  We consider the problem of storing data in a distributed manner over $T$
servers. We require the data (i) to be recoverable from the $T$ servers, and
(ii) to remain private from any $T-1$ colluding servers, where privacy is
quantified in terms of mutual information between the data and all the
information available at the $T-1$ colluding servers. For this model, we
determine (i) the fundamental trade-off between storage size and the level of
desired privacy, (ii) the optimal amount of local randomness necessary at the
encoder, and (iii)~an explicit low-complexity coding scheme that solely relies
on XOR operations and that asymptotically (with the data size) matches the
fundamental limits found.
