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Quantum Coin: Difference between revisions
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== Properties == | == Properties == | ||
* '''Parameters''': HMP<sub>4</sub>-states, Let x ∈ {0, 1}<sup>4</sup>. The corresponding HMP<sub>4</sub>-states is <math>|\alpha(x) | * '''Parameters''': HMP<sub>4</sub>-states, Let x ∈ {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''': | * '''General Features''': | ||
** No need to quantum communication for quantum coin verification. | ** No need to quantum communication for quantum coin verification. | ||
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** 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 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. | ** The number of verifications that a quantum coin can go through is limited. | ||
*'''Security Claims''': | *'''Security Claims''': | ||
**The coins are exponentially hard to counterfeit. | **The coins are exponentially hard to counterfeit. | ||
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== Protocol == | == 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. | |||
'''Stage 2: Quantum coin verification'''<br> | |||
''Input'': the identification number of the quantum coin<br> | |||
''Output'': Accept or Reject<br> | |||
<br> | |||
This stage is run as follows: | |||
* The holder sends the identification number of the quantum coin to the TTP. | |||
* 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. | |||
* 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 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 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. | |||