Private Information Delivery

We introduce the problem of private information delivery (PID), comprised of K messages, a user, and N servers (each holds M ≤ K messages) that wish to deliver one out of K messages to the user privately, i.e., without revealing the delivered message index to the user. The information theoretic capacity of PID, C, 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 K messages, N servers, M messages stored per server, and N ≥ ΓMK⌉, we provide an achievable scheme of rate 1/ΓMK⌉ and an information theoretic converse of rate M/K, i.e., the PID capacity satisfies 1/ΓKM⌉ ≤ C ≤ M/K. This settles the capacity of PID whenMKis an integer. When K/M is not an integer, we show that the converse rate of M/K is achievable if N ≥ K/gcd(K,M ) - ( M/gcd(K,M) - 1)(⌊K/M⌋- 1), and the achievable rate of 1/ΓMK⌉ is optimal if N =ΓK/M⌉. Otherwise if ΓK/M⌉ <; N<; K/gcd(K,M) -( M/gcd(K,M) -1)(⌊K/M⌋-1), we give an improved achievable scheme and prove its optimality for several small settings.