Solution: Zero-knowledge proof m π (1) m π (2) m π (N) … π1π1 π2π2 π = π 1 ◦ π 2 m1m1 m2m2 mNmN Threshold decryption Server 1 ZK proof No message changed (soundness) Server 2 ZK proof Permutation still secret (zero-knowledge)
Preventing deviation (active attacks) by keeping people honest AliceBob Yes, here is a zero- knowledge proof that everything is correct Did you follow the protocol honestly without deviation?
Cryptography 11 Problems typically arise when attackers deviate from a protocol (active attack) Zero-knowledge proofs prevent deviation and give security against active attacks
Fundamental building block 12 encryption signatures zero-knowledge Доверяй, но проверяй - Trust but verify
Zero-knowledge proofs Completeness –Prover can convince verifier when statement is true Soundness –Cannot convince verifier when statement is false Zero-knowledge –No leakage of information (except truth of statement) even if interacting with a cheating verifier 13
Exercise Argue the GI proof system is complete What is the probability of the prover cheating the verifier? (soundness) Argue the GI proof system is witness indistinguishable, i.e., when there are several isomorphisms between the two graphs it is not possible to know which one the prover has in mind 21
Perfect completeness: Pr[Accept] = 1 Accept or reject Statement x L Witness w so (x,w) R Completeness 22
Computational soundness: For ppt adversary Pr[Reject] ≈ 1 Statistical soundness: For any adversary Pr[Reject] ≈ 1 Perfect soundness: Pr[Reject] = 1 Accept or reject Statement x L Soundness 23
Arguments and proofs Argument (or computationally sound proof) –Computational soundness, holds against polynomial time adversary, relies on cryptographic assumptions Proof –Unconditional soundness, holds against unbounded adversary, and in particular without relying on cryptographic assumptions 24 Arguments can be more efficient than proofs
Zero-knowledge Zero-knowledge: –The proof only reveals the statement is true, it does not reveal anything else Defined by simulation: –The adversary could have simulated the proof without knowing the prover’s witness
view Zero-knowledge Simulator’s advantage: Can rewind adversary
Exercises Show the GI proof is perfect zero-knowledge Argue why zero-knowledge implies witness indistinguishability Give an example of a language and a proof system that is witness indistinguishable but not zero-knowledge (under reasonable assumptions) 28
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