Unprovable Security of Two-Message Zero Knowledge
Kai-Min Chung, Edward Lui, Mohammad Mahmoody, Rafael Pass · 2012
Goldreich and Oren (JoC’94) show that only trivial languages have 2-message zero-knowledge arguments. In this note we consider weaker, super-polynomial-time simulation (SPS), notions of zero-knowledge. We present barriers to using black-box reductions for demonstrating soundness of 2-message protocols with efficient prover strategies satisfying SPS zero-knowledge. More precisely, we show that assuming the existence of poly(T (n))-hard one-way functions, the following holds: • For sub-exponential (or smaller) T (·), polynomial-time black-box reductions cannot be used to prove soundness of 2-message T (·)-simulatable arguments based on any polynomialtime intractability assumption. This matches known 2-message quasi-polynomial-time simulatable arguments using a quasi-polynomial-time reduction (Pass’03), and 2-message exponential-time simulatable proofs using a polynomial-time reduction (Dwork-Naor’00, Pass’03). • poly(T (·))-time black-box reductions cannot be used to prove soundness of 2-message strong T (·)-simulatable (efficient prover) arguments based on any poly(T (·))-time intractability assumption; strong T (·)-simulatability means that the output of the simulator is indistinguishable also for poly(T (·))-size circuits. This matches known 3-message strong quasi-polynomial-time simulatable proofs (Blum’86, Canetti et al ’ 00).