Towards 3-query locally decodable codes of subexponential length
Journal of the ACM (JACM)
Communications of the ACM
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A $t$-private private information retrieval (PIR) scheme allows a user to retrieve the $i$th bit of an $n$-bit string $x$ replicated among $k$ servers, while any coalition of up to $t$ servers learns no information about $i$. We present a new geometric approach to PIR and obtain the following: (1) A $t$-private $k$-server protocol with communication $O (\frac{k^2}{t} \log k n^{1/\left \lfloor (2k-1)/t \right \rfloor})$, removing the ${k}{t}$ term of previous schemes. This answers an open question of [Y. Ishai and E. Kushilevitz, in Proceedings of the $31$st ACM Symposium on Theory of Computing, 1999, pp. 79-88]. (2) A $2$-server protocol with $O(n^{1/3})$ communication, polynomial preprocessing, and online work $O(n/\log^r n)$ for any constant $r$. This improves the $O(n/\log^2 n)$ work of [A. Beimel, Y. Ishai, and T. Malkin, J. Cryptology, 17 (2004), pp. 125-151]. (3) Smaller communication for instance hiding [D. Beaver, J. Feigenbaum, J. Kilian, and P. Rogaway, J. Cryptology, 10 (1997), pp. 17-36; Y. Ishai and E. Kushilevitz, in Proceedings of the $31$st ACM Symposium on Theory of Computing, 1999, pp. 79-88], PIR with a polylogarithmic number of servers, and robust PIR [A. Beimel and Y. Stahl, in Proceedings of the $3$rd Conference on Security in Communications Networks (SCN $2002$), Lecture Notes in Comput. Sci. 2576, Springer, Berlin, 2003, pp. 326-341].