  This work investigates the problem of demand privacy against colluding users
for shared-link coded caching systems, where no subset of users can learn any
information about the demands of the remaining users. The notion of privacy
used here is stronger than similar notions adopted in past work and is
motivated by the practical need to insure privacy regardless of the file
distribution. Two scenarios are considered: Single File Retrieval (SFR) and
Linear Function Retrieval (LFR), where in the latter case each user demands an
arbitrary linear combination of the files at the server. The main contributions
of this paper are a novel achievable scheme for LFR, referred as privacy key
scheme, and a new information theoretic converse bound for SFR. Clearly, being
SFR a special case of LFR, an achievable scheme for LFR works for SFR as well,
and a converse for SFR is a valid converse for LFR as well. By comparing the
performance of the achievable scheme with the converse bound derived in this
paper (for the small cache size regime) and existing converse bounds without
privacy constraints (in the remaining memory regime), the communication load of
the privacy key scheme turns out to be optimal to within a constant
multiplicative gap in all parameter regimes. Numerical results show that the
new privacy key scheme outperforms in some regime known schemes based on the
idea of virtual users, which also satisfy the stronger notion of user privacy
against colluding users adopted here. Moreover, the privacy key scheme enjoys
much lower subpacketization than known schemes based on virtual users.
