A new exponential separation between quantum and classical one-way communication complexity

We present a new example of a partial boolean function whose one-way quantum communication complexity is exponentially lower than its one-way classical communication complexity. The problem is a natural generalisation of the previously studied Subgroup Membership problem: Alice receives a bit string x, Bob receives a permutation matrix M, and their task is to determine whether Mx=x or Mx is far from x. The proof uses Fourier analysis and an inequality of Kahn, Kalai and Linial.