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Efficient Private Approximation Protocols Piotr Indyk David Woodruff Work in progress

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Outline 1.Private approximation of L 2 distance 2.Private near neighbor 3.Private approximate near neighbor

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1. Private approximation of L 2 distance

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a {0,1} n b {0,1} n Want to compute some function F(a,b) Security: protocol does not reveal anything except for the value F(a,b) –Semi-honest: both parties follow protocol –Malicious: parties are adversarial Efficiency: want to exchange few bits Secure communication Alice Bob

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Secure Function Evaluation (SFE) [Yao, GMW]: If F computed by circuit C, then F can be computed securely with O~(|C|) bits of communication [GMW] + … + [NN]: can assume parties semi- honest –Semi-honest protocol can be compiled to give security against malicious parties Problem: circuit size at least linear in n * O~() hides factors poly(k, log n)

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Secure and Efficient Function Evaluation Can we achieve sublinear communication? Ideally: secure computation with communication comparable to insecure case With sublinear communication, many interesting problems can be solved only approximately. What does it mean to have a private approximation?

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Private Approximation [FIMNSW01]: A protocol computing an approximation G(a,b) of F(a,b) is private, if each party can simulate its view of the protocol given the exact value F(a,b) Note: not sufficient to simulate non-private G(a,b) using SFE Example: –Define G(a,b): bin(G(a,b)) i =bin( (a,b)) i if i>0 bin(G(a,b)) 0 =a 0 –G(a,b) is a 1 -approximation of (a,b), but not private

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Concrete Pitfall: Dimension Reduction A basic problem: Hamming distance (a,b) Approximate decision version: with prob. 1-, –If (a,b)r, answer NO –If (a,b)r(1+ ), answer YES [Kushilevitz-Ostrovsky-Rabani98]: –Create m n binary matrix D, where Pr[D ij =1]= 1/(2r) for m= O~(log 1/ / 2 ) –Exchange Da, Db (mod 2) –Answer YES if wt[D(a-b)]>r, r function of r, NOTE: This protocol was not designed to be private

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Non-Privacy of KOR Let x = a – b. If, wt(x) = r, r log n ¼ m then can recover x from D, Dx in O(mn) time! Algorithm: for j=1…n, estimate Pr[ =1| d ij =1] = Pr[ =1 d ij =1]/Pr[d ij =1] –If x j =1 then Pr[ =1|d ij =1] is high –If x j =0 then Pr[ =1|d ij =1] is low

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Approximating Hamming Distance [FIMNSW01]: A private protocol with complexity O~(n 1/2 / ) –wt(x) small: compute wt(x) using O~(wt(x)) bits –wt(x) high: sample O~(n/wt(x)) x i, estimate wt(x) Our result: –Complexity: O~(1/ 2 ) bits –Works even for L 2 norm, i.e., estimates ||x|| 2 for a,b {1…M} n * O~() hides factors poly(k, log n, log M, log 1/ )

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Crypto Tools SFE of circuits [Yao86]: O~(|circuit|) communication Efficient SPIR or OT 1 n : –Alice has A[1] … A[n] 2 {0,1} m, Bob has i 2 [n] –Goal: Bob privately learns A[i] and thats it –Can be done using O~(m) communication [CMS99, NP99] Circuits with ROM [Naor, Nissim01]: –Standard AND/OR/NOT gates –Lookup gates: In: i Out: M gate [i] –Takes care of the security of computation: begin secure … end secure –Can just focus on privacy of the output Communication at most O~(m|C|)

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High-dimensional tools Random projection: –Take a random orthonormal n n matrix D, that is ||Dx|| = ||x|| for all x. –There exists c>0 s.t. for any x R n, i=1…n Pr[ (Dx) i 2 > ||Dx|| 2 /n * k] < e -ck

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Approximating ||a-b|| 2 Recall: –Alice has a 2 [M] d, Bob has b 2 [M] d –Goal: estimate ||x|| 2, x=a-b

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Algorithm Alice and Bob create random orthonormal matrix D such that, for each i=1…n (Dx) i 2 < k||x|| 2 /n T=M 2 n+1 Repeat –{Assertion: ||x|| 2 T} –Invoke P RIVATE S AMPLE to get L=O~(1/ 2 ) independent bits z i such that Pr[z i =1]=||Dx|| 2 /(Tk) –T = T/2 Until Σ i z i L/(4k) Output E= Σ i z i /L * 2Tk as an estimate of ||x|| 2 Correctness: –Unbiased estimator –High probablity from Chernoff bound SECURE!

