We study the communication complexity of symmetric XOR functions, namely functions f:01n01n01 that can be formulated as f(xy)=D(xy) for some predicate D:01n01 , where xy 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 quantum and 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.
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