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

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 ≤ 1/2 with either 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/4d ) . No XOR lemma was known even for d = 2. • For c-bit k-party protocols, the correlation drops to 2c · εm/2 . No XOR lemma was known for k ≥ 3 parties. ∗Supported by NSF grant CCR-0324906. This research was partially done while the author was a postdoctoral fellow at Harvard University, supported by NSF grant CCR-0133096, US-Israel BSF grant 2002246, and ONR grant N-00014-04-10478. †Supported by NSF grant CCR-0324906. ACM Classification: 68Q17 AMS Classification: F.2.3