  Shamir's (n, k) threshold secret sharing is an important component of several
cryptographic protocols, such as those for secure multiparty-computation and
key management. These protocols typically assume the presence of direct
communication links from the dealer to all participants, in which case the
dealer can directly pass the shares of the secret to each participant. In this
paper, we consider the problem of secret sharing when the dealer does not have
direct communication links to all the participants, and instead, the dealer and
the participants form a general network. Existing methods are based on secure
message transmissions from the dealer to each participant requiring
considerable coordination in the network. In this paper, we present a
distributed algorithm for disseminating shares over a network, which we call
the SNEAK algorithm, requiring each node to know only the identities of its
one-hop neighbours. While SNEAK imposes a stronger condition on the network by
requiring the dealer to be what we call k-propagating rather than k-connected
as required by the existing solutions, we show that in addition to being
distributed, SNEAK achieves significant reduction in the communication cost and
the amount of randomness required.
