ComputationalDifferentialPrivacy IlyaMironov (MICROSOFT) - - PowerPoint PPT Presentation

ComputationalDifferentialPrivacy

IlyaMironov

(MICROSOFT)

OmkantPandey

(UCLA)

OmerReingold

(MICROSOFT)

SalilVadhan

(HARVARD)

FocusoftheTalk

Motivation

• Achievebetterutility
• StandardMPC doesnotpreventwhatis

leakedbytheoutput

–

!"#$%&'"#()*

• Nontrivialdifferentiallyprivatemechanisms

mustberandomized

– + ,

• *

DifferentialPrivacy

[Dwork’06]

• %

" ./

D

• R ensuresε0DP ifforalladjacentdatasets

1,2 andforallsubsets ofR: D

• R ensuresε0DP ifforalladjacentdatasets

1,2 andforallsubsets ofR:

1 2

( ) ( )

Pr[ ] Pr[ ]

K D S K D S

eε

∈ ∈

≤

“adjacent” means “differinone individual’sentry” “adjacent” means “differinone individual’sentry”

PictorialRepresentation

— badoutcome — probabilitywithrecord — probabilitywithoutrecord

TowardsComputationalNotions

1 2

( ) ( )

Pr[ ] Pr[ ]

K D S K D S

eε

∈ ∈

≤

1 2

( ( )) 1 ( ( )) 1

Pr[ ] A Pr[A ]

K D K D

eε

= =

≤

Equivalently,

FirstDefinition:IND0CDP

ε0IND0CDP:Mechanism isε0IND0CDP ifforall adjacent12, forallpolynomialsizedcircuits A,andforalllargeenoughλ,itholdsthat, ε0IND0CDP:Mechanism isε0IND0CDP ifforall adjacent12, forallpolynomialsizedcircuits A,andforalllargeenoughλ,itholdsthat,

1 2

( ( )) 1 ( ( )) 1

Pr[ ] Pr[ A ] n g ) A e l(

K D K D

eε λ

= =

≤ +

Necessary

Simulation0basedApproach

c

≈

D:010110 D:010110

X

M(D) M(D)

Y

K(D) K(D) Differentially PrivateM

SecondDefinition:SIM0CDP

ε0SIM0CDP:Mechanism isε0SIM0CDP ifthere existsanε0differentially0privatemechanism suchthatforall,distributions and arecomputationallyindistinguishable. ε0SIM0CDP:Mechanism isε0SIM0CDP ifthere existsanε0differentially0privatemechanism suchthatforall,distributions and arecomputationallyindistinguishable.

1 2

, ( , ) M D D ∃ ∀

– MisnotnecessarilyaPPT mechanism – Reversingtheorderofquantifiersyields anotherdefinition,SIM∀∃

∀∃ ∀∃ ∀∃ 0CDP: 1 2

( , ), D D M ∀ ∃

ImmediateQuestions

• Arethesedefinitionsequivalent?
• Nothardtoseethat
• Mainquestion:

SIM0CDPIND0CDP

⇒

IND0CDPSIM0CDP?

⇒

ConnectionwithDenseModels

[RTTV’08,Imp’08]

• DistributionXisα0dense inYifforalltestsT,
• Xisα0pseudodenseinYifforallPPT testsT,

( ) 1 ( ) 1

1 Pr[ ] Pr[ ]

X Y

T T α

= =

≤

( ) 1 ( ) 1

1 Pr[ ] Pr[ ] negl

X Y

T T α

= =

≤ +

001"#()2%',%0%3,%0%,%1%4, .4/%5'46##(,

ConnectionwithDenseModels

[RTTV’08,Imp’08]

• DifferentialPrivacy:

– –

• Inthelanguageofdensemodels

– K(D1) iseε0dense in K(D2) – K(D2) iseε0dense inK(D1)

1 2

( ) ( )

Pr[ ] Pr[ ]

K D S K D S

eε

∈ ∈

≤

2 1

( ) ( )

Pr[ ] Pr[ ]

K D S K D S

eε

∈ ∈

≤ ε0DP:K(D1) andK(D2) aremutuallyeε0dense ε0DP:K(D1) andK(D2) aremutuallyeε0dense

ConnectionwithDenseModels

[RTTV’08,Imp’08]

• ε 0 IND0CDP:

– –

• Inthelanguageofdensemodels

– K(D1) iseε0pseudodense in K(D2) – K(D2) iseε0pseudodense inK(D1)

1 2

( )) 1 ( ) 1

Pr[ ( ] Pr[ ( ] negl

K D K D

A e A

ε

= =

≤ + ε0IND0CDP:K(D1) andK(D2) aremutuallyeε0pseudodense ε0IND0CDP:K(D1) andK(D2) aremutuallyeε0pseudodense

2 1

( )) 1 ( ) 1

Pr[ ( ] Pr[ ( ] negl

K D K D

A e A

ε

= =

≤ +

SomeNotation

X Y

(Xispseudodense inY)

X Y

(X,Yaremutually pseudodense )

X Y

(Xisdense inY)

X Y

(X,Yaremutually dense) (X,Ycomp.indistinguishable)

X Y

≈

TheDenseModelTheorem

[RTTV’08]

X1 X2 Y

Thm:If1 ispseudodensein2,thereexistsamodel (truly)densein2 suchthat1 iscomputationally indistinguishablefrom. Thm:If1 ispseudodensein2,thereexistsamodel (truly)densein2 suchthat1 iscomputationally indistinguishablefrom.

X1 X2

X1=K(D1) X2=K(D2) (IND0CDP)

Y1 Y2

Y1=M(D1) Y2=M(D2)

X1 X2

⇑ ⇑

Z1 Z2

?

ProofIdeas

E x t e n s i

• n
• f
• D

M

• T

X Y: X dense in Y, X Y: X,Y mutually dense X Y: X pseudo-dense in Y, X Y: X,Y mutually pseudo-dense (SIM0CDP)

∀∃

1 2

( , ), D D M ∀ ∃

1 2

, ( , ) M D D ∃ ∀

Z2

ToRecap

• -.0

0/ 001"#(),

• 42
• 4'72 IND0CDP SIM0CDP

⇒

? IND0CDP⇔ SIM∀∃

∀∃ ∀∃ ∀∃0CDP

Benefits:BetterUtility

CDP:EasilygetΘ Θ Θ Θ(1/ε) errorw/constantprobability.

Alice Bob

x1 x2

…

xn y1 y2

…

yn H(x,y)

2 0:8%938:;ε<

SFE

DP:Requires(n½) error![Reingold0Vadhan]

~

OtherResults

• AnewprotocolforHammingDistance:

– Differentiallyprivate(standard) – Constantmultiplicative error

• DifferentiallyPrivateTwo0PartyComputation

Thankyouforyourattention!