Strong direct product conjecture holds for all relations in public coin randomized one-way communication complexity

Let f subset of X x Y x Z be a relation. Let the public coin one-way communication complexity of f, with worst case error 1/3, be denoted R^{{1,pub}_{1/3}(f).} We show that if for computing f^k (k independent copies of f), o(k R^{{1,pub}_{1/3}(f))} communication is provided, then the success is exponentially small in k. This settles the strong direct product conjecture for all relations in public coin one-way communication complexity. We show a new tight characterization of public coin one-way communication complexity which strengthens on the tight characterization shown in [J., Klauck, Nayak 08]. We use the new characterization to show our direct product result and this may also be of independent interest.