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P2P users Mobile wireless crowds (PDAs, cell phones) What about privacy? A bidirectional approach: Finding what is actually needed Designing useful primitivesZs.<s.<hyMg22L/Cryptographic methods vs. randomization methods00(h In this talkComputing Intersections Two lists of length n O(n) communication, O(n lnln n) computation A little on approximation Computing the K th ranked item (e.g. median) Lists of values from a range of size D Two parties: log k rounds X O(log D) Multi-party: log D rounds X O(log D) Efficient security against malicious parties Z\Z-ZwZ.ZR-4     .t.8a#2i%Private Matching and Set Intersection2with Mike Freedman, NYU Kobbi Nissim, DIMACS, MSR33P Applications of Set Intersection!!( Q& ApplicationsCVS and RiteAid P2P networks Alice: I like Madonna, Britney& Bob: Madonna, Mozart, Bach Dating Genetic matching6==cOComputing the IntersectionPrivate Equality Test (PET) Alice: x. Bob: y. Output: 1 iff x=y Privacy preserving solutions: Cannot use hash functions alone Yao, [FNW], [NP] Generalization: list intersection X = x1, & , xn Y = y1, & , ynL1"L1   ><S Related Work on set intersection!!(Use a circuit Compute intersection using n2 PETs (O(n) computation [NP]) Diffie-Hellman based solutions [FHH99, AES03] Insecure against malicious adversaries Depend on a  random oracle assumption Our work: O(n lnln n) overhead.  Semi-honest adversaries - no RO assumption  Malicious adversaries  with an RO assumption j|O!^* QO!^b-ymT&The basic tool: Homomorphic Encryption''(Semantically secure public key encryption Given Enc(M1), ENC(M2), can compute (without knowing the decryption key) Enc(M1+M2) Enc(c M1) for any constant c. I.e. Enc(a0)+Enc(a1)x+& +Enc(an)xn = Enc(P(x)) Examples: El Gamal, Paillier, DJ. sY#sD   #b~3 U6Homomorphic Encryption: Modified El Gamal Encryption77$& Work in Zp* where DDH is hard. Private key a. Public key PK = ga. Encryption: Enc(M) = (gr, PK r g m) Homomorphic operations: Enc(M1+M2) = Enc(M1) Enc(M2) Enc(c M1) = [Enc(M1)] c Z;       D #@H@b4EV The ScenarioClient: X = x1, & , xn Server: Y = y1, & , yn Output: Client learns X Y. Server learns nothing.9- 8  W The Protocol~Client defines a polynomial of degree n whose roots are x1,& ,xn P(y) = (x1-y)(x2-y)& (xn-y) = anyn + & + a1y + a0 Sends to server homomorphic encryptions of coefficients Enc(an),& , Enc(a0)@89 @@@8 b=H X& The ProtocolbServer uses homomorphic properties to compute "y Enc( rP(y) + y) (r is random) Result is Enc(y) if y XY, or Enc(random) otherwise Server sends (permuted) results to C/)Z/     0 >8% 1YSecurity (semi-honest case)Client s privacy: The server only sees semantically secure encryptions. Learning about C s input = breaking enc. Server s privacy: Given the output XY alone, the client can simulate her  view in the protocol. I.e. encryptions of items in XY, and of random items. N `YZ EfficiencyClient encrypts and decrypts n values Communication is O(n) Server: For each input computes Enc(rP(y)+y), i.e. n exponentiations. Total O(n2) exponentiations TD\D*,7" 8[Improving Efficiency 1>Inputs typically from a  small domain of D values. Represented by log D bits (& 20). Horner s rule P(y)= a0 + y(a1+& y(an-1+yan)..) I.e., exponents are only logD (20) bits Overhead of exponentiation is linear in |exp|. Therefore improvement by a factor of |modulus|/logN (e.g. 1024/20).V >d6i\ Improving Efficiency 2 (Hashing)!!(^Client uses H() to hash inputs to B bins (H is independent of inputs) Let M bound max # of items in a bin. Client defines B polynomials of deg M. Each poly encodes x s mapped to its bin. Client sends poly s and H to server. For every y, server computes H(y) and evaluates a single polynomial of deg M0Z '/D>"0-]Hashing (overhead)0Communication: B M. Server: nM short exponentiations. n full exp. (computes rP(y)+y) Balanced allocations [ABKU]: Choose two bins. Map to the emptier bin. B = n/ lnln n -> M = O(lnln n) (M 5 [BM]) Communication O(n). Server: nlnln n short exp. (n full exp in practice) wZZ@D@& @ K Jt.O   (xMalicious AdversariesAgainst a malicious client: verify that total # of roots (in all B poly s) is n. Solution using cut-and-choose. Exponentially small error probability. Against a malicious server: Privacy: simple. Server receives semantically secure encryptions.>B|BCy"Security against malicious server ##(Correctness: Ensure that there is an input of n items corresponding to server s actions. Problem: server computes rP(y)+y . Solution: server evaluates two poly s. Security based on RO assumption. p N@  ?,s"%^Is Approximation easier?Suppose inputs sets are large, dense, represented as n bit vectors. Approximate size of intersection (i.e. scalar product) with sublinear overhead? Lower bound: L Approximating |XY| within 1  factor requires &!