Editing Quantum Cheque (section)

==Notation==
* <math>k</math>: Shared key between account holder and bank where <math>k \in \{0,1\}^L</math>.
* <math>\Pi</math>: A decided Digital Signature scheme, which could also be [[Quantum Digital Signature]] or any other classical digital signature scheme, where <math>\Pi=(Gen, Sign, Vrfy)</math>. Here <math>Gen</math> is the digital signature generation funciton, <math>Sign</math> is the signing function and <math>Vrfy</math> is the function which verifies if the digital signature belongs to a party.
* <math>id</math>: Account holder's identity.
* <math>pk</math>: Account holder's public key.
* <math>sk</math>: Account holder's private key.
* <math>s</math>: Unique serial number where <math>s \in \{0,1\}^{l(n)}</math>.
* <math>{|\phi^{(i)}\rangle_{GHZ}}</math> or <math>{\rho^{(i)}_{GHZ}}</math>: <math>n</math> GHZ triplet states where,
<div style="text-align: center;"><math>|\phi^{(i)}\rangle_{GHZ} = \rho^{(i)}_{GHZ} = \frac{1}{\sqrt{2}}(|0^{(i)}\rangle_{A_1}|0^{(i)}\rangle_{A_2}|0^{(i)}\rangle_{B} + |1^{(i)}\rangle_{A_1}|1^{(i)}\rangle_{A_2}|1^{(i)}\rangle_{B})</math></div>
* <math>{\rho^{(i)}_{A_1}}</math>: First entangled qubit from every GHZ triplet state which is given to account holder.
* <math>{\rho^{(i)}_{A_2}}</math>: Second entangled qubit from every GHZ triplet state which is given to account holder.
* <math>{\rho^{(i)}_{B}}</math>: First entangled qubit from every GHZ triplet state which stays with bank.
* <math>r</math>: Generated random number where <math>r \in \{0,1\}^L</math>.
* <math>M</math>: Amount of money account holder signs on the cheque.
* <math>f</math>: Quantum one way function.
* <math>x\mid\mid</math>y:  concatenation of two bit strings <math>x</math> and <math>y</math>.
* <math>{|\psi^{(i)}\rangle}</math>: quantum state prepared by the account 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>. This quantum state is also denoted as <math>\alpha_i|0\rangle + \beta_i|1\rangle</math>.
* <math>|\Phi^\pm\rangle</math>: Bell state <math>\frac{|00\rangle \pm |11\rangle}{\sqrt{2}}</math>
* <math>|\Psi^\pm\rangle</math>: Bell state <math>\frac{|10\rangle \pm |01\rangle}{\sqrt{2}}</math>
* <math>\sigma</math>: In <math>\chi</math>, output of signing with <math>s</math>.
* <math>\chi</math>: Quantum cheque where,
<div style="text-align: center;"><math>
        \chi = (id, s, r, \sigma, M, \{{\rho^{(i)}_{A_1}}\}_{i=1:n})
</math></div>
* <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.