Inverse conjecture for the gowers norm is false
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Unconditional pseudorandom generators for low degree polynomials
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Hardness amplification proofs require majority
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Improved Separations between Nondeterministic and Randomized Multiparty Communication
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
Random graphs and the parity quantifier
Proceedings of the forty-first annual ACM symposium on Theory of computing
Random Low Degree Polynomials are Hard to Approximate
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
On the structure of cubic and quartic polynomials
Proceedings of the forty-second ACM symposium on Theory of computing
Optimal Testing of Reed-Muller Code
Property testing
Optimal Testing of Reed-Muller Code
Property testing
Studies in complexity and cryptography
Random graphs and the parity quantifier
Journal of the ACM (JACM)
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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 \in \leqslant 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 \cdot\in^{m/2^k} . No XOR lemma was known for k \geqslant 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 \in \leqslant1/2 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 \in \leqslant exp (-c \cdot 2^k). In this range of \in, 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.