NDLI: Norms, XOR Lemmas, and Lower Bounds for GF(2) Polynomials and Multiparty Protocols

Content Provider	IEEE Xplore Digital Library
Author	Viola, E. Wigderson, A.
Copyright Year	2007
Description	Author affiliation: Inst. for Adv. Study Princeton, Princeton (Viola, E.; Wigderson, A.)
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. The main contributions are new XOR lemmas. We show that if a Boolean function has correlation at most epsi les 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: For GF(2) polynomials of degree d, the correlation drops to exp $(-m/4^{d}).$ No XOR lemma was known even for d = 2. For c-bit k-party protocols, the correlation drops to $2^{c}$ $ldrepsi^{m/2}$ $^{k}$ . No XOR lemma was known for k ges 3 parties. 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 epsi les 1/2 with any of the above models, we obtain the following bounds on the probability of computing m independent instances of f correctly: For GF(2) polynomials of degree d we again obtain a bound of $exp(-m/4^{d}).$ For c-bit k-party protocols we obtain a bound of $2^{-Omega(m)}$ in the special case when epsi les exp (-c ldr $2^{k}).$ In this range of epsi, our bound improves on a direct product lemma for two-parties by Parnafes, Raz, and Wigderson (STOC '97). 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.
Starting Page	141
Ending Page	154
File Size	309349
Page Count	14
File Format	PDF
ISBN	0769527809
ISSN	10930159
DOI	10.1109/CCC.2007.15
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2007-06-13
Publisher Place	USA
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Polynomials Protocols Circuits Computational modeling Boolean functions Computational complexity Mathematics Mathematical model Complexity theory Fellows
Content Type	Text
Resource Type	Article

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