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 ≤ 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/4d . No XOR lemma was known even for d = 2. • For c-bit k-party protocols, the correlation drops to 2c · 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 ≤ 1/2 with any of the above models, we obtain the following bou...

• Type: Conference
• Year: 2007
• Where: COCO
• Authors: Emanuele Viola, Avi Wigderson
• Keywords: C-bit K-party Protocols, COCO 2007, NSF Grant, Xor Lemmas