Approximate Privacy: Foundations and Quantification

1. Approximate Privacy:Foundations and Quantification Joan Feigenbaum http://www.cs.yale.edu/homes/jf Northwest Univ.; May 20, 2009 Joint work with A. D. Jaggard and M. Schapira

2. Starting Point: Agents’ Privacy in MD • Traditional goal of mechanism design: Incent agents to reveal private information that is needed to compute optimal results. • Complementary, newly important goal: Enable agents not to reveal private information that is not needed to compute optimal results. • Example (Naor-Pinkas-Sumner, EC ’99): It’s undesirable for the auctioneer to learn the winning bid in a 2nd–price Vickrey auction.

3. Privacy is Important! • Sensitive Information: Information that can harm data subjects, data owners, or data users if it is mishandled • There’s a lot more of it than there used to be! • Increased use of computers and networks • Increased processing power and algorithmic knowledge • Decreased storage costs • “Mishandling” can be very harmful. • ID theft • Loss of employment or insurance • “You already have zero privacy. Get over it.” (Scott McNealy, 1999)

4. . . . xn-1 x3 xn x2 x1 Private, MultipartyFunction Evaluation y = F (x1, …, xn) • Each i learns y. • No i can learn anything about xj • (except what he can infer from xiand y ). • Very general positive results.

5. Drawbacks of PMFE Protocols • Information-theoretically private MFE: Requires that a substantial fraction of the agents be obedient rather than strategic. • Cryptographically private MFE: Requires (plausible but) currently unprovable complexity-theoretic assumptions and (usually) heavy communication overhead. • Brandt and Sandholm (TISSEC ’08): Which auctions of interest are unconditionally privately computable?

6. Minimum Knowledge Requirements for 2nd–Price Auction 0 1 2 3 bidder 2 0 1 2 3 2, 0 1, 0 winner price bidder 1 2, 1 1, 1 2, 2 1, 2 RI (2, 0) 1, 3 Perfect Privacy Auctioneer learns only which region corresponds to the bids. ≈

7. Outline • Background • Two-party communication (Yao) • “Tiling” characterization of privately computable functions (Chor + Kushilevitz) • Privacy Approximation Ratios (PARs) • Bisection auction protocol: exponential gap between worst-case and average-case PARs • Summary of Our Results • Open Problems

8. Two-party Communication Model f: {0, 1}k x {0, 1}k {0, 1}t x1 Party 1 Party 2 x2 q1 qj{0, 1} is a function of (q1, …, qj-1) and one player’s private input. q2 ••• qr-1 qr = f(x1, x2) Δ s(x1, x2) = (q1, …, qr)

9. Example: Millionaires’ Problem 0 1 2 3 millionaire 2 0 1 2 3 millionaire 1 A(f) f(x1, x2) = 1 if x1 ≥ x2 ; else f(x1, x2) = 2

10. Bisection Protocol In each round, a player “bisects” an interval. 0 1 2 3 0 1 2 3 Example: f(2, 3)

11. Monochromatic Tilings • A region of A(f) is any subset of entries (not necessarily a submatrix). A partition of A(f) is a set of disjoint regions whose union is A(f). • Monochromatic regions and partitions • A rectangle in A(f) is a submatrix. A tiling is a partition into rectangles. • Tiling T1(f) is a refinement of partition PT2(f) if every rectangle in T1(f) is contained in some region in PT2(f).

12. A Protocol “Zeros in on” a Monochromatic Rectangle Let A(f) = R x C While R x C is not monochromatic • Party i sends bit q. • If i = 1, q indicates whether x1 is in R1 or R2, where R = R1⊔ R2. If x1 Rk, both parties set R  Rk. • If i = 2, q indicates whether x2 is in C1 or C2, where C = C1⊔ C2. If x2 Ck, both parties set C  Ck. One party sends the value of f in R x C.

13. Example: Ascending-Auction Tiling bidder 2 0 1 2 3 0 1 2 3 bidder 1 Same execution for f(1, 1), f(2, 1), and f(3, 1)

14. Perfectly Private Protocols • Protocol P for f is perfectly private with respect to party 1 if f(x1, x2) = f(x’1, x2) s(x1, x2) = s(x’1, x2) • Similarly, perfectly private wrt party 2 • P achieves perfect subjective privacy if it is perfectly private wrt both parties. • P achieves perfect objective privacy if f(x1, x2) = f(x’1, x’2) s(x1, x2) = s(x’1, x’2)

15. Ideal Monochromatic Partitions • The ideal monochromatic partition of A(f) consists of the maximal monochromatic regions. • Note that this partition is unique. • Protocol P for f is perfectly privacy-preserving iff the tiling induced by P is the ideal monochromatic partition of A(f).

16. Privacy and Communication Complexity[Kushilevitz (SJDM ’92)] • f is perfectly privately computable if and only if A(f) has no forbidden submatrix. • Note that the Millionaires’ Problem is not perfectly privately computable. • If 1 ≤ r(k) ≤ 2(2k-1), there is an f that is perfectly privately computable in r(k) rounds but not r(k)-1 rounds. X2X’2 x1 f(x1, x2) = f(x’1, x2) = f(x’1, x’2) = a, but f(x1, x’2) ≠ a x’1

17. Perfect Privacy for 2nd–Price Auction[Brandt and Sandholm (TISSEC ’08)] • The ascending-price, English-auction protocol is perfectly private. • It is essentially the only perfectly privateprotocol for 2nd–price auctions. • Note the exponential communication cost of perfect privacy.

