  This paper considers the problem of multi-server Private Linear Computation,
under the joint and individual privacy guarantees. In this problem, identical
copies of a dataset comprised of $K$ messages are stored on $N$ non-colluding
servers, and a user wishes to obtain one linear combination of a $D$-subset of
messages belonging to the dataset. The goal is to design a scheme for
performing the computation such that the total amount of information downloaded
from the servers is minimized, while the privacy of the $D$ messages required
for the computation is protected. When joint privacy is required, the
identities of all of these $D$ messages must be kept private jointly, and when
individual privacy is required, the identity of every one of these $D$ messages
must be kept private individually. In this work, we characterize the capacity,
which is defined as the maximum achievable download rate, under both joint and
individual privacy requirements. In particular, we show that when joint privacy
is required the capacity is given by ${(1+1/N+\dots+1/N^{K-D})^{-1}}$, and when
individual privacy is required the capacity is given by
${(1+1/N+\dots+1/N^{\lceil K/D\rceil-1})^{-1}}$ assuming that $D$ divides $K$,
or $K\pmod D$ divides $D$. Our converse proofs are based on reduction from two
variants of the multi-server Private Information Retrieval problem in the
presence of side information. Our achievability schemes build up on our
recently proposed schemes for single-server Private Linear Transformation and
the multi-server private computation scheme proposed by Sun and Jafar. Using
similar proof techniques, we also establish upper and lower bounds on the
capacity for the cases in which the user wants to compute $L$ (potentially more
than one) linear combinations.
