  Consider the communication efficient secret sharing problem. A dealer wants
to share a secret with $n$ parties such that any $k\leq n$ parties can
reconstruct the secret and any $z<k$ parties eavesdropping on their shares
obtain no information about the secret. In addition, a legitimate user
contacting any $d$, $k\leq d \leq n$, parties to decode the secret can do so by
reading and downloading the minimum amount of information needed. We are
interested in communication efficient secret sharing schemes that tolerate the
presence of malicious parties actively corrupting their shares and the data
delivered to the users. The knowledge of the malicious parties about the secret
is restricted to the shares they obtain. We characterize the capacity, i.e.
maximum size of the secret that can be shared. We derive the minimum amount of
information needed to to be read and communicated to a legitimate user to
decode the secret from $d$ parties, $k\leq d \leq n$. Error-correcting codes do
not achieve capacity in this setting. We construct codes that achieve capacity
and achieve minimum read and communication costs for all possible values of
$d$. Our codes are based on Staircase codes, previously introduced for
communication efficient secret sharing, and on the use of a pairwise hashing
scheme used in distributed data storage and network coding settings to detect
errors inserted by a limited knowledge adversary.
