Abstract: The Fiat-Shamir construction (Crypto 1986) is an efficient transformation in the random oracle model for creating non-interactive proof systems and signatures from sigma-protocols. In classical cryptography, Fiat-Shamir is a zero-knowledge proof of knowledge assuming that the underlying sigma-protocol has the zero-knowledge and special soundness properties. Unfortunately, Ambainis, Rosmanis, and Unruh (FOCS 2014) ruled out non-relativizing proofs under those conditions in the quantum setting.
In this paper, we show under which strengthened conditions the Fiat-Shamir proof system is still post-quantum secure. Namely, we show that if we require the sigma-protocol to have computational zero-knowledge and perfect special soundness, then Fiat-Shamir is a zero-knowledge simulation-sound proof system (but not a proof of knowledge!). Furthermore, we show that Fiat-Shamir leads to a post-quantum secure strongly unforgeable signature scheme when additionally assuming a "dual-mode hard instance generator" for generating key pairs.
Finally, we study the extractability (proof of knowledge) property of Fiat-Shamir. While we have no proof of the extractability itself, we show that if we can prove extractability, then other desired properties such as simulation-sound extractability (i.e., non-malleability), and strongly unforgeable signatures follow.