Security Analysis of Phase Encoding Continuous-Variable Quantum Key Distribution Based on K-Nearest Neighbor

Changlan Zhao; Tianyi Wang

doi:10.3788/LOP222511

Journals >Laser & Optoelectronics Progress >Volume 60 >Issue 19 >Page 1927002 > Article

Laser & Optoelectronics Progress
Vol. 60, Issue 19, 1927002 (2023)

Security Analysis of Phase Encoding Continuous-Variable Quantum Key Distribution Based on K-Nearest Neighbor

Changlan Zhao and Tianyi Wang^*

Author Affiliations

College of Big Data and Information Engineering, Guizhou University, Guiyang 550025, Guizhou, China

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DOI: 10.3788/LOP222511 Cite this Article Set citation alerts

Changlan Zhao, Tianyi Wang. Security Analysis of Phase Encoding Continuous-Variable Quantum Key Distribution Based on K-Nearest Neighbor[J]. Laser & Optoelectronics Progress, 2023, 60(19): 1927002 Copy Citation Text

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Abstract

This paper proposes a phase encoding, continuous-variable quantum key distribution quantum-state identification method based on the K-nearest neighbor algorithm. The algorithm achieves recognition using the phase features of the accepted quantum states, which are first learned from a training set consisting of known coherent states and then classified according to the phase features extracted from the unknown quantum states. Moreover, this paper derives the secure code rate of the K-nearest neighbor-based identification method under collective attack and reverse coordination and compares the performance of the method applied to four-state and eight-state protocols under different transmission distances, modulation variance, and excess noise. The results of numerical simulation results show that the method can effectively generate secure keys with a transmission distance of 250 km when the secure code rate is 10^-5 bit per symbol.

Keywords

continuous variables K-nearest neighbors phase encoding quantum key distribution

\begin{array}{l} S_{4} = \{|α e x p {(i π / 4)}^{}〉, |α e x p {(3 i π / 4)}^{}〉, \\ |α e x p {(5 i π / 4)}^{}〉, |α e x p {(7 i π / 4)}^{}〉\}, \end{array}

（1）

\begin{array}{l} S_{8} = \{|α〉, |α e x p (i π / 4)〉, |α e x p (i π / 2)〉, \\ |α e x p (3 i π / 4)〉, |α e x p (i π)〉, |α e x p (5 i π / 4)〉, \\ |α e x p (3 i π / 2)〉, |α e x p (7 i π / 4)〉\} \end{array}

，（2）

ρ_{d} = \sum_{i = 0}^{d} λ_{i} |ϕ_{i} > < ϕ_{i}|, d \in \{4,8\}

。（3）

ρ_{4} = \sum_{i = 0}^{3} λ_{i} |ϕ_{i}〉 〈ϕ_{i}|

。（4）

\{\begin{array}{l} λ_{0,2} = \frac{1}{2} e x p {(- α^{2})}^{} (c o s h α^{2} \pm c o s α^{2}) \\ λ_{1,3} = \frac{1}{2} e x p (- α^{2}) (s i n h α^{2} \pm s i n α^{2}) \end{array}

，（5）

\begin{array}{l} |ϕ_{k}〉 = \frac{e x p {(- α^{2} / 2)}^{}}{\sqrt[]{λ_{k}}} \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} α^{4 n + k}}{\sqrt[]{(4 n + k)!}} |4 n + k〉, \\ k \in \{0,1, 2,3\} 。 \end{array}

（6）

ρ_{8} = \sum_{i = 0}^{7} λ_{i} |ϕ_{i}〉 〈ϕ_{i}|

。（7）

\{\begin{array}{l} λ_{0,4} = \frac{1}{4} e x p (- α^{2}) (c o s h α^{2} + c o s α^{2} \pm 2 c o s \frac{α^{2}}{\sqrt[]{2}} c o s h \frac{α^{2}}{\sqrt[]{2}}) \\ λ_{1,5} = \frac{1}{4} e x p (- α^{2}) (s i n h α^{2} + s i n α^{2} \pm \sqrt[]{2} c o s \frac{α^{2}}{\sqrt[]{2}} s i n h \frac{α^{2}}{\sqrt[]{2}} \pm \sqrt[]{2} s i n \frac{α^{2}}{\sqrt[]{2}} c o s h \frac{α^{2}}{\sqrt[]{2}}) \\ λ_{2,6} = \frac{1}{4} e x p (- α^{2}) (c o s h α^{2} - c o s α^{2} \pm 2 s i n \frac{α^{2}}{\sqrt[]{2}} s i n h \frac{α^{2}}{\sqrt[]{2}}) \\ λ_{3,7} = \frac{1}{4} e x p (- α^{2}) (s i n h α^{2} - s i n α^{2} \mp \sqrt[]{2} c o s \frac{α^{2}}{\sqrt[]{2}} s i n h \frac{α^{2}}{\sqrt[]{2}} \pm \sqrt[]{2} s i n \frac{α^{2}}{\sqrt[]{2}} c o s h \frac{α^{2}}{\sqrt[]{2}}) \end{array}

