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Algebraic Pseudorandomness in VNC⁰

Author Robert Andrews

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ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Theory of computation → Pseudorandomness and derandomization
  • Theory of computation → Proof complexity
Keywords
  • Polynomial identity testing
  • Algebraic circuits
  • Ideal Proof System

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Abstract

We study the arithmetic complexity of hitting set generators, which are pseudorandom objects used for derandomization of the polynomial identity testing problem. We give new explicit constructions of hitting set generators whose outputs are computable in VNC⁰, i.e., can be computed by arithmetic formulas of constant size. Unconditionally, we construct a VNC⁰-computable generator that hits arithmetic circuits of constant depth and polynomial size. We also give conditional constructions, under strong but plausible hardness assumptions, of VNC⁰-computable generators that hit arithmetic formulas and arithmetic branching programs of polynomial size, respectively. As a corollary of our constructions, we derive lower bounds for subsystems of the Geometric Ideal Proof System of Grochow and Pitassi.
Constructions of such generators are implicit in prior work of Kayal on lower bounds for the degree of annihilating polynomials. Our main contribution is a construction whose correctness relies on circuit complexity lower bounds rather than degree lower bounds.

Cite As Get BibTex

Robert Andrews. Algebraic Pseudorandomness in VNC⁰. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CCC.2025.15

Author Details

Robert Andrews
  • Cheriton School of Computer Science, University of Waterloo, Canada

Funding

  • Andrews, Robert: Part of this work was done at the Institute for Advanced Study and was supported by NSF grant CCF-1900460 and the Erik Ellentuck Endowed Fellowship Fund.

Acknowledgements

We thank Ramprasad Saptharishi for a simpler formulation and proof of [Robert Andrews, 2025, Lemma 3.5], Rafael Oliveira for helpful conversations on Nullstellensatz degree bounds, and Abhibhav Garg for useful feedback that improved the presentation of this work.

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