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P RIVATE S AMPLE P=Tk/n Pick random t [n] Retrieve (Da) t, (Db) t Compute (Dx) t = (Da) t - (Db) t Define v=[(Dx) t ] 2 If v P then generate z s.t. Pr[z=1]=v/P Else output fail Output z Correct as long as (Dx) 2 i < Tk/n for each i=1…n SECURE! Generate independent bits z i with E[z i ] = ||Dx|| 2 /(Tk)

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Algorithm, again Alice and Bob create random * orthonormal ** matrix D such that, for each i=1…n (Dx) i 2 < ||x|| 2 /n * k T=M 2 n+1 Repeat –{Assertion: ||x|| 2 T} –Invoke P RIVATE S AMPLE to get L=O~(1/ 2 ) independent bits z i such that Pr[z i =1]= ||Dx|| 2 /Tk { Works as long as (Dx) 2 i < Tk/n for each i=1…n} – T=T/2 Until Σ i z i L/(4k) Output E= Σ i z i /L * 2Tk as an estimate of ||x|| 2 If Assertion not true, then Pr[z i =1]>1/(2k) E[Σ i z i ] > L/(2k) >> L/(4k)

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Simulation SIMULATION Repeat –Choose L independent bits z i such that Pr[z i =1]= ||x|| 2 /Tk –T=T/2 Until Σ i z i (L/k) Output E= Σ i z i /L * 2Tk as an estimate of ||x|| 2 ALGORITHM Repeat –{Assertion: ||x|| 2 T} –Invoke P RIVATE S AMPLE to get L independent bits z i such that Pr[z i =1]= ||Dx|| 2 /Tk –T=T/2 Until Σ i z i (L/k) Output E= Σ i z i /L * 2Tk as an estimate of ||x|| 2 Recall: ||Dx||=||x|| Communication: O~(1/ 2 )

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2. Private near neighbor

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Private Near Neighbor q 2 [U] d P = p 1, p 2, …, p n 2 {1, 2, …, U} d = [U] d Distance function: f(x,y) Correctness: Bob learns min i f(q, p i ) Privacy: Alice learns nothing, Bob learns nothing else Goal: Minimize communication AliceBob

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Private Near Neighbor f(a,b) = i f i (a i, b i ) L 2 Generalized Hamming Set Difference Previous [DA]O~(ndU)O~(nd)O~(ndU) Our ResultsO~(dU+n)O~(n+d)O~(d 2 + n)O~(n+d) [DA] needs 3 rd party, we dont Approach: homomorphic encryption + secure function evaluation (SFE) n points, dimension d, universe [U]

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Coordinate-wise distance functions q 2 [U] d P = p 1, p 2, …, p n 2 [U] d AliceBob Bob: 1. For each coordinate, create a degree-(U-1) polynomial g j (x) = i a i,j x i such that g j (u) = f j (q j, u) for all u 2 [U] 2. Generate (SK, PK) for Paillier Encryption scheme. Send PK and E PK (a i, j ) for all i,j Alice: 1. For all i, E( j g j (p i,j )) = E(f(q, p i )) SFE: Inputs: Alice – E(f(q, p i )) Bob - SK 1. Bob gets min i D SK (E(f(q, p i ))) Coordinate-wise distance functions: f(a,b) = f i (a i, b i ) E(x), E(y) -> E(x + y) E(x), c -> E(cx)

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Generic distance functions Security: 1. Replace SFE with oracle 2. Alice View indistinguishable from PK, E(0), E(0), …, E(0) – E semantically secure 3. Bob View just = output Efficiency: 1. Send polynomials = O~(dU) 2. SFE = O~(n) (simple circuit)

