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Multi-Database Classical Symmetric Private Information Retrieval with Quantum Key Distribution
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==Properties== <!-- important information on the protocol: parameters (threshold values), security claim, success probability... --> For this protocol, SPIR security definitions (see [[(Symmetric) Private Information Retrieval]]) are extended as follow: * <math>\eta_\text{cor}</math>-'''correctness''': If the user and the data centres are honest, then for any index <math>x</math> and database <math>w</math>, <math>(1-p_\text{fail})Pr[\hat{w}_x \neq w_x|\text{pass}]\leq\eta_\text{cor}</math>. * <math>\eta_{UP}</math>-'''user privacy''': If the user is honest, then for any database <math>w</math> and shared secret keys between the data centres <math>(s_5,s_6)</math>, and for any data centre <math>D_j</math>, <math>\Delta(\rho_{D_{j} E}^x,\rho_{D_{j} E}^{x'}) \leq \eta_{UP}</math> for all indexes <math>x</math> and <math>x'</math>. * <math>\eta_{DP}</math>-'''database privacy''': If the data centres are honest, then for any index <math>x</math>, randomness <math>r</math> and keys <math>(s_1^\text{dec},s_2^\text{enc},s_3^\text{dec},s_4^\text{enc})</math>, then there exists an index <math>x'</math> such that for all databases <math>w</math> and <math>w'</math> with <math>w_{x'}={w'}_{x'}</math>, <math>\Delta(\rho_{UE}^w,\rho_{UE}^{w'}) \leq \eta_{DP}</math>. In addition to the above extended SPIR security definitions, the notion of protocol secrecy is added which allows to bound the amount of information that an external eavesdropper may obtain on the database <math>w</math> or the index of the desired database element <math>x</math>. This extra definition is required for this protocol as it relies on practical QKD protocols, which means that there is a non-zero probability that an eavesdropper may learn part of the QKD keys. * <math>\eta_{PS}</math>-'''protocol secrecy''': If the user and the data centres are honest, then for all index-database pairs <math>(x,w)</math> and <math>(x',w')</math>, <math>\Delta(\rho_{E}^{x,w},\rho_{E}^{x',w'}) \leq \eta_{PS}</math>. Those extended security definitions result from the possibility of unperfect QKD keys and the non-zero probability that one of the three QKD protocols may abort. A SPIR protocol that satisfies the four above conditions is said to be <math>(\eta_\text{cor},\eta_{UP},\eta_{DP},\eta_{PS})</math>-'''secure'''.<br></br> A two-database one-round <math>(0,0,0,0)</math>-secure SPIR protocol with ideal keys replaced by <math>\epsilon</math>-secure QKD keys, where <math>\epsilon=\epsilon_\text{corr}+\epsilon_\text{sec}</math> (see [[Quantum Key Distribution#Properties|QKD security definitions]]), can be shown to be <math>(3\epsilon_\text{corr},2\epsilon,2\epsilon,4\epsilon)</math>-secure (see [https://www.mdpi.com/1099-4300/23/1/54/htm Kon and Lim (2021)] for a proof).
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