Decentralized Pliable Index Coding

Abstract

This paper studies a variant of the Pliable Index CODing (PICOD) problem, i.e., an index coding problem where a user can be satisfied by decoding any message that is not in its side information set, where communication is decentralized, i.e., it occurs among users rather than by the central server, and secure, i.e., each user is allowed to decode only one message outside its side information set and must not be able to collect any information about any other message that is not its decoded one. Given the difficulty of the general version of this problem, this paper focuses on the case where the side information sets are 's circular shifts', namely, user u's side information set is the set of messages indexed by {u, u + 1, . . . , u + s − 1} for some fixed s and where the indices are intended modulo the cardinality of the message set. This particular setting has been studied in the 'decentralized non-secure' and in the 'centralized secure' settings, thus allows one to quantify the cost of decentralized communication under security constraints on the number of transmissions. Interestingly, the decentralized vs the centralized secure setting incurs a multiplicative gap of approximately three. This is in contrast to the cases without security constraint, where the multiplicative gap is known to be at most two.

Optimal Linear Coding Schemes for the Secure Decentralized Pliable Index Coding Problem

“…The information theoretical converse was derived in [6] for the centralized case. The achievable scheme is the one for the decentralized PICOD without security constraint discussed in [4], which satisfies the security constraint because for each user, among all the messages that are involved in the encoding function, there is one and only one message that is not in its side information set. Therefore, the scheme is also information theoretically optimal with an additional security constraint.…”

“…Under linear encoding constraint, the minimum code-length of the decentralized IC is shown to be no more than twice that of its centralized counterpart [1]. The pliable version of decentralized IC was studied in [4], where information theoretical bounds on the optimal code-length were given for several cases. For the solved cases, the multiplicative gap between centralized and decentralized PICOD is usually much less than two.…”

“…4,5 ) and g 2 (W 4,5,6 ): the user with side information W 1,2,3,4 can decode W 5,6 , which is insecure; 2) g 1 (W 4,5 ) and g 2 (W 5,6,7 ): the user with side information W (W 3,5 ) and g 2 (W 5,6,7 ): the user with side information W 6,7,8,9 can decode W 3,5 , which is insecure. c) Both transmissions involve 3 consecutive messages:We have the following cases:1) g 1 (W 4,5,6 ) and g 2 (W 4,5,6 ): a linear combination of these two transmissions can generate a linear combination of W 4,5 .…”

Private Index Coding

“…Liu et al [24] extended this setup to a setting where the weak privacy constraint is required only against a subset of the messages not available to a user. Recently, some works [22], [23], [25] studied the notion of weak privacy for a variant of the index coding problem called the Pliable Index CODing (PICOD(t)) introduced by Brahma et al [26], where each user wants to obtain any t ≥ 1 messages that it does not have access to. Sasi et al [25] studied PICOD (1) with the privacy condition that each user is able to decode exactly one message outside its side information set for the special case of consecutive side information.…”

“…A generalization of [25] in terms of the form of the side information sets was investigated by Liu et al [22]. The authors further extended their study to a decentralized setting [23] where communication occurs among users rather than by a central server.…”