Editing Device-Independent Quantum Key Distribution (section)

==Protocol Description==
*'''Input: '''<math> n, \delta</math></br>
*'''Output: '''<math> K_A, K_B</math></br>
'''1.''' Distribution and measurement</br>	
#'''For''' every block <math> j \in [m]</math>
##Set <math>i=0</math> and <math>C_j=\bot</math>.
##'''While''' <math>i \leq s_{max}</math>
###Set <math>i=i+1</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>. 
### '''Else''' they choose  <math>X_i ,Y_i \in \{0,1\}</math>.
### 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>.</br>
''At this point 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>.''

'''2.''' Error Correction

''Alice and Bob apply the error correction protocol <math>EC</math> (see [[BB84 Quantum Key Distribution #References| [5]]]) , communicating script <math>O_{EC}</math> in the process. ''
# '''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>.
	
'''3.''' Parameter estimation

#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\neq X_i\cdot Y_i</math> '''then''' <math>C_i=0</math>
## '''If''' <math>T_i=0</math>  '''then''' <math>C_i=\bot</math>
# Bob 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. 
''For the summation in 3.2 we use the convention that <math>\forall x\in \{0,1,\bot\},\ x+\bot=\bot+x=x</math>, that is <math>\bot</math> acts as <math>0</math> with respect to the addition.''

'''4.''' Privacy amplification

<math>PA(\cdot,\cdot)</math> ''is a privacy amplification subroutine'' (see [[BB84 Quantum Key Distribution #References| [6]]])
# Alice and Bob run <math>PA(A_1^{n'},\tilde{B}_1^{n'})</math> and obtain secret keys <math>K_A, K_B</math>;