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

Authors Hanlin Ren , Yichuan Wang , Yan Zhong

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LIPIcs.ITCS.2026.111.pdf
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ACM Subject Classification
  • Theory of computation → Complexity classes
  • Theory of computation → Circuit complexity
  • Theory of computation → Proof complexity
  • Theory of computation → Cryptographic primitives
  • Theory of computation → Pseudorandomness and derandomization
  • Theory of computation → Expander graphs and randomness extractors
Keywords
  • Range Avoidance
  • Proof Complexity Generators

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Abstract

Given a circuit G: {0, 1}ⁿ → {0, 1}^m with m > n, the range avoidance problem (Avoid) asks to output a string y ∈ {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 → {0, 1}^m where m > n but for every y ∈ {0, 1}^m, it is infeasible to prove the statement "y ̸ ∈ 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 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’s PRG, we prove the hardness of Avoid even when the instances are constant-degree polynomials over 𝔽₂. 
- We show that the dual weak pigeonhole principle is unprovable in Cook’s theory PV₁ under the existence of demi-bits generators secure against AM/_{O(1)}, thereby separating Jeřábek’s theory APC₁ from PV₁. 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 in certain parameter regime. Pseudo-surjectivity is the strongest form of hardness considered in the literature for proof complexity generators.  Our constructions are inspired by the recent breakthroughs on the hardness of 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.

Cite As Get BibTex

Hanlin Ren, Yichuan Wang, and Yan Zhong. Hardness of Range Avoidance and Proof Complexity Generators from Demi-Bits. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 111:1-111:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ITCS.2026.111

Author Details

Hanlin Ren
  • Institute for Advanced Study, Princeton, NJ, USA
Yichuan Wang
  • Department of EECS, University of California, Berkeley, CA, USA
Yan Zhong
  • Department of Computer Science, Johns Hopkins University, Baltimore, MD, USA

Acknowledgements

We thank Yilei Chen for helpful discussions regarding the LPN assumption, Xin Li for helpful discussions about extractors, and Rahul Ilango for sending us a draft version of [Rahul Ilango, 2025]. We are also grateful to Erfan Khaniki and Iddo Tzameret for helpful suggestions that improved the presentation of this paper.

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