Structure of Protocols for XOR Functions
Hamed Hatami, Kaave Hosseini, Shachar Lovett · SIAM Journal on Computing · 2018
Let $f:\{0,1\}^n\to\{0,1\}$ be a boolean function. Its associated XOR function is the two-party function $f_{\oplus}(x,y)=f(x\oplus y)$. We show that, up to polynomial factors, the deterministic communication complexity of $f_{\oplus}$ is equal to the parity decision tree complexity of $f$. This relies on a novel technique of entropy reduction for protocols, combined with existing techniques in Fourier analysis and additive combinatorics.