Editing Device-Independent Quantum Key Distribution
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'''Tags:''' [[:Category:Two Party Protocols|Two Party]], [[:Category:Quantum Enhanced Classical Functionality|Quantum Enhanced Classical Functionality]], [[:Category:Specific Task|Specific Task]],[[Quantum Key Distribution]], [[BB84 Quantum Key Distribution|BB84 QKD]], [[Category:Multi Party Protocols]] [[Category:Quantum Enhanced Classical Functionality]][[Category:Specific Task]][[Category:Entanglement Distribution Network | A device-independent quantum key distribution protocol implements the task of [[Quantum Key Distribution]] (QKD) without relying on any particular description of the underlying system. The protocol enables two parties, Alice and Bob, to establish a classical secret key by distributing an entangled quantum state and checking for the violation of a [[Bell inequality]] in order to certify the security. The output of the protocol is a classical secret key which is completely unknown to any third party, namely an eavesdropper. | ||
'''Tags:''' [[:Category:Two Party Protocols|Two Party]], [[:Category:Quantum Enhanced Classical Functionality|Quantum Enhanced Classical Functionality]], [[:Category:Specific Task|Specific Task]],[[Quantum Key Distribution]], [[BB84 Quantum Key Distribution|BB84 QKD]], [[Category:Multi Party Protocols]] [[Category:Quantum Enhanced Classical Functionality]][[Category:Specific Task]][[Category:Entanglement Distribution Network Stage]] | |||
==Assumptions== | ==Assumptions== | ||
* | * We assume the existence of an authenticated public classical channel between the two parties | ||
* | * We assume synchronous network between parties | ||
* | * We assume security from [[coherent attacks]] | ||
==Outline== | ==Outline== | ||
A DIQKD protocol is composed by the following steps: | A DIQKD protocol is composed by the following steps: | ||
* The first phase of the protocol is | * The first phase of the protocol is the distribution. For each round of this phase: | ||
** Alice uses the source to prepare a maximally entangled state and send half of the state to Bob. | ** Alice uses the source to prepare a maximally entangled state and send half of the state to Bob. | ||
** Upon receiving the state, Bob announces that he received it, and they both use their respective devices to measure the quantum systems. They record their output in a string of bits. | ** Upon receiving the state, Bob announces that he received it, and they both use their respective devices to measure the quantum systems. They record their output in a string of bits. | ||
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* In the final phase, Alice and Bob perform [[privacy amplification]], where the not fully secure <math>n</math>-bit strings are mapped into smaller strings <math>K_A</math> and <math>K_B</math>, which represents the final keys of Alice and Bob respectively. | * In the final phase, Alice and Bob perform [[privacy amplification]], where the not fully secure <math>n</math>-bit strings are mapped into smaller strings <math>K_A</math> and <math>K_B</math>, which represents the final keys of Alice and Bob respectively. | ||
==Requirements == | ==Hardware Requirements == | ||
*'''Network Stage:''' [[:Category:Entanglement Distribution Network | *'''Network Stage:''' [[:Category: Entanglement Distribution Network Stage|Entanglement Distribution]] | ||
*'''Relevant Network Parameters:''' | *'''Relevant Network Parameters:''' <math>\epsilon_T, \epsilon_M</math> (see [[:Category: Entanglement Distribution Network Stage|Entanglement Distribution]]) | ||
* Distribution of Bell pairs, and measurement in three different bases (two basis on Alice's side and three basis on Bob's side). | * Distribution of Bell pairs, and measurement in three different bases (two basis on Alice's side and three basis on Bob's side). | ||
* | * Minimum number of rounds ranging from <math>\mathcal{O}(10^6)</math> to <math>\mathcal{O}(10^{12})</math> depending on the network parameters, for commonly used secure parameters. | ||
