An Introduction to Differential

Privacyand its Applications1

Ali Bagherzand

i Ph.D Candidate

University of California at

Irvine

1- Most slides in this presentation are from Cyntia Dwork’s talk.

Goal of Differential Privacy:Privacy-Preserving Statistical Analysis of Confidential Data

Finding Statistical CorrelationsAnalyzing medical data to learn genotype/phenotype associations

Correlating cough outbreak with chemical plant malfunctionCan’t be done with HIPAA safe-harbor sanitized data

Noticing EventsDetecting spike in ER admissions for asthma

Official StatisticsCensus

Contingency Table Release (statistics for small sets of attributes)Model fitting, regression analyses, etc.

Standard Datamining TasksClustering; learning association rules, decision trees, separators;

principal component analysis

Two Models:

Database

K?

Sanitized Database

Non-Interactive: Data are sanitized and released

Two Models:

Database

K?

Sanitized Database

Interactive: Multiple Queries, Adaptively Chosen

Achievements for Statistical Data-Bases

Defined Differential PrivacyProved classical goal (Dalenius, 1977) unachievable“Ad Omnia” definition; independent of linkage information

General Approach; Rigorous ProofRelates degree of distortion to the (mathematical) sensitivity of

the computation needed for the analysis“How much” can the data of one person affect the outcome?Cottage Industry: redesigning algorithms to be insensitive

Assorted ExtensionsWhen noise makes no sense; when actual sensitivity is much

less than worst-case; when the database is distributed; …Lower bounds on distortion

Initiated by Dinur and Nissim

Semantic Security for Confidentiality [Goldwasser-Micali ’ 82]

VocabularyPlaintext: the message to be transmittedCiphertext: the encryption of the plaintextAuxiliary information: anything else known to attacker

The ciphertext leaks no information about the plaintext.

Formalization Compare the ability of someone seeing aux and ciphertext to guess

(anything about) the plaintext, to the ability of someone seeingonly aux to do the same thing. Difference should be “tiny”.

Semantic Security for Privacy?

Dalenius, 1977Anything that can be learned about a respondent from thestatistical database can be learned without access to the

database.

Happily, Formalizes to Semantic Security

Unhappily, Unachievable [Dwork and Naor 2006]Both for not serious and serious reasons.

7

Semantic Security for Privacy?

The Serious Reason (told as a parable)Database teaches average heights of population subgroups

“Terry Tao is four inches taller than avg Swedish ♀”Access to DB teaches Terry’s height.

Terry’s height learnable from the DB, not learnable w/o.Proof extends to “any” notion of privacy breach.

Attack Works Even if Terry Not in DB!Suggests new notion of privacy: risk incurred by joining DB

Before/After interacting vs Risk when in/notin DB

Differential Privacy: Formal Definition

K gives differentialprivacy if for all values of DB, DB’differing in a single element, and all S in Range(K )

Pr[ K (DB) in S]Pr[ K (DB’) in S]

≤ eε ~ (1+ε)

ratio bounded

Pr [t]

f+ noise ?

K

f: DB Rd

K (f, DB) = f(DB) + [Noise]d

Eg, Count(P, DB) = # rows in DB with Property P42

Achieving Differential Privacy:

How Much Can f(DB + Me) Exceed f(DB - Me)?

Recall: K (f, DB) = f(DB) + noise

Question Asks: What difference must noise obscure?

Δf= maxH(DB, DB’)=1 ||f(DB) – f(DB’)||1

eg, ΔCount= 1

43

Sensitivity of Function

Calibrate Noise to Sensitivity

Δf= maxH(DB, DB’)=1 ||f(DB) – f(DB’)||1

-4R -3R -2R -R 0 R 2R 3R 4R 5R

Lap(R) : f(x) = 1/2R exp(-|x|/R)

Theorem: To achieve ε-differentialprivacy, usescaled symmetric noise [Lap(R)]d with R =

Δfε.

Increasing R flattens curve; more privacyNoise depends on f and not on the

database

Why does it work? (d=1)

Pr[ K (f, DB’) = t]

Pr[ K (f, DB) = t]= exp(-(|t- f(DB)|-|t- f(DB’)|)/R) ≤ exp(-Δf/R) ≤ e

Δf= maxDB, DB-Me |f(DB) – f(DB-Me)|

Theorem: To achieve ε-differentialprivacy, usescaled symmetric noise Lap(R) with R =

Δfε.

