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On the Degree of Boolean Functions as Polynomials over ℤ_m

Authors Xiaoming Sun, Yuan Sun, Jiaheng Wang, Kewen Wu, Zhiyu Xia, Yufan Zheng

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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Boolean function
  • polynomial
  • modular degree
  • Ramsey theory

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Abstract

Polynomial representations of Boolean functions over various rings such as ℤ and ℤ_m have been studied since Minsky and Papert (1969). From then on, they have been employed in a large variety of areas including communication complexity, circuit complexity, learning theory, coding theory and so on. For any integer m ≥ 2, each Boolean function has a unique multilinear polynomial representation over ring ℤ_m. The degree of such polynomial is called modulo-m degree, denoted as deg_m(⋅). 
In this paper, we investigate the lower bound of modulo-m degree of Boolean functions. When m = p^k (k ≥ 1) for some prime p, we give a tight lower bound deg_m(f) ≥ k(p-1) for any non-degenerate function f:{0,1}ⁿ → {0,1}, provided that n is sufficient large. When m contains two different prime factors p and q, we give a nearly optimal lower bound for any symmetric function f:{0,1}ⁿ → {0,1} that deg_m(f) ≥ n/{2+1/(p-1)+1/(q-1)}.

Cite As Get BibTex

Xiaoming Sun, Yuan Sun, Jiaheng Wang, Kewen Wu, Zhiyu Xia, and Yufan Zheng. On the Degree of Boolean Functions as Polynomials over ℤ_m. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 100:1-100:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ICALP.2020.100

Author Details

Xiaoming Sun
  • Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China
Yuan Sun
  • Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China
Jiaheng Wang
  • School of Electronics Engineering and Computer Science, Peking University, Beijing, China
Kewen Wu
  • School of Electronics Engineering and Computer Science, Peking University, Beijing, China
Zhiyu Xia
  • Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China
Yufan Zheng
  • Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China

Funding

This work was supported in part by the National Natural Science Foundation of China Grants No. 61832003, 61761136014, the 973 Program of China Grant No. 2016YFB1000201, the Strategic Priority Research Program of Chinese Academy of Sciences Grant No. XDB28000000, and K. C. Wong Education Foundation.

Acknowledgements

We thank Noga Alon and Xiaoxu Guo for pointing us to relevant references [Noga Alon and Richard Beigel, 2001; Fine and Wilf, 1965]. We also appreciate anonymous reviewers' careful feedback and constructive advice.

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