Abstract
Recently, Shi et al. (Phys Rev A 92:022309, 2015) proposed quantum oblivious set member decision protocol where two legitimate parties, namely Alice and Bob, play a game. Alice has a secret k, and Bob has a set \(\{k_1,k_2,\ldots k_n\}\). The game is designed towards testing if the secret k is a member of the set possessed by Bob without revealing the identity of k. The output of the game will be either “Yes” (bit 1) or “No” (bit 0) and is generated at Bob’s place. Bob does not know the identity of k, and Alice does not know any element of the set. In a subsequent work (Shi et al in Quant Inf Process 15:363–371, 2016), the authors proposed a quantum scheme for private set intersection (PSI) where the client (Alice) gets the intersected elements with the help of a server (Bob) and the server knows nothing. In the present draft, we extended the game to compute the intersection of two computationally indistinguishable sets X and Y possessed by Alice and Bob, respectively. We consider Alice and Bob as rational players, i.e. they are neither “good” nor “bad”. They participate in the game towards maximizing their utilities. We prove that in this rational setting, the strategy profile ((cooperate, abort), (cooperate, abort)) is a strict Nash equilibrium. If ((cooperate, abort), (cooperate, abort)) is strict Nash, then fairness and correctness of the protocol are guaranteed.
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Notes
For the brevity of notation, in the rest of the paper, we will write \(\log (.)\) instead of \(\log _2(.)\).
M qubit string can be written as the tensor product of M individual qubits. As \(j\in \mathbb {Z_N^*}, j\) can be expressed in \(M=\log N\) bits. Each bit corresponds to a qubit. Thus \(\left| j\right\rangle \) can be written as \(\left| d_M\right\rangle ^{\otimes M}\), \(d_M\in \{0,1\}\) and \(M\in [1,\log N]\).
We assume that there is no payoff for a player who deviates from the game and gets partial knowledge about the functionality. In this case, partial knowledge is considered as no knowledge or \(\perp \).
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Maitra, A. Quantum secure two-party computation for set intersection with rational players. Quantum Inf Process 17, 197 (2018). https://doi.org/10.1007/s11128-018-1968-9
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DOI: https://doi.org/10.1007/s11128-018-1968-9