Quantum Coin

Quantum Money is a unique object generated by a Trusted Third Party (TTP). Then, it is circulated among untrusted clients (Transferability property). Each client should be able to prove the authenticity of his owned quantum money to a verifier. On the other hand, an adversary must fail in counterfeiting the quantum money with overwhelmingly high probability (Unforgeability property).

Tags: Multiparty, Quantum Enhanced Classical functionality, prepare (bank) and measure (client)

Outline

In this scheme, a Trusted Third Party (TTP) and a coin holder run the following procedure for generating and verifying a quantum coin:

• Quantum coin Generation - The TTP chooses k random 4-bit strings, keeps them in secret and produce k quantum states. A newly issued quantum coin consists of a piece of paper glued to k quantum registers that hold k quantum states. The piece of paper contains a unique identification tag and k initially unmarked positions, where the i-th position has to be marked in k-bit classical register P when the corresponding quantum state is used in the verification protocol.
• 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.

Properties

• Parameters: HMP₄-states, Let x ∈ {0, 1}⁴. The corresponding HMP₄-states is $|\alpha (x)>={\dfrac {1}{2}}\sum _{1\leq i\leq 4}(-1)^{x_{i}}|i>$

• General Features:
  • No need to quantum communication for quantum coin verification.
  • The classical communication channel used for verification can be unencrypted.
  • 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.
  • Secure against an adversary who uses adaptive “attempted verifications” in order to collect information about a coin.

Protocol

\begin{algorithm} \caption{Quantum coin generation} \noindent\textbf{Input} A secret record consists of $k$ entries $x_1, . . . , x_k, x_i\in\{{0,1}\}^4$\\ \textbf{Output} A “fresh” quantum coin\\ The Trusted Third Party (TTP) chooses $x_1, . . . , x_k\in\{{0, 1}\}^4$ at random, keeps them in secret and produces quantum states $\ket{\alpha(x_1)}, . . . , \ket{\alpha(x_k)}$. A “fresh” quantum coin corresponding to this record consists of: \begin{itemize} \item $k$ quantum registers consisting of 2 qubits each, where the $i$’th register contains $\ket{\alpha(x_i)}$; \item a $k$-bit classical register $P$, that is initially set to $0^k$; \item a unique identification number. \end{itemize} \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}