  The Pliable Index CODing (PICOD) problem is a variant of the Index Coding
(IC) problem, where the desired messages by the users, who are equipped with
message side information, is part of the optimization. This paper studies the
PICOD problem where users are subject to a privacy constraint. In particular,
the following spacial class of private PICODs is investigated: 1) the side
information structure is circular, and 2) each user can decode one and only one
message. The first condition is a special case of the "circular-arc network
topology hypergraph" class of PICOD studied in [Liu and D. Tuninetti, "Tight
information theoretic converse results for some pliable index coding problems,"
ITW, 2018], for which an optimal solution was given without the privacy
constraint. The second condition was first studied in [S. Sasi and B. S. Rajan,
"On pliable index coding," arXiv:1901.05809] and was motivated by the need to
keep content privacy is some distribution networks. This paper proposes both
converse and achievable bounds. The proposed achievable scheme not only
strictly outperforms the one in [S. Sasi and B. S. Rajan, "On pliable index
coding," arXiv:1901.05809] for some values of the system parameters, but it is
also information theoretically optimal in some settings. For the remaining
cases, the proposed linear code is shown to require at most one more
transmission than the converse bound derived by restricting the sender to only
use linear codes.
