When do classical zero-knowledge protocols remain secure against quantum attacks? In this work, we develop the techniques, tools, and abstractions necessary to answer this question for foundational protocols:
1) We prove that the Goldreich-Micali-Wigderson protocol for graph non-isomorphism and the Feige-Shamir protocol for NP remain zero-knowledge against quantum adversaries. At the heart of our proof is a new quantum rewinding technique that enables extracting information from multiple invocations of a quantum adversary without disturbing its state.
2) We prove that the Goldreich-Kahan protocol for NP is post-quantum zero knowledge using a simulator that can be seen as a natural quantum extension of the classical simulator.
Our results achieve negligible simulation error, appearing to contradict a recent impossibility result due to Chia-Chung-Liu-Yamakawa (FOCS 2021). This brings us to our final contribution:
3) We introduce coherent-runtime expected quantum polynomial time, a simulation notion that (1) precisely captures all of our zero-knowledge simulators, (2) cannot break any polynomial hardness assumptions, (3) implies strict polynomial-time epsilon-simulation and (4) is not subject to the CCLY impossibility. In light of our positive results and the CCLY negative results, we propose coherent-runtime simulation to be the appropriate quantum analogue of classical expected polynomial-time simulation
