  Consider the problem of storing data in a distributed manner over $T$
servers. Specifically, the data needs to (i) be recoverable from any $\tau$
servers, and (ii) remain private from any $z$ colluding servers, where privacy
is quantified in terms of mutual information between the data and all the
information available at any $z$ colluding servers. For this model, our main
results are (i) the fundamental trade-off between storage size and the level of
desired privacy, and (ii) the optimal amount of local randomness necessary at
the encoder. As a byproduct, our results provide an optimal lower bound on the
individual share size of ramp secret sharing schemes under a more general
leakage symmetry condition than the ones previously considered in the
literature.