* <math>QBER \leq 0.071</math>, taking a depolarizing model as benchmark. Parameters satisfying <math>\epsilon_T+\epsilon_M\leq 0.071</math> are sufficient. | |||
* [[Authenticated classical channel]]. | |||
* [[Random number generator]]. | |||
==Notation== | ==Notation== | ||
* <math>n</math> expected number of rounds | * <math>n</math> expected number of rounds | ||
* <math>l</math> final key length | * <math>l</math> final key length | ||
* <math>\gamma</math> fraction of test rounds | * <math>\gamma</math> fraction of test rounds | ||
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* <math>\epsilon_{PA}</math> error probability of the privacy amplification protocol | * <math>\epsilon_{PA}</math> error probability of the privacy amplification protocol | ||
* <math>\mbox{leak}_{EC}</math> leakage in the error correction protocol | * <math>\mbox{leak}_{EC}</math> leakage in the error correction protocol | ||
==Properties== | ==Properties== | ||
Either | Either Protocol (see [[Device Independent Quantum Key Distribution#Pseudo-code|Pseudo-code]]) abort with probability higher than <math>1-(\epsilon_{EA}+\epsilon_{EC})</math>, or it generates a</br> | ||
<math>(2\epsilon_{EC}+\epsilon_{PA}+\epsilon_s)</math>-correct-and-secret key of length | <math>(2\epsilon_{EC}+\epsilon_{PA}+\epsilon_s)</math>-correct-and-secret key of length</br> | ||
<math> | <math>l\geq \frac{{n}}{\bar{s}}\eta_{opt} -\frac{{n}}{\bar{s}}h(\omega_{exp}-\delta_{est}) -\sqrt{\frac{{n}}{\bar{s}}}\nu_1 -\mbox{leak}_{EC}</math></br> | ||
<math>-3\log\Bigg(1-\sqrt{1-\Bigg(\frac{\epsilon_s}{4(\epsilon_{EA} + \epsilon_{EC})}\Bigg)^2}\Bigg)+2\log\Bigg(\frac{1}{2\epsilon_{PA}}\Bigg)</math>,</br> | |||
l\geq \frac{{n}}{\bar{s}}\eta_{opt} -\frac{{n}}{\bar{s}}h(\omega_{exp}-\delta_{est}) -\sqrt{\frac{{n}}{\bar{s}}}\nu_1 -\mbox{leak}_{EC} | where <math>\mbox{leak}_{EC}</math> is the leakage due to error correction step and the functions <math>\bar{s}</math>, <math>\eta_{opt}</math>, <math>\nu_1</math> and <math>\nu_2</math> are specified in below. | ||
The security parameters of the error correction protocol, <math>\epsilon_{EC}</math> and <math>\epsilon'_{EC}</math>, mean that if the error correction step in Protocol 1 does not abort, then <math>K_A=K_B</math> with probability at least <math>1-\epsilon_{EC}</math>, and for an honest implementation, the error correction protocol aborts with probability at most <math>\epsilon'_{EC}+\epsilon_{EC}</math>. | |||
</math></br> | |||
where <math>\mbox{leak}_{EC}</math> is the leakage due to error correction step and the functions <math>\bar{s}</math>, <math>\eta_{opt}</math>, <math>\nu_1</math> and <math>\nu_2</math> are specified below. | |||
The security parameters of the error correction protocol, <math>\epsilon_{EC}</math> and <math>\epsilon'_{EC}</math>, mean that if the error correction step | |||
*<math>\bar{s}=\frac{1-(1-\gamma)^{\left\lceil \frac{1}{\gamma} \right\rceil}}{\gamma}</math> | *<math>\bar{s}=\frac{1-(1-\gamma)^{\left\lceil \frac{1}{\gamma} \right\rceil}}{\gamma}</math> | ||
*<math>\eta_{opt}=\max_{\frac{3}{4}<\frac{{p}_t(1)}{1-(1-\gamma)^{s_{max}}}<\frac{2+\sqrt{2}}{4}} \Bigg(F_{\min}(\vec{p},\vec{p}_t)-\frac{1}{\sqrt{m}}\nu_2\Bigg)</math> | *<math>\eta_{opt}=\max_{\frac{3}{4}<\frac{{p}_t(1)}{1-(1-\gamma)^{s_{max}}}<\frac{2+\sqrt{2}}{4}} \Bigg(F_{\min}(\vec{p},\vec{p}_t)-\frac{1}{\sqrt{m}}\nu_2\Bigg)</math> | ||
*<math>F_{\min}(\vec{p},\vec{p}_t) = \frac{d}{d {p}(1)}g(\vec{p}) \Big|_{\vec{p}_t}\cdot {p}(1)+\Bigg( g(\vec{p}_t)- \frac{d}{d{p}(1)}g(\vec{p})|_{\vec{p}_t}\cdot {p}_t(1) \Bigg)</math> | *<math>F_{\min}(\vec{p},\vec{p}_t) = \frac{d}{d {p}(1)}g(\vec{p}) \Big|_{\vec{p}_t}\cdot {p}(1)+\Bigg( g(\vec{p}_t)- \frac{d}{d{p}(1)}g(\vec{p})|_{\vec{p}_t}\cdot {p}_t(1) \Bigg)</math> | ||
*<math>g({\vec{p}}) = {s}\Bigg(1-h\ | *<math>g({\vec{p}}) = {s}\Bigg(1-h\Big(\frac{1}{2}+\frac{1}{2}\sqrt{16\frac{{p}(1)}{1-(1-\gamma)^{s_{max}}}\Bigg(\frac{{p}(1)}{1-(1-\gamma)^{s_{max}}} -1}+3 )\Bigg)</math> | ||
*<math>\nu_2 =2 \ | *<math>\nu_2 =2 \Big(\log\Bigg(1+2\cdot 2^{s_{\max}}3}+\left\lceil \frac{d}{d{p}(1)}g(\vec{p})\big|_{\vec{p}_t}\right\rceil\Bigg)\sqrt{1-2\log \epsilon_s)</math> | ||
