Lower Bounds from Succinct Hitting Sets

TL;DR

This work studies how succinct and cryptographic hitting-set generators for algebraic circuits influence fundamental lower-bound questions. By developing an upper bound on annihilators via VPSPACE and expressing annihilators as determinants of explicit matrices, the authors connect explicit-hitting-set phenomena to separations such as $VP \neq VNP$ and to strong lower bounds for small-depth circuits under GRH. They introduce cryptographic hitting-set generators and show that their existence would imply notable circuit lower bounds, extending the landscape beyond VP versus VNP to NC^1 and TC^0 interactions, and even P versus PSPACE under certain hypotheses. The results hinge on translating natural-proof barriers into concrete algebraic-structure consequences, motivating a new axis of cryptographic HSGs and refining the role of VPSPACE in algebraic complexity theory.

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.

Figures (1)

• Figure 1: The annihilator is computed by using (1) repeatedly to get $C_{\log N}$, if $N$ is the length of the ABP described by (2) (\ref{['claim:ABP-to-projection-circuit']}); (2) describes the ABP computing $\det(\widetilde{M})$ using (3) (\ref{['claim:succinct-matrix-to-ABP']}); (3) describes the matrix $\widetilde{M}$, such that $\det(\widetilde{M}) \circ \mathcal{G} \equiv 0$, in terms of the generator $\mathcal{G}$ (\ref{['lem:ann-as-succinct-det']}).

Theorems & Definitions (90)

• definition 1.1: $\mathcal{C}$-Succinct HSG for $\mathcal{D}$
• theorem 1.1: Hardness from Succinct Hitting Sets
• remark 1.2
• theorem 1.2: Equations for VP
• remark 1.3
• definition 1.5
• definition 1.6: $\mathcal{C}$-Cryptographic HSG for $\mathcal{D}$
• theorem 1.6: Lower Bounds from Cryptographic HSGs
• remark 1.7
• theorem 1.7: Upper bound for annihilators of $\VP$