Languages wit h Efficient Zer o-Knowledge PCP are in SZK i SZK - PowerPoint PPT Presentation

Languages wit h Efficient Zer o-Knowledge PCP are in SZK i SZK MOHAMMAD MA HMOODY (CORNELL) DAVID XIAO (LIAFA)

Probabilistically y Checkable y Proofs (PCPs) accept / reject

Z Zero-Knowledge PC K l d PC CPs CP statisti ically close actual SIM generated generated (default (default in this talk) in this talk) execution efficiently Def : View of any efficient verifier can be efficientl Def : View of any efficient verifier can be efficientl y simulated (similar to ZK interactive proofs) y “simulated” (similar to ZK interactive proofs) • Harder to achieve zero ‐ knowledge PCPs (than pr rovers) verifier can read any PCP answers. • Easier to achieve sound PCPs (than provers). • Easier to achieve sound PCPs (than provers) [Kilian ‐ Petrank ‐ Tardos’97] NEXP has (statistical) ze ero ‐ knowledge PCPs • Inherently of super ‐ polynomial length (even for I h tl f l i l l th ( f NP ) NP )

Efficient Zero-Kn Efficient Zero-Kn nowledge PCPs nowledge PCPs accept / reject

Main Q estion A Main Question : Are there efficient ( t ti ti th efficie nt (statistical) ZK PCPs for NP ? l) ZK PCP f NP ?

Motivation: Basin Motivation Basin ng Crypto on Tamper ng Crypto on Tamper Proof Hardware [Kat07, MS08, CGS08, GKR08, GISVW10, Kol10, GIMS10, ... ]

Motivation: Reset ttable Statistical Zero-Knowledge give me answer to q answer to q FORGET q and FORGET q and d answer p d answer p answer to p

Limits of Efficie Limits of Efficie ent (statistical) ent (statistical) Zero-Knowledge PC CPs? Main Question : Are there efficient t ZK PCPs for NP ? [Ishai ‐ M ‐ Sahai’12] Any language w [Ishai M Sahai 12] Any language w with an efficient ZK PCP using a with an efficient ZK PCP using a non ‐ adaptive verifier is in co ‐ AM Corollary : No efficient ZK for NP us sing a non ‐ adaptive verifier unless the polynomial ‐ time hierarc l h l i l i hi chy collapses [BHZ’87] h ll [BHZ’87]

Our Result

Id Ideas behind th b hi d d the proof f

Approach of [IMS’ Approach of [IMS’12] 12]

Naïve Approach Naïve Approach

Our Approach Our Approach

Our Approach Our Approach

Our Approach Our Approach Conditional Entropy Approxim py pp mation: in SZK [Vadhan’04]

Putting Things To Putting Things To ogether ogether

Summary Theorem : No efficient statistical Z ZK PCP for NP unless polynomial ‐ tim hi hierarchy collapses ‐‐ removing th h ll i th h he non ‐ adaptivity barrier of [IMS’12 d ti it b i f [IMS’12 Open : Characterize languages wit th efficient ZK PCPs. Conjecture : All of SZK (sufficient to make compiler of [GOVW] efficie Open : Number of messages (2 or Open : Number of messages (2 or r 3 or 4) needed in addition to an r 3 or 4) needed in addition to an efficient PCP (hardware token) to o get statistical zero ‐ knowledge for N

Th Thank Y k You ! !