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

Let f ⊆ X × Y × Z be a relation. Let the public coin one-way communication complexity of f , with worst case error 1/3, be denoted R 1/3 (f). We show that if for computing fk (k independent copies of f), o(k ·R 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 [JKN08]. We use the new characterization to show our direct product result and this may also be of independent interest.