License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2020.8
URN: urn:nbn:de:0030-drops-126112
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12611/
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Bläser, Markus ; Pandey, Anurag

Polynomial Identity Testing for Low Degree Polynomials with Optimal Randomness

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LIPIcs-APPROX8.pdf (0.5 MB)

Abstract

We give a randomized polynomial time algorithm for polynomial identity testing for the class of n-variate poynomials of degree bounded by d over a field ?, in the blackbox setting.
Our algorithm works for every field ? with | ? | ≥ d+1, and uses only d log n + log (1/ ε) + O(d log log n) random bits to achieve a success probability 1 - ε for some ε > 0. In the low degree regime that is d ≪ n, it hits the information theoretic lower bound and differs from it only in the lower order terms. Previous best known algorithms achieve the number of random bits (Guruswami-Xing, CCC'14 and Bshouty, ITCS'14) that are constant factor away from our bound. Like Bshouty, we use Sidon sets for our algorithm. However, we use a new construction of Sidon sets to achieve the improved bound.
We also collect two simple constructions of hitting sets with information theoretically optimal size against the class of n-variate, degree d polynomials. Our contribution is that we give new, very simple proofs for both the constructions.

BibTeX - Entry

@InProceedings{blser_et_al:LIPIcs:2020:12611,
  author =	{Markus Bl{\"a}ser and Anurag Pandey},
  title =	{{Polynomial Identity Testing for Low Degree Polynomials with Optimal Randomness}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
  pages =	{8:1--8:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-164-1},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{176},
  editor =	{Jaros{\l}aw Byrka and Raghu Meka},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12611},
  URN =		{urn:nbn:de:0030-drops-126112},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2020.8},
  annote =	{Keywords: Algebraic Complexity theory, Polynomial Identity Testing, Hitting Set, Pseudorandomness}
}

Keywords: Algebraic Complexity theory, Polynomial Identity Testing, Hitting Set, Pseudorandomness
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)
Issue Date: 2020
Date of publication: 11.08.2020

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