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* <math>\kappa</math>: Threshold constant set as a security parameter by the bank in the swap test. | * <math>\kappa</math>: Threshold constant set as a security parameter by the bank in the swap test. | ||
* <math>|\psi^{,(i)}\rangle</math>: Quantum state prepared by the bank for verification. | * <math>|\psi^{,(i)}\rangle</math>: Quantum state prepared by the bank for verification. | ||
== | |||
<div style= | ==Pseudo-Code== | ||
'''Stage 1''': Gen</br> | |||
'''Output''': Account holder now holds <math>(id, pk, sk, k, s, \{|\phi^{(i)}\rangle_{A_1}|\phi^{(i)}\rangle_{A_2}\}_{i=1:n})</math> and Bank now holds <math>(id, pk, k, s, {\|\phi^{(i)}\rangle_{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>|\phi^{(i)}\rangle_{B}</math> in their private database. | |||
** Bank gives <math>|\phi^{(i)}\rangle_{A_1}|\phi^{(i)}\rangle_{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>|\phi^{(i)}\rangle_{A_1}|\phi^{(i)}\rangle_{A_2}</math> privately. | |||
* Account holder now holds <math>(id, pk, sk, k, s, \{|\phi^{(i)}\rangle_{A_1}|\phi^{(i)}\rangle_{A_2}\}_{i=1:n})</math> | |||
* Bank now holds <math>(id, pk, k, s, {|\phi^{(i)}\rangle_{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>|\phi^{(i)}\rangle_{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>|\phi^{(i)}\rangle_{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>. | |||