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Private Information Delivery

14 June 2018
Hua Sun
    MoE
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Abstract

We introduce the problem of private information delivery (PID), comprised of KKK messages, a user, and NNN servers (each holds M≤KM\leq KM≤K messages) that wish to deliver one out of KKK messages to the user privately, i.e., without revealing the delivered message index to the user. The information theoretic capacity of PID, CCC, 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 KKK messages, NNN servers, MMM messages stored per server, and N≥⌈KM⌉N \geq \lceil \frac{K}{M} \rceilN≥⌈MK​⌉, we provide an achievable scheme of rate 1/⌈KM⌉1/\lceil \frac{K}{M} \rceil1/⌈MK​⌉ and an information theoretic converse of rate M/KM/KM/K, i.e., the PID capacity satisfies 1/⌈KM⌉≤C≤M/K1/\lceil \frac{K}{M} \rceil \leq C \leq M/K1/⌈MK​⌉≤C≤M/K. This settles the capacity of PID when KM\frac{K}{M}MK​ is an integer. When KM\frac{K}{M}MK​ is not an integer, we show that the converse rate of M/KM/KM/K is achievable if N≥Kgcd⁡(K,M)−(Mgcd⁡(K,M)−1)(⌊KM⌋−1)N \geq \frac{K}{\gcd(K,M)} - (\frac{M}{\gcd(K,M)}-1)(\lfloor \frac{K}{M} \rfloor -1)N≥gcd(K,M)K​−(gcd(K,M)M​−1)(⌊MK​⌋−1), and the achievable rate of 1/⌈KM⌉1/\lceil \frac{K}{M} \rceil1/⌈MK​⌉ is optimal if N=⌈KM⌉N = \lceil \frac{K}{M} \rceilN=⌈MK​⌉. Otherwise if ⌈KM⌉<N<Kgcd⁡(K,M)−(Mgcd⁡(K,M)−1)(⌊KM⌋−1)\lceil \frac{K}{M} \rceil < N < \frac{K}{\gcd(K,M)} - (\frac{M}{\gcd(K,M)}-1)(\lfloor \frac{K}{M} \rfloor -1)⌈MK​⌉<N<gcd(K,M)K​−(gcd(K,M)M​−1)(⌊MK​⌋−1), we give an improved achievable scheme and prove its optimality for several small settings.

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