Quantum Coin: Difference between revisions

Quantum Coin (edit)

Revision as of 21:53, 17 April 2019

112 bytes added , 17 April 2019

no edit summary

Mahshid

Write

31

edits

@@ Line 11: / Line 11: @@
 == Properties ==
-* '''Parameters''': HMP<sub>4</sub>-states, Let x &isin; {0, 1}<sup>4</sup>. The corresponding HMP<sub>4</sub>-states is <math>|\alpha(x)>=\dfrac{1}{2}\sum_{1\leq i\leq4}(-1)^{x_i}|i></math>
+* '''Parameters''': HMP<sub>4</sub>-states, Let x &isin; {0, 1}<sup>4</sup>. The corresponding HMP<sub>4</sub>-states is <math>|\alpha(x)\rangle=\dfrac{1}{2}\sum_{1\leq i\leq4}(-1)^{x_i}|i\rangle</math>
 * '''General Features''':
 ** No need to quantum communication for quantum coin verification.
@@ Line 18: / Line 17: @@
 ** The database of the bank is static, and therefore many de-centralized “verification branches” can exist that do not have to communicate with one another.
 ** The number of verifications that a quantum coin can go through is limited.
 *'''Security Claims''':
 **The coins are exponentially hard to counterfeit.
@@ Line 23: / Line 23: @@
 == Protocol ==
+'''Stage 1: Quantum coin generation'''<br>
+''Input'': A secret record consists of <math>k</math> entries <math>x_1, . . . , x_k</math>,<math> x_i\in \{0,1\}^4</math><br>
+''Output'': A “fresh” quantum coin<br>
+The Trusted Third Party (TTP) chooses <math>x_1, . . . , x_k\in\{{0, 1}\}^4</math> at random, keeps them in secret and produces quantum states <math>|\alpha(x_1)\rangle, . . . , |\alpha(x_k)\rangle</math>.
+A “fresh” quantum coin corresponding to this record consists of:
+* <math>k</math> quantum registers consisting of 2 qubits each, where the <math>i</math>-th register contains <math>|\alpha(x_i)\rangle</math>;
+* a <math>k</math>-bit classical register <math>P</math>, that is initially set to <math>0^k</math>;
+* a unique identification number.
-	\begin{algorithm}
+'''Stage 2: Quantum coin verification'''<br>
-	\caption{Quantum coin generation}
+''Input'': the identification number of the quantum coin<br>
-	\noindent\textbf{Input} A secret record consists of $k$ entries $x_1, . . . , x_k, x_i\in\{{0,1}\}^4$\\
+''Output'': Accept or Reject<br>
-	\textbf{Output} A “fresh” quantum coin\\
+<br>
-	The Trusted Third Party (TTP) chooses $x_1, . . . , x_k\in\{{0, 1}\}^4$ at random, keeps them in secret and produces quantum states $\ket{\alpha(x_1)}, . . . , \ket{\alpha(x_k)}$.
+This stage is run as follows:
-	A “fresh” quantum coin corresponding to this record consists of:
+* The holder sends the identification number of the quantum coin to the TTP.
-		\begin{itemize}
+* The TTP chooses uniformly at random a set <math>L_{bn}\subset[k]</math> of size <math>t</math>, and sends it to the coin holder.
-		\item $k$ quantum registers consisting of 2 qubits each, where the $i$’th register contains $\ket{\alpha(x_i)}$;
+* The holder consults with P and chooses uniformly at random a set <math>L_{hl} \subset L_{bn}</math> consisting of <math>2t/3</math> yet unmarked positions. He sends <math>L_{hl}</math> to the bank and marks in <math>P</math> all the elements of <math>L_{hl}</math> as used.
-		\item a $k$-bit classical register $P$, that is initially set to $0^k$;
+* The TTP chooses at random <math>2t/3</math> values <math>m_i \in\{{0, 1}\}</math>, one for each <math>i \in L_{hl}</math> , and sends them to the coin holder.
-		\item a unique identification number.
+* The holder measures the quantum registers corresponding to the elements of <math>L_{hl}</math> in order to produce <math>2t/3</math> pairs <math>(a_i, b_i)</math>, such that <math>(x_i,m_i, a_i, b_i)\in HMP_4</math> for all <math>i \in L_{hl}</math>. He sends the list of <math>(a_i, b_i)</math>s to the TTP.
-		\end{itemize}
+* The TTP checks whether <math>(x_i,m_i, a_i, b_i)\in HMP_4</math> for all <math>i \in L_{hl}</math>, in which case it confirms validity of the quantum coin. Otherwise, the coin is declared to be a counterfeit.
-	\end{algorithm}
-	\begin{algorithm}
-	\caption{Quantum coin verification}
-	\noindent\textbf{Input} the identification number of the quantum coin\\
-	\textbf{Output} Accept or Reject\\
-	\renewcommand{\labelenumi}{\alph{enumi})}
-	\begin{enumerate}
-		\item The holder sends the identification number of the quantum coin to the TTP.
-		\item The TTP chooses uniformly at random a set $L_{bn}\subset[k]$ of size $t$, and sends it to the coin holder.
-		\item The holder consults with P and chooses uniformly at random a set $L_{hl} \subset L_{bn}$ consisting of $2t/3$ yet unmarked positions. He sends $L_{hl}$ to the bank and marks in $P$ all the elements of $L_{hl}$ as used.
-		\item The TTP chooses at random $2t/3$ values $m_i \in\{{0, 1}\}$, one for each $i \in L_{hl}$ , and sends them to the coin holder.
-		\item The holder measures the quantum registers corresponding to the elements of $L_{hl}$ in order to produce $2t/3$ pairs $(a_i, b_i)$, such that $(x_i,m_i, a_i, b_i)\in HMP_4$ for all $i \in L_{hl}$ . He sends the list of $(a_i, b_i)$s to the TTP.
-		\item The TTP checks whether $(x_i,m_i, a_i, b_i)\in HMP_4$ for all $i \in L_{hl}$ , in which case it confirms validity of the quantum coin. Otherwise, the coin is declared to be a counterfeit.
-		\end{enumerate}
-	\end{algorithm}