Lower Bounds for Non-Black-Box Zero Knowledge
Boaz Barak, Yehuda Lindell, and Salil Vadhan
We show new lower bounds and impossibility results for general (possibly non-black-box)
zero-knowledge proofs and arguments. Our main results are that, under reasonable
The complexity assumptions we use are not commonly used in cryptography. However, in all
cases, we show that assumptions similar to ours are necessary for the above results.
Most previously known lower bounds, such as those of Goldreich and Krawczyk (SIAM J.
Computing, 1996), were only for black-box zero knowledge. However, a result of Barak
(FOCS 2001) shows that many (or even most) of these black-box lower bounds do not
extend to the case of general zero knowledge.
- There does not exist a two-round zero-knowledge proof system with perfect completeness
for an NP-complete language. The previous impossibility result for two-round zero
knowledge, by Goldreich and Oren (J. Cryptology, 1994) was only for the case of
auxiliary-input zero-knowledge proofs and arguments.
- There does not exist a constant-round zero-knowledge strong proof or argument of
knowledge (as defined by Goldreich (2001)) for a nontrivial language.
- There does not exist a constant-round public-coin proof system for a nontrivial
language that is resettable zero knowledge. This result also extends to "bounded-resettable"
zero knowledge, in which the number of resets is a priori bounded by a
polynomial in the input length and prover-to-verifier communication. In contrast, we show
that under reasonable assumptions, there does exist such a (computationally sound)
argument system that is bounded-resettable zero knowledge.
Extended abstract in Proceedings of the 44th Annual Symposium on Foundations
of Computer Science (FOCS `03), pages 384-393, Cambridge, MA, October 2003.
Electronic Colloquium on Computational Complexity (ECCC), technical
report TR04-083. September 2004. [ECCC
Cryptology ePrint Archive, Report 2004/226. September 2004.
Journal of Computer and System Sciences, 72(2):321--391,
March 2006. Special Issue on FOCS `03. [official
back to Salil Vadhan's research]