lA},\hat{S}_{lB})\) source based on the LP synthesizer, which can be much simpler than the MELPe synthesizer. Namely, the complexity of the MELPe synthesizer originates from the complex process of forming the excitation signal in order to meet the demanding criteria of intelligibility and naturalness of the synthesized speech. This requirement has no significance in the formation of synthesized signals \(\hat{S}_{lA}\) and \(\hat{S}_{lB}\). Therefore, the excitation is greatly simplified and consists of a periodic train of unit pulses, with possibly controlled jittering and the addition of locally generated purely random noise. For this purpose, in the system experimental evaluation, we used vTRNG based on a natural process entropy source with a built-in randomness checking system; see Figure 6. Figures 10 and 11 show a generic scheme generator of locally synthesized signals based on random LSP parameters, periodic pulse input, and additive noise at the input and the output to the LP synthesizer filter, respectively. The LP synthesis filter is provided by the transfer function  
\[H(z)=\frac{1}{A(z)}=\frac{1}{1-\sum_{i=1}^{p}a_{i}z^{-i}}, \tag{7}\]  
where \(p\) is the order of the LP filter  
\[A(z)=1-\sum_{i=1}^{p}a_{i}z^{-i}. \tag{8}\]  
Figure 9: Logarithm ratio of \(\frac{CrossCorr(S_{l},\,\hat{S})}{CrossCorr(S,\,\hat{S})}\) for all 24 test samples of the speech signals. Only in the case of test sample No. 3 is this ratio less than 1, which means that the cross-correlation \(CrossCorr(\hat{S}_{l},\,\hat{S})\) is almost always higher than the cross-correlation \(CrossCorr(S,\,\hat{S})\). If the LP filter is of the minimum phase, i.e., if all its zeros are inside the unit circle in the Z plane, the LP synthesis filter \(H(z)\) is stable. The coefficients of the LP filter \(\{a_{i}\}\) are obtained based on the loaded set of LSP parameters from the FIFO memory  
\[\left\{\varphi_{1},\theta_{1},\varphi_{2},\theta_{2},\ldots,\varphi_{\frac{F}{2 }},\theta_{\frac{F}{2}}\right\} \tag{9}\]  
subject to restriction  
\[0<\varphi_{1}<\theta_{1}<\varphi_{2},<\theta_{2},\ldots,\varphi_{\frac{F}{2}}< \theta_{\frac{F}{2}}<\pi. \tag{10}\]  
As is known [20], the LP filter \(A(z)\) can be decomposed in the form  
\[A(z)=\frac{1}{2}[P(z)+Q(z)] \tag{11}\]  
where \(P(z)\) and \(Q(z)\) are the so-called even and odd polynomials defined by LS frequencies (10)  
\[P(z)=(1+z^{-1})\prod_{i=1}^{\frac{F}{2}}(1-e^{-j\varphi_{i}}z^{-1})(1-e^{j \varphi_{i}}z^{-1}) \tag{12}\]  
\[Q(z)=(1-z^{-1})\prod_{i=1}^{\frac{F}{2}}(1-e^{-j\varphi_{i}}z^{-1})(1-e^{j \varphi_{i}}z^{-1}). \tag{13}\]  
If we know the LSP parameters (from (9)) by replacing (12) and (13) in (11), and then in (7), we obtain the LP synthesis filter \(H(z)\). This allows for the synthesis of the signals \(\hat{S}_{IA}\) and \(\hat{S}_{IB}\), used in the SKD protocol in the protected phase of system operation. Figure 10: Generator of locally synthesized signals \(\hat{S}_{IA}\) based on random LSP parameters, periodic pulse input \(\delta\), and additive noise \(\xi\) at the input to the LP synthesizer filter. **Remark 2**. : _If and only if the condition (10) is strictly satisfied, i.e., if all zeros of the polynomials \(P(z)\) and \(Q(z)\) alternate in the range \((0,\pi)\), it can be shown that the LP filter \(A(z)\) generated in this way is of minimum phase [34; 35].