  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.
