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Efficient Zero-Knowledge Proof SystemsPowerPoint Presentation

Efficient Zero-Knowledge Proof Systems

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No! It is a trade secret.

Did I lose all my money?Show me the current portfolio!

Hedge fund Investor

Zero-knowledge proof

Statement

Zero-knowledge:Nothing but truth revealed

Soundness:Statement is true

Witness

Prover Verifier

Internet voting

Tally without decrypting individual votes

Vote

Encrypts vote to keep it private

Ciphertext

Voter Election authorities

Election fraud

Not Bob

Encrypts -100 votes for Bob

Is the encrypted vote valid?

Ciphertext

Voter Election authorities

Zero-knowledge proof as solution

Zero-knowledge:Vote is secret

Soundness:Vote is valid

Ciphertext

Zero-knowledge proof for valid vote encrypted

Voter Election authorities

Mix-net: Anonymous message broadcast

Threshold decryption

mπ(1)

mπ(2)

mπ(N)

π = π1◦π2

π2

m1

m2

mN

π1

…

Solution: Zero-knowledge proof

Threshold decryption

mπ(1)

mπ(2)

mπ(N)

π = π1◦π2

Server 2 ZK proofPermutation still secret(zero-knowledge)

π2

Server 1 ZK proofNo message changed(soundness)

m1

m2

mN

π1

…

Preventing deviation (active attacks) by keeping people honest

Yes, here is a zero-knowledge proof that everything is correct

Did you follow the protocol honestly without deviation?

Alice Bob

Cryptography honest

Problems typically arise when attackers deviate from aprotocol (active attack)

Zero-knowledge proofs prevent deviation and give security against active attacks

Fundamental building block honest

Доверяй, но проверяй

- Trust but verify

zero-knowledge

signatures

encryption

Zero-knowledge proofs honest

- 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

Parameters honest

- Efficiency
- Communication (bits)
- Prover’s computation (seconds)
- Verifier’s computation (seconds)
- Round complexity (number of messages)

- Security
- Setup
- Cryptographic assumptions

Zero-knowledge proof efficiency honest

cost

non-interactive zero-knowledge proofs

interactive zero-knowledge proofs

rest of the protocol

1985 2014

Vision honest

- Main goal
- Efficient and versatile zero-knowledge proofs

- Vision
- Negligible overhead from using zero-knowledge proofs
- Security against active attacks standard feature

zero-knowledge

core

core

Statements honest

SAT

0

- Statements are for a given NP-language
- Prover knows witness such that
- But prover wants to keep the witness secret!

1

1

Encrypted valid vote

0

Circuit SAT

Hamiltonian

Proof system honest

- A proof system for an NP-relation consists of a prover and a verifier
- We consider efficient proof systems: prover and verifier are probabilistic polynomial time interactive algorithms
- Both prover and verifier get a statement as input
- The prover gets a witness such that
- They interact and finally the verifier accepts or rejects

Graph isomorphism honest

Exercise honest

- 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

Witness w honestso (x,w)R

CompletenessStatement xL

Accept or reject

Perfect completeness: Pr[Accept] = 1

Soundness honest

Statement xL

Accept or reject

Computational soundness: For ppt adversary Pr[Reject] ≈ 1

Statistical soundness: For any adversary Pr[Reject] ≈ 1

Perfect soundness: Pr[Reject] = 1

Arguments and proofs honest

Arguments can be more efficient than 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

Zero-knowledge honest

- 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

Zero-knowledge honest

view

Simulator’s advantage: Can rewind adversary

view

ZK:

Can be computational, statistical or perfect

Exercises honest

- 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)

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