*<math>\nu_1=2 \Big(\log 7 +\left\lceil\frac{|h'(\omega_{exp}+\delta_{est})|}{1-(1-\gamma)^{s_{\max}}}\right\rceil\Big)\sqrt{1-2\log\epsilon_s}</math> | *<math>\nu_1=2 \Big(\log 7 +\left\lceil\frac{|h'(\omega_{exp}+\delta_{est})|}{1-(1-\gamma)^{s_{\max}}}\right\rceil\Big)\sqrt{1-2\log\epsilon_s}</math> | ||
== | ==Pseudocode== | ||
*'''Input: '''<math> n, \delta</math></br> | *'''Input:'''<math> n, \delta</math></br> | ||
*'''Output: '''<math> K_A, K_B</math></br> | *'''Output:'''<math> K_A, K_B</math></br> | ||
'''1.''' Distribution and measurement</br> | '''1.''' Distribution and measurement</br> | ||
#'''For''' every block <math> j \in [m]</math> | #'''For''' every block <math> j \in [m]</math> | ||
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### Alice and Bob choose a random bit <math>T_i \in \{0,1\}</math> such that <math>P(T_i=1)=\gamma</math>. | ### Alice and Bob choose a random bit <math>T_i \in \{0,1\}</math> such that <math>P(T_i=1)=\gamma</math>. | ||
### '''If''' <math>T_i=0</math> '''then''' Alice and Bob choose inputs <math>(X_i, Y_i)=(0,2)</math>. | ### '''If''' <math>T_i=0</math> '''then''' Alice and Bob choose inputs <math>(X_i, Y_i)=(0,2)</math>. | ||
### '''Else''' they choose <math>X_i ,Y_i \in \{0,1\}</math>. | ### '''Else''' they choose <math>X_i ,Y_i \in \{0,1\}</math> (the observables for the CHSH test). | ||
### Alice and Bob use their devices with the respective inputs and record their outputs, <math>A_i</math> and <math>B_i</math> respectively. | ### Alice and Bob use their devices with the respective inputs and record their outputs, <math>A_i</math> and <math>B_i</math> respectively. | ||
### '''If''' <math>T_i=1</math> they set <math>i=s_{max}+1</math>. | ### '''If''' <math>T_i=1</math> they set <math>i=s_{max}+1</math>. | ||
*At this stage Alice holds strings <math>X_1^n, A_1^n</math> and Bob <math>Y_1^n, B_1^n</math>, all of length <math>n</math>. | |||
''Alice and Bob apply the error correction protocol <math>EC</math> | '''2.''' Error Correction</br> | ||
* Alice and Bob apply the error correction protocol <math>EC</math>, communicating script <math>O_{EC}</math> in the process. | |||
# '''If''' <math>EC</math> aborts, they abort the protocol | # '''If''' <math>EC</math> aborts, they abort the protocol | ||
# '''Else''' they obtain raw keys <math>\tilde{A}_1^n</math> and <math>\tilde{B}_1^n</math>. | # '''Else''' they obtain raw keys <math>\tilde{A}_1^n</math> and <math>\tilde{B}_1^n</math>. | ||
'''3.''' Parameter estimation | '''3.''' Parameter estimation</br> | ||
#Using <math>B_1^n</math> and <math>\tilde{B}_1^n</math>, Bob sets <math>C_i</math> | #Using <math>B_1^n</math> and <math>\tilde{B}_1^n</math>, Bob sets <math>C_i</math> | ||
##'''If''' <math>T_i=1</math> and <math>A_i\oplus B_i=X_i\cdot Y_i</math> '''then''' <math>C_i=1</math> | ##'''If''' <math>T_i=1</math> and <math>A_i\oplus B_i=X_i\cdot Y_i</math> '''then''' <math>C_i=1</math> | ||
##'''If''' <math>T_i=1</math> and <math>A_i\oplus B_i | ##'''If''' <math>T_i=1</math> and <math>A_i\oplus B_i=X_i\cdot Y_i</math> '''then''' <math>C_i=0</math> | ||
## '''If''' <math>T_i= | ## '''If''' <math>T_i=1</math> and <math>A_i\oplus B_i=X_i\cdot Y_i</math> '''then''' <math>C_i=\bot</math> | ||
# | # He aborts '''If''' <math>\sum_j C_{j}<m\times (\omega_{exp}-\delta_{est})(1-(1-\gamma)^{s_{\max}})</math>, i.e., if they do not achieve the expected violation. | ||
'''4.''' Privacy amplification</br> | |||
*<math>PA(\cdot,\cdot)</math> is a privacy amplification subroutine | |||
'''4.''' Privacy amplification | |||
<math>PA(\cdot,\cdot)</math> | |||
# Alice and Bob run <math>PA(A_1^{n'},\tilde{B}_1^{n'})</math> and obtain secret keys <math>K_A, K_B</math>; | # Alice and Bob run <math>PA(A_1^{n'},\tilde{B}_1^{n'})</math> and obtain secret keys <math>K_A, K_B</math>; | ||
==Further Information== | ==Further Information== | ||
<div style='text-align: right;'>''contributed by Gláucia Murta''</div> | <div style='text-align: right;'>''contributed by Gláucia Murta''</div> |