Editing Arbitrated Quantum Digital Signature (section)

==Notation==
* <math>n</math>: Total number of qubits of message.
* <math>f</math>: public function to obtain public key from user's email-id
* <math>k_{pub}</math>: Seller's public key, where <math>k_{pub} \in \{0,1\}^n</math>.
* <math>k_{pri}</math>: Seller's private, where <math>k_{pri} \in \{0,1\}^n</math>.
* <math>k_r</math>: Random OTP number selected by PKG to denote each of Seller's signatures, where <math>k_{r} \in \{0,1\}^n</math>.
* <math>VC(x,y)</math>: function VC performs one time pads 'y' using quantum pad key 'x' via [[Arbitrated Quantum Digital Signature#References|Quantum Vernam Cipher (1), (2)]].
* <math>k_{at}</math>: Shared key between the Seller and PKG where <math>k_{at} \in \{0,1\}^n</math>.
* <math>E_{k_{at}}</math>: Quantum Vernam cipher encrypted state which uses <math>k_{at}</math>.
* <math>G</math>: PKG's master key which is a one way function where <math>\{0,1\}^n \xrightarrow{}\{0,1\}^n</math> .
* <math>F</math>: Public quantum one way function selected by Seller to generate quantum digest.
* <math>m</math>: Message sent by Seller to the Buyer, where <math>m \in \{0,1\}^n</math>.
* <math>s</math>: Random string of uniform distribution selected by the Seller, where <math>s \in \{0,1\}^n</math>.
* <math>t</math>: Random string of uniform distribution selected by the Seller, where <math>t \in \{0,1\}^n</math>.
*<math>l</math>: qubit address
* <math>|\phi\rangle_{a_l,b_l}</math>: Quantum state which is defined by
<math>|\phi\rangle_{a_l,b_l} := H^{a_l}U_{\frac{\pi}{4}}H^{b_l}|0\rangle</math>
* <math>|\phi\rangle_{a_l,b_l,c_l}</math>: Quantum state which is defined by
<math>|\phi\rangle_{a_l,b_l,c_l} := Y^{c_l}|\phi\rangle_{a_l,b_l}</math>
* <math>|S\rangle_{k_{pri},m}</math>: Signature quantum state for message <math>m</math> which is the quantum state
<math>|S\rangle_{k_{pri},m} = \bigotimes^{n}_{l=1} H^{k_{pub_l}\oplus k_{pri_l}} |\phi\rangle_{s_l,t_l\oplus m_l, m_l}</math>
* <math>|P\rangle</math>: Private key quantum state where <math>|P\rangle \in \{|+\rangle, |-\rangle, |1\rangle, |0\rangle\}^n</math> and it is the quantum state:
<math>|P\rangle := H^{k_{pri}}|\phi\rangle_{s, t\oplus m}</math>
* <math>P</math>: Classical 2n-bit for <math>n</math>-qubit <math>|P\rangle</math> where <math>|+\rangle</math> is encoded to 10, <math>|-\rangle</math> to 11, <math>|1\rangle</math> to 00 and <math>|0\rangle</math> is encoded to 01.
* <math>B_l</math>: This is the set of the basis of each <math>l^th</math> qubit in <math>|P\rangle</math>.
<math> B_l \in \{+,\times \}</math>
*<math>B_l(|P_l\rangle)</math>: Measurement of <math>l^{th}</math> qubit in basis <math>B_l</math>
*<math>b_l</math>: measurement result of <math>l^{th}</math> qubit in the concerned quantum state
* <math>|F\rangle</math>: Quantum digital digest received by PKG.
* <math>|F\rangle'</math>: Quantum digital digest generated by Buyer.
* <math>u</math>: The most number of Buyer in this scheme.
* <math>w</math>: Safety parameter threshold for acceptance.
* <math>w_0</math>: Security threshold decided in advance.
* <math>w'</math>: Number of times SWAP test is performed.
* <math>|V\rangle_{m, k_{pub},S}</math>: A quantum state, where
<math>|V\rangle_{m, k_{pub},S} := Y^m H^{k_{pub}}|S\rangle_{k_{pri}, m}</math>
This state is also expressed as <math>\beta|\phi\rangle_{k_{pri}\oplus s, t\oplus m}</math> where <math>\beta \in \{1, -1, \iota, -\iota\}</math>
* <math>Q</math>: Classical bit string denoted as <math>Q \in \{00, 01, 10, 11\}^n</math>. It is proven that <math>P=Q</math>.
*<math>g(Q)</math>: g is a classical function which when takes classical 2n bit string Q, gives seller's random string t as output. This function can be calculated.
* <math>\delta</math>: <math>\langle F|F\rangle'</math>, where <math>\delta \in [0,1)</math>.