2012 IEEE 27th Conference on Computational Complexity (2007)

San Diego, California

June 13, 2007 to Mar. 16, 2007

ISSN: 1093-0159

ISBN: 0-7695-2780-9

pp: 141-154

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/CCC.2007.15

Emanuele Viola , Institute for Advanced Study, USA

Avi Wigderson , Institute for Advanced Study, USA

ABSTRACT

This paper presents a unified and simple treatment of basic questions concerning two computational models: multiparty communication complexity and GF(2) polynomials. The key is the use of (known) norms on Boolean functions, which capture their approximability in each of these models. <p>The main contributions are new XOR lemmas. We show that if a Boolean function has correlation at most \in \leqslant 1/2 with any of these models, then the correlation of the parity of its values on m independent instances drops exponentially with m. More specifically:</p> <p> For GF(2) polynomials of degree d, the correlation drops to exp (-m/4^d). No XOR lemma was known even for d = 2.</p> <p> For c-bit k-party protocols, the correlation drops to 2^c \cdot \in^{m/2^k} . No XOR lemma was known for k \geqslant 3 parties.</p> <p>Another contribution in this paper is a general derivation of direct product lemmas from XOR lemmas. In particular, assuming that f has correlation at most \in \leqslant 1/2 with any of the above models, we obtain the following bounds on the probability of computing m independent instances of f correctly:</p> <p> For GF(2) polynomials of degree d we again obtain a bound of exp (-m/4^d).</p> <p> For c-bit k-party protocols we obtain a bound of 2^{-\Omega(m)} in the special case when \in \leqslant exp (-c \cdot 2^k). In this range of \in, our bound improves on a direct product lemma for two-parties by Parnafes, Raz, and Wigderson (STOC '97).</p> <p>We also use the norms to give improved (or just simplified) lower bounds in these models. In particular we give a new proof that the Mod_m function on n bits, for odd m, has correlation at most exp(-n/4^d) with degree-d GF(2) polynomials.</p>

INDEX TERMS

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CITATION

Emanuele Viola,
Avi Wigderson,
"Norms, XOR Lemmas, and Lower Bounds for GF(2) Polynomials and Multiparty Protocols",

*2012 IEEE 27th Conference on Computational Complexity*, vol. 00, no. , pp. 141-154, 2007, doi:10.1109/CCC.2007.15