  We consider the following question of bounded simultaneous messages (BSM)
protocols: Can computationally unbounded Alice and Bob evaluate a function
$f(x,y)$ of their inputs by sending polynomial-size messages to a
computationally bounded Carol? The special case where $f$ is the mod-2
inner-product function and Carol is bounded to AC$^0$ has been studied in
previous works. The general question can be broadly motivated by applications
in which distributed computation is more costly than local computation,
including secure two-party computation.
  In this work, we initiate a more systematic study of the BSM model, with
different functions $f$ and computational bounds on Carol. In particular, we
give evidence against the existence of BSM protocols with polynomial-size Carol
for naturally distributed variants of NP-complete languages.
