Norms, XOR lemmas, and lower bounds for GF(2) polynomials and multiparty protocols

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 ε ≤ 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 · 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 ≤ 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 G F (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^-Ω(m) in the special case when ε ≤ exp (-c · 2^k). In this range of ε, 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 degreed GF (2) polynomials.