One-way functions are essential for non-trivial zero-knowledge

If one-way functions exist, then there are zero-knowledge proofs for every language in PSPACE. The authors prove that unless very weak one-way functions exist, zero-knowledge proofs can be given only for languages in BPP. For average-case definitions of BPP they prove an analogous result under the assumption that uniform one-way functions do not exist. Thus, very loosely speaking, zero-knowledge is either useless (exists only for `easy´ languages), or universal (exists for every provable language)