ZKP11.3 Correlation Intractability

ZKP学习笔记

ZK-Learning MOOC课程笔记

Lecture 11: From Practice to Theory (Guest Lecturer: Alex Lombardi)

11.3 What can we do without random oracle model

• Falsifiable Assumptions
  • Prove security assuming that some concrete algorithmic task is infeasible:
    • Computing discrete logarithms is hard.
    • Solving random noisy linear equations (LWE) is hard.
    • SHA256 is collision-resistant
  • Many cryptographic constructions use random oracles to get better efficiency, but can be based on falsifiable assumptions.
    • CCA-secure public key encryption
    • Identity-based encryption.
    • Non-interactive zero knowledge
  • Can (ZK-)SNARKs for NP be built based on falsifiable assumptions?
    • (minor caveats but) No!
    • No way to extract a long witness from a short proof. Need assumption (RO, “knowledge assumption”) that guarantees adversary “knows” a long string given a short commitment.
  • Can (ZK-)SNARGs for NP be built based on falsifiable assumptions?
    • It’s complicated. (We don’t know)
    • Significant barriers [Gentry-Wichs ‘11]
    • The community is still trying to understand this.
• SNARGs for limited computations from falsifiable assumptions (LWE)
  • Two tools/techniques
    • Correlation-intractable hash functions [CCHLRRW19,PS19,HLR21]
      • Used to instantiate Fiat-Shamir without random oracles, for “nice enough” interactive protocols
    • Somewhere extractable commitments [HW15]
      • Used to make a “nice enough” interactive protocol
      • Special variant of the typical IOP-based approach.
• Correlation Intractability

  • Function
    

  • Binary relations
    

  • Weren’t these impossible to build?

    • Restrict to fixed input length (necessary)
    • Restrict to fixed running time on f (unclear if necessary)

  • CI Construction

    • A simple construction [CLW18] using Fully Homomorphic Encryption (FHE)
      

• Security Analysis
  

• Correlation Intractability: what we know