Device-Independent Quantum Key Distribution: Difference between revisions

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==Pseudocode==
==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>
<u>'''Stage 1'''</u> 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>
##Set <math>i=0</math> and <math>C_j=\bot</math>.
##Set <math>i=0</math> and <math>C_j=\bot</math>.
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*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>.
*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>.


<u>'''Stage 2'''</u> Error Correction</br>
'''2.''' Error Correction</br>
* Alice and Bob apply the error correction protocol <math>EC</math>, communicating script <math>O_{EC}</math> in the process.  
* 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>.
<u>'''Stage 3'''</u> Parameter estimation</br>
'''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>  
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# He aborts '''If''' <math>\sum_j C_{j}<m\times (\omega_{exp}-\delta_{est})(1-(1-\gamma)^{s_{\max}})</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.  
* ''i.e.'', if they do not achieve the expected violation.  
 
'''4.''' Privacy amplification</br>
<u>'''Stage 4'''</u> Privacy amplification</br>
*<math>PA(\cdot,\cdot)</math> is a privacy amplification subroutine
*<math>PA(\cdot,\cdot)</math> is a privacy amplification subroutine
# 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>;
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