(n) communication (const ). True even for randomized algorithms. Proof: reduction to Razborov s lower bound for Disjointness. ZZS @@e@Bj :,<  c(Approximation Protocol based on Sampling))(b(Approximation Protocol based on Sampling))(xA picks maskA{0,1}, rotA[1,n]. Adds maskA to every bit, shifts X by rotA B does the same with maskB, rotB A uses PIR to learn bit rotA in rotated Y (i.e. masked bit rotA + rotB of original Y). B uses PIR to learn bit rotB in rotated X. SFE computes  and of (results xor masks) Repeat enough times to approx |XY| "=Z          M&).-j,Secure Computation of the Kth-ranked element$-.with Gagan Aggarwal, Stanford Nina Mishra, HPL//k-Secure Computation of the Kth-ranked element*.$ $$zInputs: A: SA B: SB Large sets of unique items (D). There s also the multi-party scenario Output: x SA SB s.t. |{y | y<x, ySASB}| = k-1 Median: k = (|SA| + |SB|) / 2 \"  *  ,} *m MotivationBasic statistical analysis of distributed data E.g. histogram of salaries in all CS departments Sometimes the parties might want to hide the size of their inputs&n$Some information is always revealed %%(The Kth-ranked element reveals some information Suppose SA = x1,& ,x1000 Median of SA SB = x400 Party A now learns that SB contains at least 200 elements smaller than x400 But she shouldn t learn moreHZZiZ$"  ..M uoResults, and previous work8Previous work: generic constructions  overhead at least linear in k. (also, [NN]) New results: Two-party: reduction to log k secure comparisons of log D bit numbers. Multi-party: reduction to log D simple computations with log D bit numbers. Also, security against malicious parties`E   %  p"Insecure two-party median protocol"#"$ q#Two-party median: why does it work?$$ r Secure two-party median protocol !$$sProof of securitySimulation: Given the protocol s output, each party can simulate the execution of the protocol__t"Security against malicious parties##$\Comparison protocol should be secure against malicious parties. Composition thm [C]: in this case, can consider a  hybrid model where comparisons are done by a trusted partyL`zF& Security against malicious parties$$$xVerify that parties inputs to comparisons are consistent. I.e., prevent Round 1: mA = 1000. Is told to delete all x>1000. Round 2: mA = 1100& Implement  reactive computation [C,CLOS]. I.e., each round sends secure  state to next round (i.e., boundaries for parties inputs). Can implement in a single circuitJJ/   3   ,U2{F& Security against malicious parties$$$Why is consistency sufficient? Should generate equivalent input to TTP A s strategy is a binary tree. Each node is an input to a comparison Consistency: nodes are ordered. Each item in A s input corresponds to one internal node. Value used in a node depends only on whether node material for median decision. Simulate A s operation and learn tree..Z>Z>vArbitrary inputs, arbitrary k |Hiding size of inputsCan search for Kth element without revealing size of input sets. However, k=median reveals input size Solution: Let S be a bound on input size A adds to her input (S-|SA|)/2 values of +, and of -. Same for B. Sets of size S. Median value is unchanged.p3%8.% 3 8uDuplicate itemsThe problem: x is the median. Appears twice in SA, and possibly in SB. A does not know which  instance of x was output, and cannot simulate the alg. Solution: create unique items by adding a suffix of logn +1 bits. The protocol is private wrt new inputs.V $Q b>6# wThe multi-party caseLInput: Pi has Si, i=1..n. (unique val s [a,b]) Output: Kth element of S1 & Sn Protocol: Set m = (b-a)/2. Pi inputs to a secure computation Li=|{ySi, y<m}|, Bi= 1 iff mSi. If Li k, set b=m, m=(m-a)/2. If Li=k-1, Bi=1, stop. Kth element is m. Otherwise, set a=m, m = m+(b-m)/2.nIp $ #    @@@ ) F  2} Conclusions6Efficient privacy preserving primitives for basic tasks Open problems Intersection: approximate matching? Median: clustering? There s a lot more to be done6F8F8/hz%&' ` ` ̙33` 333MMM` ff3333f` f` f` 3>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> (    6I "P  T Click to edit Master title style! !$  0K "  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  0HR "``  >*  0 W "`   @*  0[ "`   @*H  0޽h ? f Default Design 0  (      Nkյkյ x$   n*  J&&JJpp   N@:kյkյ  <$  p*  J&&JJppd   c $ ?UI  4   Nkյkյ  L%  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     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