We show that if a language has an interactive proof of logarithmic statistical knowledge-complexity, then it belongs to the class AM ∩ co-AM. Thus, if the polynomial time hierarchy does not collapse, then NP-complete languages do not have logarithmic knowledge complexity. Prior to this work, there was no indication that would contradict NP languages being proven with even one bit of knowledge. Our result is a common generalization of two previous results: The first asserts that statistical zero knowledge is contained in AM ∩ co-AM, while the second asserts that the languages recognizable in logarithmic statistical knowledge complexity are in BPPNP . Next, we consider the relation between the error probability and the knowledge complexity of an interactive proof. Note that reducing the error probability via repetition is not free: it may increase the knowledge complexity. We show that if the negligible error probability ε(n) is less than 2-3k(n) (where k(n) is the knowledge complexity) then the language proven is in the third level of the polynomial time hierarchy (specifically, it is in AMNP). In the standard setting of negligible error probability, there exist PSPACE-complete languages which have sub-linear knowledge complexity. However, if we insist, for example, that the error probability is less than 2-n2, then PSPACE-complete languages do not have sub-quadratic knowledge complexity, unless PSPACE = σ3p. In order to prove our main result, we develop an AM protocol for checking that a samplable distribution D has a given entropy h. For any fractions ε, δ, the verifier runs in time polynomial in 1/δ and log(1/ε) and fails with probability at most e to detect an additive error δ in the entropy. We believe that this protocol is of independent interest. Subsequent to our work Goldreich and Vadhan established that the problem of comparing the entropies of two samplable distributions if they are noticeably different is a natural complete promise problem for the class of statistical zero knowledge (S Ζ K).
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
- Computational Mathematics