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).
* 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 $\mathbb{F}_2$.
* We show that the dual weak pigeonhole principle is unprovable in Cook's theory $\mathrm{PV}_1$ under the existence of demi-bits generators secure against $\mathbf{AM}$, thereby separating Jerabek's theory $\mathrm{APC}_1$ from $\mathrm{PV}_1$. Previously, Ilango, Li, and Williams (STOC '23) obtained the same separation under different (and arguably stronger) cryptographic assumptions.
* We transform demi-bits generators to proof complexity generators that are *pseudo-surjective* with nearly optimal parameters. Pseudo-surjectivity is the strongest form of hardness considered in the literature of proof complexity generators.

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.