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* '''Quantum coin Verification''' - To verify a quantum coin through classical communication with the TTP, its holder sends the identification number of the quantum coin to the TTP. Then, the TTP and the coin holder exchange some classical information for choosing some quantum registers. The coin holder measures the chosen registers and sends their corresponding classical information to the TTP. The TTP verifies the authenticity of the coin by the secret information he possesses. | * '''Quantum coin Verification''' - To verify a quantum coin through classical communication with the TTP, its holder sends the identification number of the quantum coin to the TTP. Then, the TTP and the coin holder exchange some classical information for choosing some quantum registers. The coin holder measures the chosen registers and sends their corresponding classical information to the TTP. The TTP verifies the authenticity of the coin by the secret information he possesses. | ||
==Notations== | ==Notations== | ||
* <math>HMP_4</math>-states: <math>|\alpha(x) | * <math>HMP_4</math>-states: <math>|\alpha(x)e=\dfrac{1}{2}\sum_{1\leq i\leq4}(-1)^{x_i}|i\rangle</math>, <math>x\in\{0, 1\}</math> | ||
* <math>HMP_4</math>-queries: An <math>HMP_4</math>-query is an element <math>m \in \{0, 1\}</math>. A valid answer to the query w.r.t. <math>x \in \{0, 1\}^4</math> is a pair <math>(a, b) \in \{0, 1\} \times \{0, 1\}</math>, such that <math>(x, m, a, b) \in HMP_4</math>. An <math>HMP_4</math> -state can be used to answer an <math>HMP_4</math> -query with certainty: If <math> m = 0 </math>, let | |||
<math> v_1 \overset{def}{=}\dfrac{|1\rangle+|2\rangle}{\sqrt{2}} </math> <math> v_2 \overset{def}{=}\dfrac{|1\rangle-|2\rangle}{\sqrt{2}} </math> <math> v_3 \overset{def}{=}\dfrac{|3\rangle+|4\rangle}{\sqrt{2}} </math> <math> v_4 \overset{def}{=}\dfrac{|3\rangle-|4\rangle}{\sqrt{2}} </math> | |||
otherwise (m = 1), let | |||
<math> v_1 \overset{def}{=}\dfrac{|1\rangle+|3\rangle}{\sqrt{2}} </math> <math> v_2 \overset{def}{=}\dfrac{|1\rangle-|3\rangle}{\sqrt{2}} </math> <math> v_3 \overset{def}{=}\dfrac{|2\rangle+|4\rangle}{\sqrt{2}} </math> <math> v_4 \overset{def}{=}\dfrac{|2\rangle-|4\rangle}{\sqrt{2}} </math> | |||
Measure <math>|\alpha(x_i)\rangle</math> in the basis <math>{v_1, v_2, v_3, v_4}</math>, and let <math>(a, b)</math> be <math>(0, 0)</math> if the outcome is <math>v_1</math>; <math>(0, 1)</math> in the case of <math>v_2</math>; <math>(1, 0)</math> in the case of <math>v_3</math>; <math>(1, 1)</math> in the case of <math>v_4</math>. Then <math>(x, m, a, b) \in HMP_4</math> always. | |||
==Requirements== | ==Requirements== | ||
*Network stage: [[:Category: Quantum Memory Network Stage|quantum memory network]][[Category:Quantum Memory Network Stage]]. | *Network stage: [[:Category: Quantum Memory Network Stage|quantum memory network]][[Category:Quantum Memory Network Stage]]. | ||
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* 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. | * 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. | ||
* 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. | * 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. | ||
* 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. | * 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> (refer to <math>HMP_4</math>-queries in Notations), 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. | ||
* 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. | * 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. | ||