Volume 4 (2008) Article 7 pp. 137-168

Norms, XOR Lemmas, and Lower Bounds for Polynomials and Protocols

by Emanuele Viola and Avi Wigderson

Received: July 24, 2007
Published: November 18, 2008
DOI: 10.4086/toc.2008.v004a007

Abstract: [Plain Text Version]

This paper presents a unified and simple treatment of basic questions concerning two computational models: multiparty communication complexity and polynomials over GF(2). The key is the use of (known) norms on Boolean functions, which capture their proximity to each of these models (and are closely related to property testers of this proximity).

The main contributions are new XOR lemmas. We show that if a Boolean function has correlation at most ε ≤ ½ 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 polynomials over GF(2) 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 · ε^m/2k. No XOR lemma was known for k ≥ 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 ε ≤ ½ with any of the above models, we obtain the following bounds on the probability of computing m independent instances of f correctly:

• For polynomials over GF(2) 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^-Ω(m) in the special case when
  ε ≤ exp(-c · 2^k).

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 polynomials over GF(2).