The traditional definition of quantum zero-knowledge stipulates that the knowledge gained by any quantum polynomial-time verifier in an interactive protocol can be simulated by a quantum polynomial-time algorithm. One drawback of this definition is that it allows the simulator to consume significantly more computational resources than the verifier. We argue that this drawback renders the existing notion of quantum zero-knowledge not viable for certain settings, especially when dealing with near-term quantum devices. In this work, we initiate a fine-grained notion of post-quantum zero-knowledge that is more compatible with near-term quantum devices. We introduce the notion of $(s,f)$ space-bounded quantum zero-knowledge. In this new notion, we require that an $s$ -qubit malicious verifier can be simulated by a quantum polynomial-time algorithm that uses at most $f(s)$ -qubits, for some function $f(\cdot)$ , and no restriction on the amount of the classical memory consumed by either the verifier or the simulator. We explore this notion and establish both positive and negative results: - For verifiers with logarithmic quantum space $s$ and (arbitrary) polynomial classical space, we show that $(s,f)$ -space-bounded QZK, for $f(s)=2s$ , can be achieved based on the existence of post-quantum one-way functions. Moreover, our protocol runs in constant rounds. - For verifiers with super-logarithmic quantum space $s$ , assuming the existence of post-quantum secure one-way functions, we show that $(s,f)$ -space-bounded QZK protocols, with fully black-box simulation (classical analogue of black-box simulation) can only be achieved for languages in BQP.The traditional definition of quantum zero-knowledge stipulates that the knowledge gained by any quantum polynomial-time verifier in an interactive protocol can be simulated by a quantum polynomial-time algorithm. One drawback of this definition is that it allows the simulator to consume significantly more computational resources than the verifier. We argue that this drawback renders the existing notion of quantum zero-knowledge not viable for certain settings, especially when dealing with near-term quantum devices. In this work, we initiate a fine-grained notion of post-quantum zero-knowledge that is more compatible with near-term quantum devices. We introduce the notion of $(s,f)$ space-bounded quantum zero-knowledge. In this new notion, we require that an $s$-qubit malicious verifier can be simulated by a quantum polynomial-time algorithm that uses at most $f(s)$-qubits, for some function $f(\cdot)$, and no restriction on the amount of the classical memory consumed by either the verifier or the simulator. We explore this notion and establish both positive and negative results: - For verifiers with logarithmic quantum space $s$ and (arbitrary) polynomial classical space, we show that $(s,f)$-space-bounded QZK, for $f(s)=2s$, can be achieved based on the existence of post-quantum one-way functions. Moreover, our protocol runs in constant rounds. - For verifiers with super-logarithmic quantum space $s$, assuming the existence of post-quantum secure one-way functions, we show that $(s,f)$-space-bounded QZK protocols, with fully black-box simulation (classical analogue of black-box simulation) can only be achieved for languages in BQP.

arXiv:2210.06093Oct 12, 20222 citations

Post-Quantum Zero Knowledge, Revisited or: How to Do Quantum Rewinding Undetectably

Alex Lombardi, Fermi Ma, Nicholas Spooner

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 (a) precisely captures all of our zero-knowledge simulators, (b) cannot break any polynomial hardness assumptions, (c) implies strict polynomial-time ε-simulation and (d) 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.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 (a) precisely captures all of our zero-knowledge simulators, (b) cannot break any polynomial hardness assumptions, (c) implies strict polynomial-time ε-simulation and (d) 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.

arXiv:2111.12257Annual Symposium on Foundations of Computer ScienceOct 1, 202231 citations

(s,f)

space-bounded quantum zero-knowledge. In this new notion, we require that an

s

-qubit malicious verifier can be simulated by a quantum polynomial-time algorithm that uses at most

f(s)

-qubits, for some function

f(\cdot)

, and no restriction on the amount of the classical memory consumed by either the verifier or the simulator. We explore this notion and establish both positive and negative results: - For verifiers with logarithmic quantum space

s

and (arbitrary) polynomial classical space, we show that

(s,f)

-space-bounded QZK, for

f(s)=2s

, can be achieved based on the existence of post-quantum one-way functions. Moreover, our protocol runs in constant rounds. - For verifiers with super-logarithmic quantum space

s

, assuming the existence of post-quantum secure one-way functions, we show that

(s,f)

-space-bounded QZK protocols, with fully black-box simulation (classical analogue of black-box simulation) can only be achieved for languages in BQP.