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Private Near Neighbor Pointwise distance L 2 Generalized Hamming Set Difference Previous [DA]O~(ndU)O~(nd)O~(ndU) Our ResultsO~(dU+n)O~(n+d)O~(d 2 + n)O~(n+d) n points, dimension d, universe [U] (homomorphic tricks) Alice x 1, …, x n 2 {0,1} d, Bob y 1, …, y n 2 {0,1} d, Threshold t Bob gets all x i s.t. (x i, y j ) < t for some j Communication: O~(n 2 + nd 2 ). Resolves open question of [FNP04]: [FNP04] achieve O~((d choose t)nt) May be superpolynomial in n

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3. Private Approximate Near Neighbor

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Private Near Neighbor Drawback: Protocols depend linearly on # points n Necessary? Not if algebraically homomorphic E exists Our approach: solve the approximate problem

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Private c-Approximate Near Neighbor Alice has P = {p 1, …, p n } {0,1} d, Bob has q {0,1} d PrPr P cr Notation: P r = P B(q, r) Correctness: P r nonempty Bob learns some element of P cr Privacy: Bobs view simulatable given q and P cr

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Private Approximate Near Neighbor Definition Remarks: Privacy: Dont care what Bob gets as long as it follows from P cr Simulator gets P cr Correctness: Dont specify anything if P r empty, but view still simulatable Our results: - O~(n 1/2 + d) - If Bob just wants some coordinate of an element of P cr, then improve to O~(n 1/2 + polylog(d))

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Private Approximate Near Neighbor Two approaches: 1. Dimensionality Reduction in Hamming Cube [KOR98] 2. Locality Sensitive Hashing [IM98] This talk: protocol using #1

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Dimensionality Reduction [KOR]: Let A be random m times d binary matrix, m = O(log d / 2 ) Then there is a separator r s.t. with probability 1-1/n 2, for any p,q {0,1} d 1. (p,q) > cr (Ap, Aq) > r 2. (p,q) · r (Ap, Aq) < r Idea: Alice 1. Applies A to P dimension small 2. Enumerates all w {0,1} m, forms array: B[w]={p 2 P s.t. (Ap, w) < r} 3. Use Oblivious ROM

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Dimensionality reduction protocol 2. Agree on k matrices A 1, …, A k 3. Create array B i based on A i 4. B i [p] contains any n 1/2 points p 2 P s.t. (A i p, p) < r 5. Alice sets ROM to be the B i s P cr 1. Randomly sample O~(n 1/2 ) points P 1 2. If |P cr | > n 1/2, then P 1 Å P cr ;, w.h.p. Protocol: 6. If P 1 Å P cr ;, SFE outputs a random element of P 1. Otherwise, SFE uses [ i B i [A i q] to output a random element of P r

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Dimensionality Reduction Analysis Properties: 1. If |P cr | > n 1/2, we output random element of P cr,w.h.p. 2. If |P cr | < n 1/2, by properties of A, for any p P r, Pr A [8 p 2 P r, (Ap, Aq) r] > 1- 1/n 3. Since bucket size is n 1/2 and |P cr | < n 1/2, p B i [A i q], P r i B i [A i q] Correctness: If |P cr | > n 1/2, output element from P cr Else output an element from P r

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Dimensionality Reduction Analysis Communication: 1. Sampling O~(n 1/2 ) elements to ensure |P cr | < n 1/2 2. OT on O~(1) buckets of size n 1/2 Thus, balanced steps 1 & 2 O~(dn 1/2 ) total communication Simulatability: Output either a random element of P cr, or a random element of P r

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Dimensionality Reduction Analysis Dependence on d: 1. Homomorphic encryption: O~(d + n 1/2 ) 1. Bob sends E(q 1 ), …, E(q d ) 2. Alice computes E( (p i, q)) - Uses these for sampling and bucketing 2. Reduce to O~(polylog(d) + n 1/2 ) if Bob just wants a coordinate of point in P cr – use approximations

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Conclusions Extensions: Can achieve O(n 1/3 + d) communication if you allow the protocol to leak k bits of information Open problems: 1. Polylogarithmic Private Approximation of other distances 2. More efficient protocols for exact near neighbor. Tricks for PIR may be useful 3. Polylogarithmic c-approx NN protocol

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