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Optimal Pseudorandom Generators for Low-Degree Polynomials over Moderately Large Fields

Authors Ashish Dwivedi , Zeyu Guo , Ben Lee Volk

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  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • Pseudorandom Generators
  • Low Degree Polynomials

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Abstract

We construct explicit pseudorandom generators that fool n-variate polynomials of degree at most d over a finite field 𝔽_q. The seed length of our generators is O(d log n + log q), over fields of size exponential in d and characteristic at least d(d-1)+1. Previous constructions such as Bogdanov’s (STOC 2005) and Derksen and Viola’s (FOCS 2022) had either suboptimal seed length or required the field size to depend on n.
Our approach follows Bogdanov’s paradigm while incorporating techniques from Lecerf’s factorization algorithm (J. Symb. Comput. 2007) and insights from the construction of Derksen and Viola regarding the role of indecomposability of polynomials.

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Ashish Dwivedi, Zeyu Guo, and Ben Lee Volk. Optimal Pseudorandom Generators for Low-Degree Polynomials over Moderately Large Fields. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 44:1-44:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.44

Author Details

Ashish Dwivedi
  • Department of Computer Science and Engineering, The Ohio State University, Columbus, OH, USA
Zeyu Guo
  • Department of Computer Science and Engineering, The Ohio State University, Columbus, OH, USA
Ben Lee Volk
  • Efi Arazi School of Computer Science, Reichman University, Israel

Funding

  • Volk, Ben Lee: The research leading to these results has received funding from the Israel Science Foundation (grant number 843/23).

Acknowledgements

We thank Jesse Goodman and Pooya Hatami for helpful discussions. Part of this work was carried out while the first two authors were visiting the Simons Institute for the Theory of Computing at UC Berkeley. We thank the institute for its support and hospitality.

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