1

Approximate Privacy:Foundations and

Quantification

Joan Feigenbaumhttp://www.cs.yale.edu/homes/jfNorthwest 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

Private, MultipartyFunction Evaluation

. . .

x1

x2

x 3 x n-1

x n

y = F (x 1, …, x n)

• Each i learns y.• No i can learn anything about xj

(except what he can infer from xi and 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

2, 1

winnerprice

2, 01, 0

1, 1

1, 2 2, 2

1, 3

0

1

2

3

bidder 1

bidder 2

PerfectPrivacy

Auctioneer learns only whichregion corresponds to the bids.

≈

0 1 2 3

RI (2, 0)

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

qj {0, 1}is a functionof (q1, …, qj-1)and one player’sprivate input.

s(x1, x2) = (q1, …, qr)Δ

qr = f(x1, x2)

qr-1

••

• q2

q1

9

Example: Millionaires’ Problem

0

1

2

3

0 1 2 3

millionaire 1

millionaire 2

A(f)

f(x1, x2) = 1 if x1 ≥ x2 ; else f(x1, x2) = 2

10

Bisection Protocol

0

1

2

3

0 1 2 3

In each round, a player “bisects” an interval.

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

0

1

2

3

0 1 2 3

Same execution for f(1, 1), f(2, 1), and f(3, 1)

bidder 1

bidder 2

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.

f(x1, x2) = f(x’1, x2) = f(x’1, x’2) = a, but f(x1, x’2) ≠ a

x1

x’1

X2 X’2

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 private protocol 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

P

MAX (x1, x2)

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

PED [ ]

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

0 1 2 3

(Ri defined analogously for protocol P)P

R1 (0, 1) = R1 (0, 2) = R1 (0, 3)I I I

R1 (1, 2) = R1 (1, 3)I I

|R1 (x1,x2)| = 1for all other (x1,x2)

I

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)|

I

P

MAX(x1, x2)

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

0 1 2 3 4 5 6 7

0

1

2

3

4

5

6

7

Bisection Auction Protocol

A(f)

Example: f(7, 4)

bidder 1

bidder 2

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

k/2

2

26

Proof (1)

The monochromatic tiling induced by the Bisection Auction Protocol for k=4

• ak = number of rectangles in induced tiling for BAP(k).

• a0=1, ak = 2ak-1+2k

ak = (k+1)2k

2k-1

2k-1

2k-100

2k-1

Δ

27

Proof (2)

• 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 = R js

Δ

Δ

Δ

Δ

s

s

s

k

28

Proof (3)

PAR =

= =

122k

(x1,x2)

|RI(x1,x2)|

|RBAP(k)(x1,x2)|

122k

Rs

|RI|

|Rs|

s .|Rs|122k

Rs

s|RI|

(+)

contribution to (+)

of one (x1,x2) in Rs

number of (x1,x2)’s in Rs

29

Proof (4)

The monochromatic tiling induced by the Bisection Auction Protocol for k=4

• bk = bk-1+(bk-1+ak-12k-1)

+ ( i ) + ( i )

• b0=0, bk =2bk-1+(k+1)22(k-1)

bk = k22k-1

2k-1

2k-1

2k-100

2k-1

i=0

2k-1-1

i=1

2k-1

30

Proof (5)

= (2k-js)

= (ak2k-bk)

= ( (k+1)22k- k22k-1 )

= k+1-

= + 1

122k s|RI| 1

22k

122k

122k

k2

k2

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 (k + 2k-g(k)).

8 4

33

Average-Case Objective PARs for 2nd-price Auction Protocols

English Auction 1

Bounded Bisection Auction, g(k)=1 7 – 1

Bounded Bisection Auction, g(k)=2 19 - 3 k+1

Bounded Bisection Auction, g(k)=3 47 – 7 k+1

Bounded Bisection Auction, general g(k)

(1+g(k))

Bisection Auction k

Sealed-Bid Auction 2k+1 + 1

4 2k+1

8 2

16 2

2

+1

3

(3*2k)

34

Average-Case PARs for the Millionaires Problem

2

+1

Obj. PAR Subj. PAR

Any protocol ≥ 2k - + 2-

(k+1)

Bisection Protocol

3*2k-1 - k 2

1

2

1

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)]