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Quantum Private Queries Protocol Based on Quantum Oblivious Key Distribution
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==Protocol Description== <!-- Mathematical step-wise protocol algorithm helpful to write a subroutine. --> '''Inputs:''' * User (Alice): index <math>i</math> of the desired database element. * Server (Bob): classical database <math>X</math> with <math>N</math> elements: <math>X_1,...,X_N</math>. * Both (Alice & Bob): a security parameter <math>k</math>. '''Output:''' * User (Alice): desired database element <math>X_i</math>. '''Protocol:''' * '''Quantum oblivious key distribution:''' # Bob generates a random bit string of length <math>k\times N</math>. # Bob encodes each bit of the string in the state of a qubit belonging to one of two bases <math>\updownarrow := \{|\uparrow\rangle ,|\downarrow \rangle \}</math> or <math>\leftrightarrow :=\{|\leftarrow\rangle,|\rightarrow\rangle\}</math>: for each bit, if it is <math>0</math>, Bob chooses between <math>|\uparrow\rangle</math> and <math>|\downarrow\rangle</math>, if it is <math>1</math>, Bob chooses between <math>|\leftarrow\rangle</math> and <math>|\rightarrow\rangle</math>. # Bob sends the prepared qubits to Alice. # Alice measures each state in the <math>\updownarrow </math> basis or <math>\leftrightarrow </math> basis at random. # Alice announces to Bob in which instances she detected the sent qubit. # For each qubit that Alice successfully detected, Bob announces a pair of two states including the state that was sent. The other state is chosen at random in the other basis. # Partial reconstruction of the key: Using her measurement outcomes and Bob’s announced pairs of qubits, Alice can deduce the bit value of part of the key (<math>1/4</math> of conclusive results). ''At this stage, Alice and Bob share a secret key of length <math>k\times N</math> which is known partially (<math>{1/4}^\text{th}</math>) to Alice and entirely to Bob. Bob does not know which bits of the key are known to Alice.'' # Reduction of the key length: The key is cut into <math>k</math> substrings of length <math>N</math>. Then pieces of the key are added bitwise to create a new key of length <math>N</math>, <math>K^f</math>. This reduces the number of bits of the new key that Alice knows to roughly one bit (ideally, Alice should know exactly one bit of the new key). *'''Query:''' Alice announces to Bob a shift <math>s=j-i</math> where <math>i</math> is the index of the desired database element and <math>j</math> is the index of a bit of the key known to Alice. *'''Answer:''' Bob announces to Alice <math>N</math> bits <math>C_n=X_n\oplus K_{n+s}^f</math> corresponding to the encoded database. *'''Retrieval:''' Alice recovers the desired database element <math>X_i</math> by adding her known bit of the key, <math>K_j^f</math> to the <math>i^\text{th}</math> received bit <math>C_i</math>: <math>C_i\oplus K_j^f=X_i\oplus K_{i+s}^f\oplus K_j^f=X_i\oplus K_{i+j-i}^f\oplus K_j^f=X_i</math>.
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