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Private Information Delivery

Abstract

We introduce the problem of private information delivery (PID), comprised of KK messages, a user, and NN servers (each holds MKM\leq K messages) that wish to deliver one out of KK messages to the user privately, i.e., without revealing the delivered message index to the user. The information theoretic capacity of PID, CC, is defined as the maximum number of bits of the desired message that can be privately delivered per bit of total communication to the user. For the PID problem with KK messages, NN servers, MM messages stored per server, and NKMN \geq \lceil \frac{K}{M} \rceil, we provide an achievable scheme of rate 1/KM1/\lceil \frac{K}{M} \rceil and an information theoretic converse of rate M/KM/K, i.e., the PID capacity satisfies 1/KMCM/K1/\lceil \frac{K}{M} \rceil \leq C \leq M/K. This settles the capacity of PID when KM\frac{K}{M} is an integer. When KM\frac{K}{M} is not an integer, we show that the converse rate of M/KM/K is achievable if NKgcd(K,M)(Mgcd(K,M)1)(KM1)N \geq \frac{K}{\gcd(K,M)} - (\frac{M}{\gcd(K,M)}-1)(\lfloor \frac{K}{M} \rfloor -1), and the achievable rate of 1/KM1/\lceil \frac{K}{M} \rceil is optimal if N=KMN = \lceil \frac{K}{M} \rceil. Otherwise if KM<N<Kgcd(K,M)(Mgcd(K,M)1)(KM1)\lceil \frac{K}{M} \rceil < N < \frac{K}{\gcd(K,M)} - (\frac{M}{\gcd(K,M)}-1)(\lfloor \frac{K}{M} \rfloor -1), we give an improved achievable scheme and prove its optimality for several small settings.

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