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**Polylogarithmic Private Approximations and Efficient Matching**

David Woodruff MIT, Tsinghua Piotr Indyk MIT TCC 2006

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**Secure communication a {0,1}n b {0,1}n**

Alice Bob 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

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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? With sublinear communication, many interesting problems can be solved only approximately. What does it mean to have a private approximation? Efficiency: want SFE with communication comparable to insecure case

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**Private Approximation**

[FIMNSW’01]: 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) 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=a0 G(a,b) is a 1 -approximation of (a,b), but not private Popular protocols for approximating (a,b), e.g., [KOR98], are not private

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**Approximating Hamming Distance**

[FIMNSW01]: A private protocol with complexity O~(n1/2/ ) (a,b) small: compute (a,b) exactly in O~((a,b)) bits (a,b) high: sample O~(n/(a,b)) (a-b)i, estimate (a,b) Our main result: Complexity: O~(1/2) bits Works even for L2 norm, i.e., estimates ||a-b||2 for a,b {1…M}n * O~() hides factors poly(k, log n, log M, log 1/)

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**Crypto Tools Efficient OT1n:**

P1 has A[1] … A[n] 2 {0,1}m , P2 has i 2 [n] Goal: P2 privately learns A[i], P1 learns nothing Can be done using O~(m) communication [CMS99, NP99] Circuits with ROM [NN01] (augments [Yao86]) Standard AND/OR/NOT gates Lookup gates: In: i Out: Mgate[i] 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 nn matrix D, that is ||Dx|| = ||x|| for all x. There exists c>0 s.t. for any xRn, i=1…n Pr[ (Dx)i2 > ||Dx||2/n * k] < e-ck

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**Approximating ||a-b||**

Recall: Alice has a 2 [M]d, Bob has b 2 [M]d Goal: privately estimate ||a-b||, x=a-b Suffices to estimate ||a-b||2

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Protocol Intuition Alice and Bob agree upon a random orthonormal matrix D Efficient by exchanging a seed of a PRG Alice and Bob rotate vectors a,b, obtaining Da, Db ||Da-Db|| = ||a-b|| D “spreads the mass” of the difference vector uniformly across the n coordinates. Can now try obliviously sampling coordinates as in [FIMNSW01]

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**Protocol Intuition Con’d**

Alice and Bob agree upon random orthonormal D Alice and Bob rotate a,b, obtaining Da, Db Use secure circuit with ROMs Da and Db to: Circuit obtains (Da)i and (Db)i for many random indices i Problem: Now what? Samples leak a lot of info! Fix: - Suppose you know upper bound T with T ¸ ||a-b||2 - Flip a coin z with heads probability n((Da)i – (Db)i)2/(kT) - Then E[z] = n||Da-Db||2/(nkT) = ||a-b||2/(kT) - E[z] only depends on ||a-b||, and z only depends on E[z]!

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**Protocol Intuition Con’d**

Alice and Bob agree upon random orthonormal D Alice and Bob rotate a,b, obtaining Da, Db Use secure circuit with ROMs Da, Db, to: Obtain (Da)i and (Db)i for L random i Generate Bernoulli z1, … , zL with E[zi] = ||a-b||2/(kT) Output kT zi/L Privacy: View only depends on ||a-b|| Problem: Correctness! A priori bound T=M2 n, but ||a-b||2 may be (1), so (n) samples required. Fix: Private binary search on T

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**Protocol Intuition Con’d**

… Use secure circuit with ROMs Da, Db to: Obtain (Da)i and (Db)i for L random i Generate Bernoulli z1, … , zL with E[zi] = ||a-b||2/(kT) Output kT zi/L Fix: - Private binary search on T - If many zi = 0, then intuitively can replace T with T/2 - Eventually T = ~(||a-b||2) - We will show: final choice of T is simulatable!

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**One last detail Want to show final choice of T is simulatable**

Estimate is kT zi/L and we stop when “many” zi = 1 Recall E[zi] = ||a-b||2/(kT) Key Observation: Since orthonormal D is uniformly random, can guarantee that if many zi = 0, then ||a-b||2 << T. Note: - Suppose didn’t use D, and a = (M, 0, …, 0), b = (0, …, 0) - Then ||a-b||2 = M2 is large, but almost always zi = 0, so you’ll choose T < ||a-b||2. - Not simulatable since T depends on the structure of a, b

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**Algorithm vs. Simulation**

Repeat Generate L independent bits zi such that Pr[zi=1]= ||a-b|| 2/Tk T=T/2 Until Σi zi ≥ (L/k) Output E= Σi zi /L * 2Tk as an estimate of ||a-b||2 ALGORITHM Repeat Generate L independent bits zi such that Pr[zi=1]= ||D(a-b)|| 2/Tk T=T/2 Until Σi zi ≥ (L/k) Output E= Σi zi /L * 2Tk as an estimate of ||a-b||2 Recall:||D(a-b)||=||a-b|| Communication = O~(L) = O~(1/2)

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Other Results Use homomorphic encryption tricks to get better upper bounds for private nearest neighbor and private all-pairs nearest neighbors. Define private approximate nearest neighbor problem: Requires a new definition of private approximations for functionalities that can return sets of values. Achieve small communication in this setting.

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