ZKP Mathematical Building Blocks 1

MIT IAP 2023 Modern Zero Knowledge Cryptography课程笔记

Lecture 3: Mathematical Building Blocks (Yufei Zhao)

• Example: I (Prover) want to convince you (Verifier) that I can distinguish two colors that you see as identical
  

  • A Similar Example: How to prove two colors are different to a blind verifier
    

  • What is a proof?

    • A proof is something that could convince someone else
    • Properties: completeness, soundness, zero-knowledge

  • What is the prover and the verifier (based on blockchain)

    • Prover: run on the regular computer (much more powerful than the verifier)
    • Verifier: run on the smart contract

• Example: Hamilton cycle [Blum '87]

  • Hamilton cycle: a cycle can go through every vertex of the graph exactly once and return to the start
  • Everybody knows a graph
  • P knows a Hamilton cycle in the graph without revealing any additional information
  • Protocol
    

• ZKP Properties

  • Completeness: If everyone behaves then protocol accepts
  • Soundness: If there is no Ham cycle, then no matter what P does, V rejects with the probability of $\ge \frac{1}{2}$
    • There is a stronger requirement called knowledge soundness which says that even if the graph has a Ham Cycle , the prover doesn’t know it, the protocol will still fail. The precise definition involves an extractor with rewinding abilities.
  • Zero-knowledge: If V accepts then it learns no addl into from the interaction because V could have simulated the entire dialog by itself.