Hardness of Range Avoidance and Proof Complexity Generators from Demi-Bits

TL;DR

The paper develops a cohesive theory linking demi-bits cryptographic primitives to hardness of range avoidance and the existence of proof-complexity generators. It shows that demi-bits generators imply $ ext{Avoid}$ is hard for nondeterministic search (and, under stronger assumptions, even for restricted circuit classes), and that demi-bits can be transformed into proof-complexity generators with pseudo-surjectivity properties. It also establishes an unprovability separation between bounded arithmetic theories, showing $ ext{dwPHP}( ext{PV})$ is not provable in $ ext{PV}_1$ under demi-bits secure against $ ext{AM}$, thereby separating $ ext{APC}_1$ from $ ext{PV}_1$. The work leverages randomness extractors to simplify the constructions and frames the results as average-case to best-case reductions in proof complexity, with concurrent efforts offering complementary viewpoints. Overall, the results push toward a minicrypt foundation for range-avoidance hardness and illuminate how nondeterministic cryptographic assumptions interact with proof complexity landscapes.

Abstract

Given a circuit $G: \{0, 1\}^n \to \{0, 1\}^m$ with $m > n$, the *range avoidance* problem ($\text{Avoid}$) asks to output a string $y\in \{0, 1\}^m$ that is not in the range of $G$. Besides its profound connection to circuit complexity and explicit construction problems, this problem is also related to the existence of *proof complexity generators* -- circuits $G: \{0, 1\}^n \to \{0, 1\}^m$ where $m > n$ but for every $y\in \{0, 1\}^m$, it is infeasible to prove the statement "$y\not\in\mathrm{Range}(G)$" in a given propositional proof system. This paper connects these two problems with the existence of *demi-bits generators*, a fundamental cryptographic primitive against nondeterministic adversaries introduced by Rudich (RANDOM '97). $\bullet$ We show that the existence of demi-bits generators implies $\text{Avoid}$ is hard for nondeterministic algorithms. This resolves an open problem raised by Chen and Li (STOC '24). Furthermore, assuming the demi-hardness of certain LPN-style generators or Goldreich' PRG, we prove the hardness of $\text{Avoid}$ even when the instances are constant-degree polynomials over $\mathbb{F}_2$. $\bullet$ We show that the dual weak pigeonhole principle is unprovable in Cook's theory $\mathsf{PV}_1$ under the existence of demi-bits generators secure against $\mathbf{AM}$, thereby separating Jerabek's theory $\mathsf{APC}_1$ from $\mathsf{PV}_1$. $\bullet$ We transform demi-bits generators to proof complexity generators that are *pseudo-surjective* with nearly optimal parameters. Our constructions build on the recent breakthroughs on the hardness of $\text{Avoid}$ by Ilango, Li, and Williams (STOC '23) and Chen and Li (STOC '24). We use *randomness extractors* to significantly simplify the construction and the proof.

Theorems & Definitions (56)

• Theorem 1.1: Main
• Theorem 1.2
• Corollary 1.3: Informal
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Definition 2.1: Demi-Bits Generators
• Lemma 2.2: GS86, also see AB09
• Definition 2.3: Breaking demi-bits generators by $\AM$ adversaries