  An XOR function is a function of the form g(x,y) = f(x + y), for some boolean
function f on n bits. We study the quantum and classical communication
complexity of XOR functions. In the case of exact protocols, we completely
characterise one-way communication complexity for all f. We also show that,
when f is monotone, g's quantum and classical complexities are quadratically
related, and that when f is a linear threshold function, g's quantum complexity
is Theta(n). More generally, we make a structural conjecture about the Fourier
spectra of boolean functions which, if true, would imply that the quantum and
classical exact communication complexities of all XOR functions are
asymptotically equivalent. We give two randomised classical protocols for
general XOR functions which are efficient for certain functions, and a third
protocol for linear threshold functions with high margin. These protocols
operate in the symmetric message passing model with shared randomness.