-4R -3R -2R -R 0 R 2R 3R 4R 5R

Differential Privacy: ExampleID Claim

(‘000$)ID Claim

(‘000$)1 9.91 7 10.63

2 9.16 8 12.54

3 10.59 9 9.29

4 11.27 10 8.92

5 10.50 11 206 11.89 12 8.55

… … … …

Query : Average claim

Aux input : A fraction C of entries.

Without Noise : Can learn my claim w/ prob. C.

What privacy level is enough for me?

I feel safe if I’m hidden among n people Pr [identify] = 1/n

If n=100, c=0.01 in this example = 1.

In most applications, same gives me more feeling of privacy. e.g. adv. Identification itself is probabilistic. = 0.1, 1.0, ln(2)

Differential Privacy: Example

Query : Average claim

Aux input : 12 entries

= 1 hide me among 1200 people

Δf = maxDB, DB-Me |f(DB) – f(DB-Me)| = maxDB, DB-Me |Avg(DB) – Avg(DB-Me)|

= 11.10-10.29 = 0.81

Add noise : Lap(f/) = Lap(0.81/1) = Lap(0.81)

ID Claim (‘000$)

ID Claim(‘000$)

1 9.91 7 10.63

2 9.16 8 12.54

3 10.59 9 9.29

4 11.27 10 8.92

5 10.50 11 206 11.89 12 8.55

… … … …

Var = 1.31 Roughly: respond w/ ± 1.31 true value

Avg. not a sensitive function!

Differential Privacy: Example

Query : max claim

Aux input : 12 entries

= 1 hide me among 1200 people

Δf = maxDB, DB-Me |f(DB) – f(DB-Me)| = maxDB, DB-Me |Avg(DB) – Avg(DB-Me)|

= 20-12.54 = 7.46

Add noise : Lap(f/) = Lap(7.46/1) = Lap(7.46)

ID Claim (‘000$)

ID Claim(‘000$)

1 9.91 7 10.63

2 9.16 8 12.54

3 10.59 9 9.29

4 11.27 10 8.92

5 10.50 11 206 11.89 12 8.55

… … … …

Var = 1.31 Roughly: respond w/ ± 111.30 true value

Max. function is very sensitive!

Useless : error > sampling error

For all DB, DB’ differing in at most one element for all subset S of Range(K ),

Pr[ K (DB) in S] ≤ ePr[ K (DB’) in S] + T

where (n) is negligible.

Cf : DifferentialPrivacy is unconditional, independent of n

A Natural Relaxation: (,T)-Differential Privacy

Database D = {d1, … ,dn}rows in {0,1}k

T “counting queries” q = (S,g)S subset [1..n], g: rows{0,1}; “How many rows in S satisfy g?”

q(D) = ∑i in S g(di)

Privacy Mechanism: Add Binomial Noiser = q(D) + B(f(n), ½)

SuLQ Theorem: (δ, T)-Differential Privacy whenT(n)=O(nc), 0< c <1; δ=1/O(nc’), 0< c’ < c/2f(n) = (T(n) /δ2) log6 (n)

If (√T(n)/δ)<O(√n) then |noise| < |sampling error|!

Feasibility Results: [Dwork Nissim ’04] (Here: Revisionist Version)

e.g. : T =n3/4 and δ= 1/10 then √T/δ=10n3/8 << √n

Blatant Non-Privacy: Adversary guesses correctly o(n) rows

Blatant non-privacy if:all / *cn / (1/2 + c’n

answers are within o(√n) of the true answer,

n / cn / c’n queries andpoly(n) / poly(n) / exp(n) computation.

[Y] / [DMT] / [DMT]

Results are independent of how noise is distributed.A variant model permits poly(n) computation in the final case [DY].

Impossibility Results:Line of Research Initiated by Dinur and Nissim ’03

2n queries: noise E permits correctly guessing n-4E rows [DiNi].Depressing implications for non-interactive case.

even against an adversary restricted to

Thank You!