Editing Quantum Cheque (section)

==Protocol Description==

'''Stage 1''': Gen</br>
'''Output''':  Account holder now holds <math>(id, pk, sk, k, s, \{\rho^{(i)}_{A_1}\rho^{(i)}_{A_2}\}_{i=1:n})</math> and Bank now holds <math>(id, pk, k, s, \{\rho^{(i)}_{B}\}_{i=1:n})</math> in their private databases.

* Account holder and Bank create <math>k</math>.
* Account holder and Bank agree on <math>\Pi</math>.
* Account holder submits <math>pk</math> to Bank.
* Account holder secretly stores <math>sk</math>.
* For <math>i = 1, 2, ...n</math>:
** Bank generates GHZ triple state <math>|\phi^{(i)}\rangle_{GHZ}</math>
** Bank stores <math>\rho^{(i)}_{B}</math> in their private database.
** Bank gives <math>\rho^{(i)}_{A_1}\rho^{(i)}_{A_2}</math> to account holder.
* Bank prepares <math>s</math> and shares it with the account holder.
* For <math>i = 1, 2, ...n</math>:
** Account holder stores <math>\rho^{(i)}_{A_1}\rho^{(i)}_{A_2}</math> privately.
* Account holder now holds <math>(id, pk, sk, k, s, \{\rho^{(i)}_{A_1}\rho^{(i)}_{A_2}\}_{i=1:n})</math>
* Bank now holds <math>(id, pk, k, s, \{\rho^{(i)}_{B}\}_{i=1:n})</math>


'''Stage 2''': Sign </br>
'''Output''': Account holder produces <math>\chi</math>

* Account holder generates <math>r</math>.
* For <math>i = 1, 2, ...n</math>:
** Account holder prepares <math>|\psi^{(i)}\rangle = f(k \mid\mid id\mid\mid r\mid\mid M\mid\mid i)</math>.
** Account holder encodes <math>|\psi^{(i)}\rangle</math> with <math>|\phi^{(i)}\rangle_{GHZ}</math> by combining <math>|\psi^{(i)}\rangle</math> with <math>\rho^{(i)}_{A_1}</math> and performing a bell measurement on the two.
<div style="text-align: center;"><math>
                |\phi^{(i)}\rangle =  |\psi^{(i)}\rangle \otimes |\phi^{(i)}\rangle_{GHZ}
</math></div>
** Based on the measurement, account holder performs the suitable error correction, by applying the corresponding Pauli matrix, on <math>\rho^{(i)}_{A_1}</math>:
         <div style="text-align: center;"><math>
            |\Phi^{(+)}\rangle  \xrightarrow{} I,
            |\Phi^{(-)}\rangle  \xrightarrow{} \sigma_z,
            |\Psi^{(+)}\rangle  \xrightarrow{} \sigma_x,
            |\Phi^{(-)}\rangle  \xrightarrow{} \sigma_y,
           </math></div>
* Account holder signs the serial number <math>s</math> using <math>\Pi</math>, as <math>\sigma \xleftarrow{} Sign_{sk}(s)</math>.
* Account holder produces <math>\chi</math>.

'''Stage 3''': Verify </br>
'''Output''': Cheque gets accepted or the process is aborted and the cheque is destroyed.

* Third party produces the <math>\chi</math> at a valid branch of the bank.
* Branch communicates with main branch of the bank and checks validity of <math>(id, s)</math> and runs <math>{Vrfy}_{pk}(\sigma, s)</math>. 
** If invalid:
*** Bank aborts the process and destroys the cheque.
** else:
*** Bank continues the verification process.
* Main branch of the bank performs the measurement in Hadamard basis on its copy of <math>\rho^{(i)}_{B}</math> and obtains outcome <math>|+\rangle</math> or <math>|-\rangle</math>. 
* Main branch communicates this result with the local branch.
* For <math>i = 1, 2, ...n</math>:
** Based on outcome, branch performs the corresponding Pauli matrix operation on  <math>\rho^{(i)}_{A_2}</math> to recover <math>|\Psi^{(i)}\rangle</math> : <math>|+\rangle \xrightarrow{} I, |-\rangle \xrightarrow{} \sigma_z</math>
* For <math>i = 1, 2, ...n</math>:
** Bank computes <math>|\Psi^{,(i)}\rangle </math>, where,
<div style="text-align: center;"><math>          
               |\Psi^{,(i)}\rangle = f(k \mid\mid id\mid\mid r\mid\mid M\mid\mid i)
</math></div>
* For <math>i = 1, 2, ...n</math>:
** Bank performs swap test on <math>|\Psi^{(i)}\rangle</math> and <math>|\Psi^{,(i)}\rangle</math>. 
*** If <math>\langle\Psi^{(i)}|\Psi^{,(i)}\rangle \geq \kappa</math>:
**** Bank aborts the process and destroys the cheque.
*** else:
**** Bank continues the verification process.
* Bank accepts the cheque.