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Constant-Round Private Database Queries Nenad Dedic and Payman Mohassel Boston UniversityUC Davis

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Outline Introduction Element rank protocol Other protocols Equivalence to one-round PIR Open problems

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Succinct Computation y Client Server x q = Q(x) a = A(q,y) Dec(a) = f(x,y) Computing f(x,y) One round of interaction Communication Complexity |q| +|a| = O(poly(log(|x|), log(|y|), |f(x,y)|, s)) Or linear in |f(x,y)|

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Privacy Computational setting Client side For any x, x, Q(x) and Q(x) are indistinguishable Server side Simulator S, simulates A(x,y) given x and f(x,y) Semi-honest adversaries

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Private Database Queries Servers input is a database Clients input is a query Private information retrieval (PIR) f(i, (x 1,x 2,…,x n )) = x i Private Keyword search (PKS) f(w, {(x 1,v 1 ),…,(x n,v n )}) = v a if there is x a = w otherwise

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Existing Solutions PIR / SPIR [KO97], [Lipmaa05], … One-round, sublinear communication PKS [FIPR05] One-round, polylog(n) communication PIR and homomorphic encryption How about more general queries?

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More General Queries General MPC Not efficient Circuits with look-up tables [NN01] Communication efficient High round complexity One-round secure computation [CCKM00] Round efficient High comm. Computing BP on encrypted data [IP07] Independent work Round and communication efficient Strong assumption

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Private Element Rank Interval Labeling f(b, (x 1,x 2,…,x n,v 1,…,v n )) = v i such that b є (x i, x i+1 ] Element Rank Add x 0 = - and x n+1 =+ v i = i Applications Ranking in auctions Online testing services Use to design other protocols

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Interval Labeling Protocol b, x 1,x 2,…,x n є {0,1} k Run a PKS for every prefix of b j th query = j-bit prefix of b Create and use a database D

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Interval Labeling Protocol v1v1 v2v2 v4v4 v0v0 v1v1 v2v2 v2v2 v3v3 x2x2 x1x1 x3x3 x4x D = {(000,v 0 ),(001,v 1 ),(0100,v 1 ), (0101,v 2 ),(011,v 2 ),(100,v 2 ),(101,v 3 ),(11,v 4 )}

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Interval Labeling Protocol v1v1 v2v2 v4v4 v0v0 v1v1 v2v2 v2v2 v3v3 x2x2 x1x1 x3x3 x4x D = {(000,v 0 ),(001,v 1 ),(0100,v 1 ), (0101,v 2 ),(011,v 2 ),(100,v 2 ),(101,v 3 ),(11,v 4 )} b = 1000 b 1 = 1 b 2 =10 b 3 =100 b 4 =1000

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Interval Labeling Protocol w is w with last bit flipped Database D, where |D| 2kn For every 1 j k, let w be j-bit prefix of x i : 1. Add (w,v i ) to D if: [w||0 k-j, w||1 k-j ] [x i,x i+1 ], but not true for w 2. Add (w,v i ) to D if: [w||0 k-j, w||1 k-j ] [x t,x t+1 ], but not true for w Prefixes of x i s and/or their siblings

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Interval Labeling r i = PKS A (b i,D) for 1 i k Randomly permute (r 1, r 2, …,r k ) and send Decode; retrieve the only r i in the list One round, polylog(n) communication Reduced to PKS

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Other Protocols Private Rectangle Labeling Which rectangle is query point in? Extension to higher dimensions One round Private Range Queries Retrieve all the points in the range On a line or in a plane Constant round Comm. proportional to number of retrieved points

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Other Protocols m th ranked element Alice holds database A Bob holds database B Find m th ranked element in (A U B) [AMP04], O(log(m)) rounds, and sublinear comm. We use our rank protocol as subprotocol O(log(log(m))) rounds Still sublinear comm.

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PKS to PIR [FIPR05] Database Hash function h : {0,1} n {0,1} n/log(n) Hash keywords (x i s) to n/log(n) bins Create degree log(n) polynomials for each bin Client Compute h(w) Send E(h(w)), E(h(w) 2 ), …, E(h(w) log(n) ) Database evaluates all polynomials at h(w) Client gets one result via PIR v a if there is x a = w otherwise f(w, {(x 1,v 1 ),…,(x n,v n )}) =

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PKS to PIR Assumption: One-round PIR Replace polynomials with Yaos garbled circuit Circuit of size O(polylog(n)) size Yaos protocol Pseudorandom function, OT Can be reduced to one-round PIR [CMO00], [BIKM99] One-round PKS one-round PIR One-round Rank one-round PKS

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Open Problems Succinct Computation of Branching programs (not length-bounded) General circuits Reduction to one-round PIR Any special functionality Decision trees Branching programs

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Thank you!

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