  We prove a lower bound on the information leakage of any classical protocol
computing the equality function in the simultaneous message passing (SMP)
model. Our bound is valid in the finite length regime and is strong enough to
demonstrate a quantum advantage in terms of information leakage for practical
quantum protocols. We prove our bound by obtaining an improved finite size
version of the communication bound due to Babai and Kimmel, relating randomized
communication to deterministic communication in the SMP model. We then relate
information leakage to randomized communication through a series of reductions.
We first provide alternative characterizations for information leakage,
allowing us to link it to average length communication while allowing for
shared randomness (pairwise, with the referee). A Markov inequality links this
with bounded length communication, and a Newman type argument allows us to go
from shared to private randomness. The only reduction in which we incur more
than a logarithmic additive factor is in the Markov inequality; in particular,
our compression method is essentially tight for the SMP model with average
length communication.
