Theory of Cryptography

Volume 7785 of the series Lecture Notes in Computer Science pp 182-201

Why “Fiat-Shamir for Proofs” Lacks a Proof

  • Nir BitanskyAffiliated withTel Aviv University
  • , Dana Dachman-SoledAffiliated withMicrosoft Research New England
  • , Sanjam GargAffiliated withUCLA
  • , Abhishek JainAffiliated withMIT and BU
  • , Yael Tauman KalaiAffiliated withMicrosoft Research New England
  • , Adriana López-AltAffiliated withNYU
  • , Daniel WichsAffiliated withIBM Research, T.J. Watson

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The Fiat-Shamir heuristic [CRYPTO ’86] is used to convert any 3-message public-coin proof or argument system into a non-interactive argument, by hashing the prover’s first message to select the verifier’s challenge. It is known that this heuristic is sound when the hash function is modeled as a random oracle. On the other hand, the surprising result of Goldwasser and Kalai [FOCS ’03] shows that there exists a computationally sound argument on which the Fiat-Shamir heuristic is never sound, when instantiated with any actual efficient hash function. This leaves us with the following interesting possibility: perhaps we can securely instantiates the Fiat-Shamir heuristic for all 3-message public-coin statistically sound proofs, even if we must fail for some computationally sound arguments. Indeed, this has been conjectured to be the case by Barak, Lindell and Vadhan [FOCS ’03], but we do not have any provably secure instantiation under any “standard assumption”. In this work, we give a broad black-box separation result showing that the security of the Fiat-Shamir heuristic for statistically sound proofs cannot be proved under virtually any standard assumption via a black-box reduction. More precisely:

–If we want to have a “universal” instantiation of the Fiat-Shamir heuristic that works for all 3-message public-coin proofs, then we cannot prove its security via a black-box reduction from any assumption that has the format of a “cryptographic game”.

–For many concrete proof systems, if we want to have a “specific” instantiation of the Fiat-Shamir heuristic for that proof system, then we cannot prove its security via a black box reduction from any “falsifiable assumption” that has the format of a cryptographic game with an efficient challenger.