ZKP11.4 Use CI to instantiate Fiat-Shamir

ZKP学习笔记

ZK-Learning MOOC课程笔记

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

11.4 Use CI to instantiate Fiat-Shamir

  • Avoid Bad Challenges

    • Def: Given false claim x x x and a first message α \alpha α, a challenge β \beta β is "bad" if there exists a prover message g a m m a gamma gamma making V V V accept
    • We want to say: if the (3 message) interactive protocol is sound, then (for all x x x, α \alpha α) most β \beta β are not bad. True for statistically sound IPs.
    • Exactly what CI is good for! Define relation R x = α , β : β i s b a d R_x = {\alpha, \beta: \beta is bad} Rx=α,β:βisbad. Then if h h h is CI for R x R_x Rx (when x ∉ L x \notin L x∈/L), Π F S \Pi_{FS} ΠFS is sound using h h h!
    • Protocols with more than 3 messages: round-by-round soundness (each round has a type of "bad challenge" to avoid).
    • Main technical challenges:
      • Sometimes our IP doesn't have statistical soundness.
      • We can only build CI for relations R R R that can be decided efficiently
  • Important example: SNARGs via IOPs (PCPs)

    • SNARGs from PCPs [Kilian, Micali]

      • Candidate SNARG: apply Fiat-Shamir to this protocol!
      • Simplified (less efficient) version of modern SNARKs you've learned about.
      • Not statistically sound, so it's not clear how to analyze FS without random oracles.
    • SNARGs for Batch NP

    • Interactive Batch Arguments from PCPs [CJJ21]

      • SSB Commitments

      • Interactive Batch Arguments from PCPs [CJJ21]

  • Summary of Fiat-Shamir without RO

    • Use hash functions that are CI for appropriate functions/relations
      • CCHLRRW19,PS19,BKM20,JJ21,HLR21

    • Carefully show that FS-soundness for protocols of interest follows from compatible forms of CI
      • CCHLRRW19\]: (non-succinct) NIZK

      • CJJ21\]: batch NP arguments

      • Characterize which protocols can be FS-compiled (we know it doesn't work in general [Bar01, GK03])
      • SNARGs for NP from falsifiable assumptions?
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