18. Objective PAR (1) • Worst-case objective privacy-approximation ratio of protocol P for function f: • Worst-case PAR of f is the minimum, over all P for f, of worst-case PAR of P. |R (x1, x2)| |R (x1, x2)| I MAX (x1, x2) P

19. Objective PAR (2) • Average-case objective privacy-approximation ratio of P for f with respect to distribution D on {0, 1}k x {0,1}k : • Average-case PAR of f is the minimum, over all P for f, of average-case PAR of P. [ ] |R (x1, x2)| |R (x1, x2)| I ED P

20. Subjective PARs (1) • The 1-partition of region R in matrix A(f): { Rx1 = {x1} x {x2 s.t. (x1, x2)  R} } (similarly, 2-partition) • The i-induced tiling of protocol P for f is obtained by i-partitioning each rectangle in the tiling induced by P. • The i-ideal monochromatic partition of A(f) is obtained by i-partitioning each region in the ideal monochromatic partition of A(f).

21. Example: 1-Ideal Monochromatic Partition for 2nd–Price Auction 0 1 2 3 I I I R1 (0, 1) = R1 (0, 2) = R1 (0, 3) 0 1 2 3 I I R1 (1, 2) = R1 (1, 3) I |R1 (x1,x2)| = 1 for all other (x1,x2) P (Ri defined analogously for protocol P)

22. Subjective PARs (2) • Worst-case PAR of protocol P for f wrt i: • Worst-case subjective PAR of P for f: maximize over i  {1, 2} • Worst-case subjective PAR of f: minimize over P • Average-case subjective PAR with respect to distribution D: use ED instead of MAX |Ri (x1, x2)| |Ri (x1, x2)| MAX (x1, x2) I P

23. Bisection Auction Protocol (BAP)[Grigorieva, Herings, Muller, & Vermeulen (ORL’06)] • Bisection protocol on [0,2k-1] to find an interval [L,H] that contains lower bid but not higher bid. • Bisection protocol on [L,H] to find lower bid p. • Sell the item to higher bidder for price p.

24. Bisection Auction Protocol bidder 2 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 bidder 1 A(f) Example: f(7, 4)

25. Objective PARs for BAP(k) • Theorem: Average-case objective PAR of BAP(k) with respect to the uniform distribution is+1. • Observation: Worst-case objective PAR of BAP(k) is at least 2 . k 2 k/2

26. Proof (1) 0 2k-1 2k-1 • ak= number of rectangles in induced tiling for BAP(k). • a0=1, ak = 2ak-1+2k ak = (k+1)2k 0 Δ 2k-1 2k-1 The monochromatic tiling induced by the Bisection Auction Protocol for k=4

27. Proof (2) Δ • R = {R1,…,Ra } is the set of rectangles in the BAP(k) tiling • RI = rectangle in the ideal partition that contains Rs • js= 2k - |RI| • bk=SR js k Δ s Δ s Δ s

28. Proof (3) |RI(x1,x2)| (+) 1 PAR = S = S = S 22k |RBAP(k)(x1,x2)| (x1,x2) . |RI| 1 1 s |Rs| |RI| 22k 22k |Rs| s Rs Rs number of (x1,x2)’s in Rs contribution to (+) of one (x1,x2) in Rs

29. Proof (4) 0 2k-1 2k-1 • bk = bk-1+(bk-1+ak-12k-1) + ( S i ) + ( S i ) • b0=0, bk=2bk-1+(k+1)22(k-1) bk = k22k-1 0 2k-1-1 2k-1 2k-1 i=0 i=1 2k-1 The monochromatic tiling induced by the Bisection Auction Protocol for k=4

30. Proof (5) S= S (2k-js) = (ak2k-bk) = ( (k+1)22k- k22k-1 ) = k+1- = + 1 1 1 |RI| 22k s 22k 1 22k 1 22k k 2 k 2 QED

31. Bounded Bisection Auction Protocol (BBAP) • Parametrized by g: N -> N • Do at most g(k) bisection steps. • If the winner is still unknown, run the ascending English auction protocol on the remaining interval. • Ascending auction protocol: BBAP(0)Bisection auction protocol: BBAP(k)

32. Average-Case Objective PAR • Theorem: For positive g(k), the average-case objective PAR of BBAP(g(k)) with respect to the uniform distribution satisfies 3g(k)+6 ≥ PAR ≥ g(k) + 1 (for g(k)=0, this PAR is exactly 1) • Observation: BBAP(g(k)) has communication complexity Q(k + 2k-g(k)). 8 4

33. Average-Case Objective PARs for 2nd-price Auction Protocols 4 2k+1 8 2 16 2 +1 2 (3*2k) 3

34. Average-Case PARs for the Millionaires Problem 1 2 k 1 +1 2 2

35. Open Problems • Upper bounds on non-uniform average-case PARs • Lower bounds on average-case PARs • PARs of other functions • Extension to n-party case • Relationship between PARs and h-privacy [Bar-Yehuda, Chor, Kushilevitz, and Orlitsky (IEEE-IT ’93)]