# Quantum Stream Cipher Based on Holevo-Yuen Theory
## 4 Survey of the Mathematical Security Analysis  
### The Main Story of Security  
In the conventional symmetric key cipher, we have  
\[H(C\mid X_{s}f(K_{s}))=0 \tag{2}\]  
where \(X\) is plaintext, \(K_{s}\) is secret key, \(f(K_{s})\) corresponds to running key and \(|f(K_{s})|\gg|K_{s}|\), and \(C\) is ciphertext. However, in physical cipher system, the eavesdropper cannot do anything without obtaining the ciphertext from the physical signal. In the case of the Y-00 scheme, the eavesdropper has no other way but to observe the non-orthogonal signal, because the Y-00 signals corresponding to the ciphertext in the symmetric key cipher are an ensemble of non-orthogonal quantum states. Thus, the ciphertext that the eavesdropper can obtain are randomized by its quantum nature for any quantum processing by several quantum no-go theorems developed by [PERSON] and [PERSON]. This result means that the ciphertext cannot be determined correctly, even if the eavesdropper obtains the secret key \(K_{s}\) and the plaintext \(X\). That is,  
\[H(C\mid X,f(K_{s}))\
eq 0 \tag{3}\]  
This is the definition of so called \"Random Cipher\". Thus, Y-00 scheme is a typical example of the random cipher. Here, let us describe the security evaluation in the practical setting based on two issues. (i) The first issue:  
The first issue was raised by the community of cryptology. The question of the cryptocommunity is how to formulate the error or correct estimation of ciphertext based on closeness between the sequence of ciphertext from the Y-00 signals received by the eavesdropper and a true random number sequence. Let us consider a quantum trace distance between density operators on the tensor product Hilbert space that corresponds to the ideal random sequence and the random sequence received by the eavesdropper. It can be denoted by following form, based on the Holevo-Yuen theory on quantum detection:  
\[\Delta_{q}=\max_{\Pi\Pi}Tr\Pi(\sum_{y}p(y)\rho_{C^{\dagger}C^{ \sharp}}^{y}-\rho_{C^{\dagger}}\otimes\rho_{C^{\sharp}}) \tag{4}\] \[\Pi:POVM\]  
In this case, \(C^{\dagger}\) is the ideal ciphertext, and \(C^{E}\) is the output of the [PERSON]'s receiver. Then, \(\rho_{C^{\dagger}}\) corresponds to the density operator for ideal randomness, and that of [PERSON] is \(\rho_{C^{\sharp}}\) which depends on the randomization based on quantum noise effect and the artificial scheme designed in the Y-00 scheme. Closeness of the ciphertext sequence of the eavesdropper to a true random number based on the above equation is evaluated as follows [18]:  
**Theorem 1**. : _Trace distance is bounded by Holevo information, as follows:_  
\[\Delta_{q}^{2}\leq B\chi(\epsilon) \tag{5}\]  
_where \(B\) is a constant depending on the definition of relative entropy, and \(\chi(\epsilon)\) is Holevo information from the channel to the eavesdropper._  
\[\chi(\epsilon)=S(\rho_{C^{\sharp}})-\sum_{y}p(y)S(\rho_{C^{\sharp}}^{y}) \tag{6}\]  
_where \(\mathcal{S}(\rho)\) is the von Neumann entropy. The above Holevo information is a decrease function by the appropriate randomization technique under the fixed \(M\)._  
_Next, the probability that an eavesdropper can estimate the ciphertext \(y=\alpha_{k}(X,f_{g}(K_{s}),R_{p})\) of Y-00 quantum stream cipher is given as follows. Let \(\Delta_{q}\) be the trace distance of the quantum density operators between an actual protocol and the ideal one. Then the average guessing probability for ciphertext of Y-00 cipher is bounded as follows:_  
\[\frac{1}{N}\leq P_{guess}\leq\frac{1}{N}+\Delta_{q}\leq\frac{1}{N}+\sqrt{B \chi(\epsilon)} \tag{7}\]  
_where \(N=2^{|C_{y}|}\). \(|C_{y}|\) is the length of binary sequence converted from 2M-ary signal with the length \(m\) (number of slot). Thus, the guessing probability for the ciphertext \(y=\alpha_{k}(X,f_{g}(K_{s}),R_{p})\) is controlled by Holevo information. In conclusion, under the fixed number of \(N\), one can try to design the randomization technique such that \(\chi(\epsilon)\to 0\), and \(P_{guess}\to 1/N\).