# A Class of Perfectly Secret Autonomous Low-Bit-Rate Voice Communication Systems
## 4 LSP-Based LP Synthesizer
Then, the noise entropy \(H(\xi)\) per one sample of the signal is equal to_  
\[H(z)=\frac{1}{A(z)}=\frac{1}{1-\sum_{i=1}^{p}a_{i}z^{-i}}, \tag{7}\]6  
\[H(z)=\frac{1}{A(z)}=\frac{1}{1-\sum_{i=1}^{p}a_{i}z^{-i}}, \tag{7}\]7  
_where \(n\) is the number of bits used to encode signal samples, \(\xi\) is the noise in the interval \(\big{[}-A_{\xi},A_{\xi}\big{]}\), \(h_{LSP}\) is the impulse response of the LP synthesis filter, and \(\|\cdot\|\) has the meaning of the Euclidean norm operator._  
Based on Theorem 8.3.1 [38], the entropy of a continuous Riemann integrable random variable \(X\), of probability density \(f(x)\), quantized with n bits is  
\[H(z)=\frac{1}{A(z)}=\frac{1}{1-\sum_{i=1}^{p}a_{i}z^{-i}}, \tag{7}\]8Let us first prove (18). Let the noise \(\xi\) be uniformly distributed in the interval \(\big{[}-A_{\xi},\ A_{\xi}\big{]}\). Then the first term in (20) is equal to  
\[H(z)=\frac{1}{A(z)}=\frac{1}{1-\sum_{i=1}^{p}a_{i}z^{-i}}, \tag{7}\]9  
while the second term is equal to the number of bits encoding the noise signal in the range \(\big{[}-A_{\xi},\ A_{\xi}\big{]}\). Since it is equal to the number of occupied quantization levels, we obtain  
\[A(z)=1-\sum_{i=1}^{p}a_{i}z^{-i}. \tag{8}\]0  
We have thus proved the correctness of statement (18). To prove statement (19), it is sufficient to directly follow the definition of SNR [dB], namely  
\[A(z)=1-\sum_{i=1}^{p}a_{i}z^{-i}. \tag{8}\]1  
since  
\[A(z)=1-\sum_{i=1}^{p}a_{i}z^{-i}. \tag{8}\]2  
From (23), solving for \(A_{\xi}\), we obtain (19), which completes the proof. \(\square\)  
The log notation specifically refers to \(\log_{2}\), unless explicitly stated otherwise. **Remark 4**. : _Based on Lemma 2, it follows that with the appropriate choice of SNR, we can control the size of the innovative entropy \(H(\xi)\) and its dominance in relation to \(H(LSP)\). Note that \(H(LSP)\) is a fixed quantity equal to the number of bits used to encode the LSP parameters. In the case of the MELPe vocoder, its corresponding rate \(R_{MELPe}\) (6) is equal to 1.2 kb/s._  
* Is there an [PERSON] strategy that provides her additional information about the generated keys compared to the classical sources of common randomness (CR)? In systems with the application of the SKD protocol over classic CR sources, the basic quality criterion of the system is the amount of information \(I(K_{i},e_{i})\) that [PERSON] can obtain about the generated secret keys following the communication over the public channel. Formally,  
\[A(z)=1-\sum_{i=1}^{p}a_{i}z^{-i}. \tag{8}\]3  
where \(w_{i}\) is a random n-bit string with uniform distribution over \(\left\{0,1\right\}^{n}\) at the output of the optimal Huffman encoder [39], \(e_{i}\) is a particular value of optimal [PERSON]'s estimate of \(w_{i}\), while \(K_{i}=G(w_{i})\) is a distilled secret key. The \(G\) is chosen at random from a universal class of hash functions from \(\left\{0,1\right\}^{n}\) to \(\left\{0,1\right\}^{|K_{i}|}\)[40]. According to well-known results from [41], specifically Corollary 4, [PERSON]'s information about \(K_{i}\) for specific \(e_{i}\) and \(G\) decreases exponentially in the excess compression \(c-|K_{i}|\)  
\[A(z)=1-\sum_{i=1}^{p}a_{i}z^{-i}. \tag{8}\]4  
where \(c\) is the lower bound of [PERSON]'s conditional Renyi entropy of order two (so-called collision entropy) about \(w_{i}\), i.e.,  
\[A(z)=1-\sum_{i=1}^{p}a_{i}z^{-i}. \tag{8}\]5However, since, in the APS-VCS system, \(\hat{S}_{IA}\) and \(\hat{S}_{IB}\) depend on previously distilled secret keys, it is necessary to examine whether [PERSON]'s information \(I(K_{i-1},e_{i})\) about the \(K_{i-1}\) is also negligibly small. According to (26), it holds  
\[A(z)=1-\sum_{i=1}^{p}a_{i}z^{-i}. \tag{8}\]6  
where \(c^{*}\) is the lower bound of [PERSON]'s conditional Renyi entropy of order two about \(w_{i}\), i.e.,  
\[A(z)=1-\sum_{i=1}^{p}a_{i}z^{-i}. \tag{8}\]7  
Note that, in (28), by \(K_{i}^{*}\), we denote the distilled secret keys, when [PERSON] possesses some information about \(K_{i-1}\).