Tight bounds on the randomized communication complexity of symmetric XOR functions in one-way and SMP models

We study the communication complexity of symmetric XOR functions, namely functions $f:\{0,1{\}}^n\times \{0,1{\}}^n\to \{0,1\}$ that can be formulated as $f(x,y)=D(\mathrm{\mid }x\oplus y\mathrm{\mid })$ for some predicate $D: {0,1,…,n} \rightarrow {0,1}$,where$|x\oplus y|$ is the Hamming weight of the bitwise XOR of $x$ and $y$. We give a public-coin randomized protocol in the Simultaneous Message Passing (SMP) model, with the communication cost matching the known lower bound for the \emph{quantum} and \emph{two-way} model up to a logarithm factor. As a corollary, this closes a quadratic gap between quantum lower bound and randomized upper bound for the one-way model, answering an open question raised in Shi and Zhang \cite{SZ09}.