Precise Concurrent Zero Knowledge

Precise Concurrent Zero Knowledge

On the Composition of PublicCoin Zero-Knowledge Protocols Rafael Pass (Cornell) Wei-Lung Dustin Tseng (Cornell) Douglas Wiktrm (KTH) 1 Zero Knowledge [GMR85] Interactive protocol between a Prover and a Verifier where the Verifier learns nothing except the proof statement Prover

Verifier Fundamental construct of cryptography Used in secure MPC, authentication, etc, etc 2 Zero Knowledge [GMR85] For every PPT V* (adversary) there is a PPT simulator S: Prover Verifier V*

Simulator S View generated by S View of V* with Prover Indistinguishable 3 Black-Box Zero Knowledge [GO90] Universal S interacts with and rewinds V*

Equivalently: Output View Most known and all practical ZK are BB This talk: Focus on BB ZK 4 Composition of ZK [GKr90] Parallel [FS90, GKr90] Concurrent [FS90, DNS04]

Do ZK protocols stay ZK when composed? 5 Composition of ZK [GKr90] In general: ZK breaks even under 2 parallel executions [FS90, GKr90] Specific protocols: Secure under both parallel and concurrent composition (e.g., [GKa96, FS90, RK99, KP01, PRS02]) But these protocols use something new:

Private Coins 6 Public vs. Private Coins Private-coin: Public-coin: Prover Verifier The original ZK protocols are all public-coin [GMR85,GMW91, Blum87]

Why care about public-coin protocols? Theory: Understand original protocols e.g. IP(Poly) = AM(Poly) [GS86] Simpler to implement Practice: V resilient to leakage and side channel attacks 7 The Question: Are private coins necessary for composing ZK (even just) in parallel?

First studied by Goldreich-Krawczyk in 1990 Partial result: No constant round public-coin BB ZK w/ neg. soundness error (L BPP) Known O(1) round public-coin BB ZK (with big soundness error) not secure in parallel 8 Our Results 1. Any public-coin protocol is not BBZK if repeated sufficiently in parallel (L BPP). 2. For every m, there is a public-coin proof for NP that is BBZK up to m concurrent sessions, assuming OWF.

[Bar01]: Public-coin constant round boundedconcurrent non-BB ZK argument assuming CRH. 9 The Goldreich-Krawczyk framework [GKr90]: If the verifier uses PRF to generate its messages in a constant round public-coin protocol Protocol is resettably-sound [BGGL01] Prover PRF( )

Verifier + PRF The Goldreich-Krawczyk framework [GKr90]: If the verifier uses PRF to generates it messages in a constant round public-coin protocol Protocol is resettably-sound [BGGL01] Resetting P* Verifier V + PRF

Goal: Accepting execution for x L The Goldreich-Krawczyk framework [GKr90]: If the verifier uses PRF to generates it messages in a constant round public-coin protocol Protocol is resettably-sound [BGGL01] If protocol is resettably-sound and BB ZK for L L BPP (decided by S) [GK90, BGGL01]: x L S(x) gives accepting view (ZK) x L S(x) gives rejecting view (resettable-sound) 12 Main Lemma

Any public-coin protocol (where V uses PRF for its messages) is resettably-sound when repeated sufficiently in parallel. Compare with soundness amplification Recent work: Parallel repetition amplifies soundness of public-coin arguments [PV07, HPPW08]: From poly(n) Our work: Quality of soundness also improves From standard sound resettably sound Can use soundness amplification techniques 13

Proof Idea Reduction R: Resettable P* normal P Reduction R Resetting P* Verifier V R tries to forward messages that P* utilize for an accepting execution Possible to continue simulation due to public-coin 14 Which Message to Forward?

[GKr90] For constant round protocols, choose random messages to forward Guess correctly w.p. 1/poly each round Doesnt work when there are more rounds Our approach: Do a test run to see which msg shouldve been forwarded. Forward it and continue simulation If P* doesnt use forwarded msg, rewind P* until it does 15 Example

Start: Two rounds are already forwarded Reduction R Verifier V Resetting P* FAIL Acc. Acc. Repeat Process Case: SForwarded fails to produce

msg is not accepting inin accepting accepting view. view view Found Rewind! next message to forward 16

The Reduction Again 1. In a test run of P*, find the msg used by P* to form an accepting view. 2. Forward the msg to V and receive a fixed reply. 3. Keep rewinding P* until the forwarded msg is used in an accepting view The next msg in view gets forwarded. Repeat. Reduction idea analogous to [HPPW08] Reduction always works! Is it poly time? 17

Analysis Sketch If we can rewind external V: Case: P* chooses which branch to use in view randomly. Then poly rewinds are enough This is actually the worst case But we cant rewind external V: Forwarded messages are fixed. Might fix a BAD message Reduction: Resettable standalone parallel P*normal standalone P New picture!

18 Analysis Sketch Can almost rewind the Verifier Results in a statistically close distribution! Technically shown by relying on Razs Lemma Technique used in soundness amplification of 2-prover games [Raz98] and public-coin arguments [HPPW08] Reduction R Resetting P* Verifier V

19 Conclusion Any public-coin protocol, with enough parallel repetitions, is resettably-sound so not BB ZK unless L BPP Elucidate connection between hardness amplification and BB ZK lower bounds New set of techniques for BB lower bounds 20

Corollary Bare Public-Key setup More efficient (private-coin) concurrent ZK Model studied in the soundness amplification literature [IW97, BIN97, HPPW08] Using [BIN97, HPPW08] techniques, we can extend our impossibility result to BPK too 21 Thank You!

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