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 \leqslant 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 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.