Lower Bounds from Succinct Hitting Sets

TL;DR

This paper explores the implications of the existence of succinct hitting sets for algebraic circuits, showing they would lead to major complexity class separations or lower bounds, and introduces the concept of cryptographic hitting sets.

Contribution

It establishes a connection between succinct hitting sets and class separations, providing new bounds and introducing cryptographic hitting sets as a novel concept.

Findings

Existence of VP-succinct hitting sets implies VP ≠ VNP or strong lower bounds assuming GRH.

Efficiently describable hitting set generators relate to separations between classes and VPSPACE.

Sub-polynomially explicit hitting sets imply VP ≠ VNP or P ≠ PSPACE.

Abstract

We investigate the consequences of the existence of ``efficiently describable'' hitting sets for polynomial sized algebraic circuit ($\mathsf{VP}$), in particular, \emph{$\mathsf{VP}$-succinct hitting sets}. Existence of such hitting sets is known to be equivalent to a ``natural-proofs-barrier'' towards algebraic circuit lower bounds, from the works that introduced this concept (Forbes \etal (2018), Grochow \etal (2017)). We show that the existence of $\mathsf{VP}$-succinct hitting sets for $\mathsf{VP}$ would either imply that $\mathsf{VP} \neq \mathsf{VNP}$, or yield a fairly strong lower bound against $\mathsf{TC}^0$ circuits, assuming the Generalized Riemann Hypothesis (GRH). This result is a consequence of showing that designing efficiently describable ($\mathsf{VP}$-explicit) hitting set generators for a class $\mathcal{C}$, is essentially the same as proving a separation between $\mathcal{C}$ and $\mathsf{VPSPACE}$: the algebraic analogue of \textsf{PSPACE}. More formally, we prove an upper bound on \emph{equations} for polynomial sized algebraic circuits ($\mathsf{VP}$), in terms of $\mathsf{VPSPACE}$. Using the same upper bound, we also show that even \emph{sub-polynomially explicit hitting sets} for $\mathsf{VP}$ -- much weaker than $\mathsf{VP}$-succinct hitting sets that are almost polylog-explicit -- would imply that either $\mathsf{VP} \neq \mathsf{VNP}$ or that $\mathsf{P} \neq \mathsf{PSPACE}$. This motivates us to define the concept of \emph{cryptographic hitting sets}, which we believe is interesting on its own.