  Secret sharing is an instrumental tool for sharing secret keys in distributed
systems. In a classical threshold setting, this involves a dealer who has a
secret/key, a set of parties/users to which shares of the secret are sent, and
a threshold on the number of users whose presence is needed in order to recover
the secret. In secret sharing, secure links with no leakage are often assumed
between the involved parties. However, when the users are nodes in a
communication network and all the links are physical links, e.g., wireless,
such assumptions are not valid anymore. In order to study this critical
problem, we propose a statistical leakage model of secret sharing, where some
noisy versions of all the secret shares might be independently leaked to an
adversary. We then study the resilience of the seminal Shamir's secret sharing
scheme with statistical leakage, and bound certain measures of security (i.e.,
semantic security, mutual information security), given other parameters of the
system including the amount of leakage from each secret share. We show that for
an extreme scenario of Shamir's scheme, in particular when the underlying field
characteristic is $2$, the security of each bit of the secret against leakage
improves exponentially with the number of users. To the best of our knowledge,
this is the first attempt towards understanding secret sharing under general
statistical noisy leakage.
