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Multi-Database Classical Symmetric Private Information Retrieval with Quantum Key Distribution
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==Notation== <!-- Connects the non-mathematical outline with further sections. --> * <math>U</math>: user. * <math>D_i</math>: <math>i^{\text{th}}</math> data centre. * <math>E</math>: Eve, an eavesdropper. * <math>R</math>: user’s local randomness (random variable); <math>r</math>: given randomness (<math>R=r</math>). * <math>X</math>: input random variable; <math>x</math>: given input (<math>X=x</math>). * <math>W</math>: database random variable; <math>w</math>: given database (<math>W=w</math>). * <math>W_X</math>: random variable which represents the database entry that the user wants to retrieve; <math>w_x</math>: given desired database entry (<math>W_X=w_x</math>). * <math>f_{\text{query},i}</math>: query function, used by the user to generate his query to the <math>i^{\text{th}}</math> data centre. Input and output are random variables. * <math>f_{\text{ans},i}</math>: answer function, used by the <math>i^{\text{th}}</math> data centre to generate his answer to the user’s query. Input and output are random variables. * <math>f_{\text{dec}}</math>: decoding function, used by the user to decode the desired database element. Input and output are random variables. * <math>S_i</math>: QKD key random variable (the user has <math>S_2</math> and <math>S_4</math> to communicate with data centres 1 and 2 respectively; data centre 1 has <math>S_1</math> and <math>S_5</math> to communicate with the user and data centre 2 respectively; data centre 2 has <math>S_3</math> and <math>S_6</math> to communicate with the user and data centre 1 respectively); <math>s_i</math>: given QKD key (<math>S_i=s_i</math>). * <math>S_i^\text{enc}</math>: half of the key <math>S_i</math> used for encryption (random variable); <math>s_i^\text{enc}</math>: given key (<math>S_i^\text{enc}=s_i^\text{enc}</math>). * <math>S_i^\text{dec}</math>: half of the key <math>S_i</math> used for decryption (random variable); <math>s_i^\text{dec}</math>: given key (<math>S_i^\text{dec}=s_i^\text{dec}</math>). * <math>Q_i</math>: query generated by the user for the <math>i^{\text{th}}</math> data centre. * <math>\tilde{Q}_i</math>: query received by the <math>i^{\text{th}}</math> data centre (may be different from <math>Q_i</math>). * <math>C_{Ui}</math>: secure channel connecting the user with the <math>i^{\text{th}}</math> data centre. * <math>A_i</math>: reply generated by the <math>i^{\text{th}}</math> data centre for the user after receiving <math>\tilde{Q}_i</math>. * <math>\tilde{A}_i</math>: reply from the <math>i^{\text{th}}</math> data centre received by the client (may be different from <math>A_i</math>). * <math>\hat{w}_x</math>: database element retrieved by the user at the end of the protocol. * <math>p_\text{fail}</math>: probability that at least one of the three QKD protocols aborted (<math>p_\text{fail}=1-(1-p_{\perp,U1})(1-p_{\perp,U2})(1-p_{\perp,12})</math>). * <math>p_{\perp,AB}</math>: probability that the QKD protocol between parties <math>A</math> and <math>B</math> aborts, where <math>A,B\in\{U,1,2\}</math> (<math>U</math>: user; <math>1</math>: data centre 1; <math>2</math>: data centre 2). * <math>\text{pass}</math>: this is the event “none of the three QKD protocols aborted”. * <math>\Delta(.,.)</math>: trace distance between two quantum systems (<math>\Delta(\rho,\sigma)=\frac{1}{2} ||\rho-\sigma||_1</math> where <math>||.||_1</math> is the trace norm). * <math>\rho_{UE}^w</math>: total view of the user <math>U</math> and the eavesdropper <math>E</math> conditioned by <math>w</math>. * <math>\rho_{D_j E}^x</math>: total view of data centre <math>D_j</math> and the eavesdropper <math>E</math> conditioned by <math>x</math>. * <math>\rho_E^{x,w}</math>: total view of the eavesdropper <math>E</math> conditioned by <math>x</math> and <math>w</math>. <!--==Knowledge Graph==--> <!-- Add this part if the protocol is already in the graph --> <!--{{graph}}-->
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