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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2025.134

A Pseudorandom Generator for Functions of Low-Degree Polynomial Threshold Functions

Authors: Penghui Yao and Mingnan Zhao

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract

Developing explicit pseudorandom generators (PRGs) for prominent categories of Boolean functions is a key focus in computational complexity theory. In this paper, we investigate the PRGs against the functions of degree-d polynomial threshold functions (PTFs) over Gaussian space. Our main result is an explicit construction of PRG with seed length poly(k, d, 1/ε)⋅log n that can fool any function of k degree-d PTFs with probability at least 1 - ε. More specifically, we show that the summation of L independent R-moment-matching Gaussian vectors ε-fools functions of k degree-d PTFs, where L = poly(k, d, 1/ε) and R = O(log kd/ε). The PRG is then obtained by applying an appropriate discretization to Gaussian vectors with bounded independence.

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Penghui Yao and Mingnan Zhao. A Pseudorandom Generator for Functions of Low-Degree Polynomial Threshold Functions. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 134:1-134:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{yao_et_al:LIPIcs.ICALP.2025.134,
  author =	{Yao, Penghui and Zhao, Mingnan},
  title =	{{A Pseudorandom Generator for Functions of Low-Degree Polynomial Threshold Functions}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{134:1--134:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.134},
  URN =		{urn:nbn:de:0030-drops-235112},
  doi =		{10.4230/LIPIcs.ICALP.2025.134},
  annote =	{Keywords: Pseudorandom generators, polynomial threshold functions}
}

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A Pseudorandom Generator for Functions of Low-Degree Polynomial Threshold Functions

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