**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 $\mathsf{NP}/\mathsf{poly}$.

**Category / Keywords: **foundations / TFNP, round-collapse, average-case hardness

**Date: **received 25 Jun 2019

**Contact author: **muthuv at cs rochester edu

**Available format(s): **PDF | BibTeX Citation

**Version: **20190626:064205 (All versions of this report)

**Short URL: **ia.cr/2019/754

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