Editing Measurement Device Independent Quantum Digital Signature (MDI-QDS) (section)

==Protocol Description==

<u>'''Stage 1'''</u> Distribution
*'''Input''' Key Length (<math>L</math>), Threshold values (<math>s_a, s_v</math>), error threshold (<math>E_{tol}</math>), error rate parameter (<math>e</math>)
*'''Output''' Seller: <math>S_B^0,S_B^1,S_V^0,S_V^1</math> Buyer: <math>B^0,B^1</math>; Verifier: <math>V^0,V^1</math>
**'''Key Distribution:'''
#For k = 0,1
##<math>S_B^k=B^k=</math>'''MDI-KGP'''(Seller, Buyer, Arbitrator,k)
##<math>S_V^k=V^k=</math>'''MDI-KGP'''(Seller, Verifier, Arbitrator,k)

**'''Symmetrisation'''
#For k = 0,1
##	Buyer chooses <math>I\subset_R\{1,2,...,L\}, |I|=[L/2]</math>
##	<math>\forall i\epsilon I</math>, Buyer sends Verifier <math>(k,i,B^k_i)</math> 
##	Verifier chooses <math>J\subset_R\{1,2,...,L\}, |J|=[L/2]</math>
##	<math>\forall j\epsilon J</math>, Verifier sends Buyer <math>(k,j,V^k_j)</math> 

<u>'''Stage 2'''</u> Messaging
 
*'''Input''' Seller: Message <math>m</math>, Private Key for <math>m</math>: <math>S^B_m,S^V_m</math>
*'''Output''' Buyer: accept or abort, Verifier: accept or abort
**'''Signing:''' 
#	Seller sends Buyer (m,<math>S^B_m,S^V_m</math>)
#	For l = 1,2,..,L
##If '''<math>l\epsilon \{1,2,...,L\}-I</math>'''
###Buyer counts the number of mismatches (<math>B^m_l!=S^m_l</math>), b=b+1
##If '''<math>l\epsilon J</math>'''
###Buyer counts the number of mismatches (<math>V^m_l!=S^m_l</math>), v=v+1
#	If <math>(b < s_aL/2)</math>&&<math>(v < s_aL/2)</math>, Buyer accepts m else he aborts
**'''Transfer'''
#	Buyer sends Verifier (m,<math>S^B_m,S^V_m</math>) 
#	For l = 1,2,..,L
##If '''<math>l\epsilon \{1,2,...,L\}-J</math>'''
###Verifier counts the number of mismatches (<math>V^m_l!=S^m_l</math>), v=v+1
##If '''<math>l\epsilon I</math>'''
###Verifier counts the number of mismatches (<math>B^m_l!=S^m_l</math>), b=b+1
#	If <math>(b < s_vL/2)</math>&&<math>(v < s_vL/2)</math>, Verifier accepts m else he aborts
===Subroutine: '''MDI-KGP'''(Seller, Receiver R, Arbitrator,i)===
*For k=0,L
#Seller chooses <math>s_{\text{basis}}\epsilon \{X,Z\}</math> and generates <math>|a\rangle</math>
#Receiver chooses <math>r_{\text{basis}}\epsilon \{X,Z\}</math> and generates <math>|b\rangle</math>
#Seller.send(Arbitrator,<math>|a\rangle</math>)
#Receiver.send(Arbitrator,<math>|b\rangle</math>)
#<math>|\Psi\rangle</math>=Arbitrator.'''BSM'''(<math>|a\rangle</math>,<math>|b\rangle</math>)
#If (<math>|\Psi\rangle!={}</math>)&&(<math>s_{\text{basis}}=r_{\text{basis}}</math>)
###<math>A^i_R(k)=a</math>
###If (<math>s_{\text{basis}}=r_{\text{basis}}=Z</math>)
####If (<math>|\Psi\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle))||(|Psi\rangle=\frac{1}{\sqrt{2}}(|00\rangle-|11\rangle)</math>) '''then''' <math>R^i(k)=b</math>
####If (<math>|\Psi\rangle=\frac{1}{\sqrt{2}}(|01\rangle+|10\rangle))||(|Psi\rangle=\frac{1}{\sqrt{2}}(|01\rangle-|10\rangle)</math>) '''then''' <math>R^i(k)=\tilde b</math>
###If (<math>s_{\text{basis}}=r_{\text{basis}}=X</math>)
####If (<math>|\Psi\rangle=\frac{1}{\sqrt{2}}(|++\rangle+|--\rangle))||(|Psi\rangle=\frac{1}{\sqrt{2}}(|+-\rangle+|-+\rangle)</math>) '''then''' <math>R^i(k)=b</math>
####If (<math>|\Psi\rangle=\frac{1}{\sqrt{2}}(|++\rangle-|--\rangle))||(|Psi\rangle=\frac{1}{\sqrt{2}}(|01\rangle-|+-\rangle)</math>) '''then''' <math>R^i(k)=\tilde b</math>
*Error rate calculation
#Seller and Buyer choose <math>I\subset_R\{1,2,...,L\}, |I|=e</math>
#<math>\forall i\epsilon I</math>, <math>E_k=\frac{1}{e}\sum_i(A^k_R(i)\oplus R^k(i))</math>
#If <math>E_k>E_{tol}</math> '''then''' abort '''else'''
# '''return''' (<math>A^k_R,R^k</math>)
* For more detail on error rate and MDI-KGP Pseudo code, refer [[MDI-QKD]]