The Capacity of Private Information Retrieval with Disjoint Colluding Sets

An extension of private information retrieval (PIR) with colluding servers is considered. The N servers are partitioned into M disjoint sets, such that collusion can only occur between servers that belong to the same set. Specifically, the m-th set is comprised of N
_m
servers, of which any T
_m
can collude. The capacity of this PIR problem is shown to be C = (1 + (Σ
_{m = 1}
^M
N
_m
/T
_m
)-1 + ⋯ + (Σ
_{m = 1}
^M
N
_m
/T
_m
)
^-(κ-1)
)
^-1
.