# A Class of Perfectly Secret Autonomous Low-Bit-Rate Voice Communication Systems
## 4 LSP-Based LP Synthesizer
Therefore, if we randomly select \(p\) LSPs from the range\((0,\pi)\) and sort them in ascending order, then conduct the above procedure to form the polynomials \(P(z)\), \(Q(z)\), and \(A(z)\), the resulting LP synthesis filter \(H(z)\) will be stable._  
Information-Theoretic Analysis of the Source of Common Randomness Based on Randomly Selected LSP Parameters  
If the proposed system provides negligible leakage of distilled secret keys to Eve (see Figures 4 and 12), then there is simultaneously negligible leakage of Vernam cipher secret keys in the main channel, i.e., the system is able to maintain perfect secrecy. \[H(z)=\frac{1}{A(z)}=\frac{1}{1-\sum_{i=1}^{p}a_{i}z^{-i}}, \tag{7}\]0 Generator of locally synthesized signals \(\hat{S}_{IA}\) based on random LSP parameters, periodic pulse input \(\delta\), and additive noise \(\xi\) at the output of the LP synthesizer filter. \[H(z)=\frac{1}{A(z)}=\frac{1}{1-\sum_{i=1}^{p}a_{i}z^{-i}}, \tag{7}\]1 Information flows of importance for the analysis of distilled secret keys in the system. The SKD protocol consists of the BP algorithm for the AD phase, the Winnow algorithm for the IR phase, and an optimal Huffman encoder. The PA phase is based on universal hash functions. To justify this claim, an appropriate information-theoretic analysis should answer the following two questions:  
* What is the entropy of the signal \(\hat{S}_{IA}\) and \(\hat{S}_{IB}\) at the output of the LP synthesizer, and what is its structure? Namely, the uncertainty of these signals originates from the uncertainty of the applied LSP coefficients, as well as additive purely random locally generated noise, regardless of whether it is located at the input or output of the synthesizer. Due to the closed loop that includes the LP synthesizer, SKD block, secret key FIFO buffer, and the LP synthesizer, if the local source of pure randomness had a negligible contribution to the entropy of the LP synthesizer output, the effective Key rate of generated secret keys for Vernam cipher would, over time, trend toward zero. Formally, it is valid  
\[H(z)=\frac{1}{A(z)}=\frac{1}{1-\sum_{i=1}^{p}a_{i}z^{-i}}, \tag{7}\]2  
where \(h_{LSP}\) is the impulse response of the LP synthesis filter \(H(z)\), \(\delta\) is periodic pulse input, \(\xi\) is additive pure random noise, and \(*\) is the convolutional operator. Based on classical results [36; 37], since  
\[H(z)=\frac{1}{A(z)}=\frac{1}{1-\sum_{i=1}^{p}a_{i}z^{-i}}, \tag{7}\]3  
it follows that the synthesis filter \(H(z)\) preserves the input entropy, i.e., that it holds  
\[H(z)=\frac{1}{A(z)}=\frac{1}{1-\sum_{i=1}^{p}a_{i}z^{-i}}, \tag{7}\]4  
regardless of whether there is additive noise at the input or output of the filter \(H(z)\). Further,  
\[H(z)=\frac{1}{A(z)}=\frac{1}{1-\sum_{i=1}^{p}a_{i}z^{-i}}, \tag{7}\]5  
since there is a 1-1 correspondence between the LSP and the \(\{a_{i}\}\) parameters of the filter \(H(z)\)[34]. **Remark 3**: _Considering (17), it is clear that \(H(\xi)\) must be significantly dominant with respect to \(H(\text{LSP})\) for \(\hat{S}_{IA}\) to maintain a sufficient level of \"innovation\" entropy necessary for the distillation of perfectly secret keys for the Vernam cipher in the main channel._  
The total entropy of synthesized signals (17) can be expressed as a function of Signal-to-Noise Ratio (SNR). **Lemma 2**: _Let Signal-to-Noise Ratio (SNR) be provided in dB.