，（8）

\begin{array}{l} |ϕ_{k}〉 = \frac{e x p (- α^{2} / 2)}{\sqrt[]{λ_{k}}} \sum_{n = 0}^{\infty} \frac{α^{8 n + k}}{\sqrt[]{(8 n + k)!}} |8 n + k〉, \\ k \in \{0,1, 2, \dots, 7\} 。 \end{array}

（9）

Γ_{d} = (\begin{matrix} V I & Z_{d} σ_{z} \\ Z_{d} σ_{z} & V I \end{matrix})

，（10）

p^{'} = \sqrt[]{T} α c o s φ + N (0, T ε),

（11）

q^{'} = \sqrt[]{T} α s i n φ + N (0, T ε) 。

（12）

θ_{a, b} = a r c t a n \frac{y_{1}}{x_{1}} - a r c t a n \frac{y_{2}}{x_{2}} 。

（13）

C_{j} = \sum_{(|x^{*}〉, Y^{*}) \in N (|x〉)} ⟦y_{j} \in Y^{*}⟧

，（14）

h (|x〉) = \{y_{j} ∣ P (H_{j} ∣ C_{j}) / P ({\bar{H}}_{j} ∣ C_{j}) > t (|x〉), 1 ⩽ j ⩽ l\}

。（15）

f (|x〉, y_{j}) = \frac{P (H_{j} ∣ C_{j})}{P ({\bar{H}}_{j} ∣ C_{j})} = \frac{P (H_{j}) \cdot P (C_{j} ∣ H_{j})}{P ({\bar{H}}_{j}) \cdot P (C_{j} ∣ {\bar{H}}_{j})},

（16）

P (H_{j}) = \frac{s + \sum_{i = 1}^{m} ⟦y_{j} \in Y_{i}⟧}{s \times 2 + m}, (1 \leq j \leq l),

（17）

P ({\bar{H}}_{j}) = 1 - P (H_{j}), (1 \leq j \leq l),

（18）

\{\begin{matrix} ς_{j} [r] = \sum_{i = 1}^{m} ⟦y_{j} \in Y_{i}⟧ \cdot ⟦ψ_{j} (|x_{i}) = r⟧, (0 \leq r \leq k) \\ {\bar{ς}}_{j} [r] = \sum_{i = 1}^{m} ⟦y_{j} \notin Y_{i}⟧ \cdot ⟦ψ_{j} (|x_{i}) = r⟧, (0 \leq r \leq k) \end{matrix}

。（19）

ψ_{j} (|x_{i}〉) = \sum_{(|x^{*}〉, Y^{*}) \in N (|x^{*}〉)} ⟦y_{j} \in Y^{*}⟧,

（20）

\{\begin{matrix} P (C_{j} ∣ H_{j}) = \frac{s + ς_{j} [C_{j}]}{s \times (k + 1) + \sum_{r = 0}^{k} ς_{j} [r]} \\ P (C_{j} ∣ {\bar{H}}_{j}) = \frac{s + {\bar{ς}}_{j} [C_{j}]}{s \times (k + 1) + \sum_{r = 0}^{k} {\bar{ς}}_{j} [r]} \end{matrix}

，（21）

K = β I_{A B} - χ,

（22）

I_{A B} = l o g_{2} \frac{V + χ_{t o t}}{1 + χ_{t o t}},

（23）

χ = \sum_{i = 1}^{2} G (\frac{λ_{i} - 1}{2}) - \sum_{i = 3}^{5} G (\frac{λ_{i} - 1}{2}),

（24）

\{\begin{array}{l} λ_{1,2}^{2} = \frac{1}{2} [A \pm \sqrt[]{A^{2} - 4 B}] \\ λ_{3,4}^{2} = \frac{1}{2} [C \pm \sqrt[]{C^{2} - 4 D}] \end{array}

。（25）

\{\begin{array}{l} A = V^{2} + T^{2} {(V + χ_{l i n e})}^{2} - 2 T Z^{2} \\ B = T^{2} {(V^{2} + V χ_{l i n e} - Z^{2})}^{2} \\ C = \frac{1}{T^{2} {(V + χ_{t o t})}^{2}} \{A χ_{h e t}^{2} + B + 1 + 2 χ_{h e t} [V \sqrt[]{B} + T (V + χ_{l i n e}) + 2 T Z^{2}]\} \\ D = {[\frac{V + \sqrt[]{B} χ_{h e t}}{T (V + χ_{t o t})}]}^{2} \end{array}

，（26）

K_{k n n} = Λ (T) β I_{A B} - χ_{k n n},

（27）

χ_{k n n} = S (ρ_{E}) - \sum_{y_{i}}^{m} p (y_{i}) S (ρ_{E ∣ y_{i}}),

（28）

ρ_{E} = \sum_{y_{i}}^{m} p (y_{i}) ρ_{E ∣ y_{i}}

，（29）

Changlan Zhao, Tianyi Wang. Security Analysis of Phase Encoding Continuous-Variable Quantum Key Distribution Based on K-Nearest Neighbor[J]. Laser & Optoelectronics Progress, 2023, 60(19): 1927002

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