Paper 2019/754
A Round-Collapse Theorem for Computationally-Sound Protocols; or, TFNP is Hard (on Average) in Pessiland
Rafael Pass and Muthuramakrishnan Venkitasubramaniam
Abstract
Consider the following two fundamental open problems in complexity theory: 1) Does a hard-on-average language in $\mathsf{NP}$ imply the existence of one-way functions? 2) Does a hard-on-average language in $\mathsf{NP}$ imply a hard problem in $\mathsf{TFNP}$ (i.e., the class of \emph{total} $\mathsf{NP}$ search problem)? We show that the answer to (at least) one of these questions is yes. In other words, in Impagliazzo's Pessiland (where $\mathsf{NP}$ is hard-on-average, but one-way functions do not exist), $\mathsf{TFNP}$ is unconditionally hard (on average). This result follows from a more general theory of interactive average-case complexity, and in particular, a novel round-collapse theorem for computationally-sound protocols, analogous to Babai-Moran's celebrated round-collapse theorem for information-theoretically sound protocols. As another consequence of this treatment, we show that the existence of $O(1)$-round public-coin non-trivial arguments (i.e., argument systems that are not proofs) imply the existence of a hard-on-average problem in $\mathsf{NP}/\mathsf{poly}$.
Metadata
- Available format(s)
- Category
- Foundations
- Publication info
- Preprint. MINOR revision.
- Keywords
- TFNPround-collapseaverage-case hardness
- Contact author(s)
- muthuv @ cs rochester edu
- History
- 2020-04-15: last of 2 revisions
- 2019-06-26: received
- See all versions
- Short URL
- https://ia.cr/2019/754
- License
-
CC BY