We consider the problem of private linear computation (PLC) in a distributed storage system. In PLC, a user wishes to compute a linear combination of f messages stored in noncolluding databases while revealing no information about the coefficients of the desired linear combination to the databases. In extension of our previous work we employ linear codes to encode the information on the databases. We show that the PLC capacity, which is the ratio of the desired linear function size and the total amount of downloaded information, matches the maximum distance separable (MDS) coded capacity of private information retrieval for a large class of linear codes that includes MDS codes. In particular, the proposed converse is valid for any number of messages and linear combinations, and the capacity expression depends on the rank of the coefficient matrix obtained from all linear combinations.

We study the problem of private function retrieval (PFR) in a distributed storage system. In PFR the user wishes to retrieve a linear combination of M messages stored in noncolluding (N, K) MDS coded databases while revealing no information about the coefficients of the intended linear combination to any of the individual databases. We present an achievable scheme for MDS coded PFR with a rate that matches the capacity for coded private information retrieval derived recently,where Rc = K N is the rate of the MDS code. This achievable rate is tight in some special cases.

Private computation in a distributed storage system (DSS) is a generalization of the private information retrieval (PIR) problem. In such setting a user wishes to compute a function of f messages stored in noncolluding coded databases while revealing no information about the desired function to the databases. We consider the problem of private polynomial computation (PPC). In PPC, a user wishes to compute a multivariate polynomial of degree at most g over f variables (or messages) stored in multiple databases. First, we consider the private computation of polynomials of degree g = 1, i.e., private linear computation (PLC) for coded databases. In PLC, a user wishes to compute a linear combination over the f messages while keeping the coefficients of the desired linear combination hidden from the database. For a linearly encoded DSS, we present a capacity-achieving PLC scheme and show that the PLC capacity, which is the ratio of the desired amount of information and the total amount of downloaded information, matches the maximum distance separable coded capacity of PIR for a large class of linear storage codes. Then, we consider private computation of higher degree polynomials, i.e., g > 1. For this setup, we construct two novel PPC schemes. In the first scheme we consider Reed-Solomon coded databases with Lagrange encoding, which leverages ideas from recently proposed star-product PIR and Lagrange coded computation. The second scheme considers the special case of coded databases with systematic Lagrange encoding. Both schemes yield improved rates compared to the best known schemes from the literature for a small number of messages, while asymptotically, as f → ∞, the systematic scheme gives a significantly better computation rate compared to all known schemes up to some storage code rate that depends on the maximum degree of the candidate polynomials.

Abstract-We study the problem of strong coordination of actions of two agents X and Y that communicate over a noisy communication channel such that the actions follow a given joint probability distribution. We propose two novel schemes for this noisy strong coordination problem, and derive inner bounds for the underlying strong coordination capacity region. The first scheme is a joint coordination-channel coding scheme that utilizes the randomness provided by the communication channel to reduce the local randomness required in generating the action sequence at agent Y . The second scheme exploits separate coordination and channel coding where local randomness is extracted from the channel after decoding. Finally, we present an example in which the joint scheme is able to outperform the separate scheme in terms of coordination rate. I. INTRODUCTIONThe problem of communication-based coordination of multi-agent systems arises in numerous applications including mobile robotic networks, smart traffic control, and distributed computing such as distributed games and grid computing [1]. Several theoretical and applied studies on multi-agent coordination have targeted questions on how agents exchange information and how their actions can be correlated to achieve a desired overall behavior. Two types of coordination have been addressed in the literature -empirical coordination where the histogram of induced actions is required to be close to a prescribed target distribution, and strong coordination, where the induced sequence of joint actions of all the agents is required to be statistically close (i.e., nearly indistinguishable) from a chosen target probability mass function (pmf).Recently, the capacity regions of several empirical and strong coordination network problems have been established [1]- [6]. Bounds for the capacity region for the point-to-point case were obtained in [7] under the assumption that the nodes communicate in a bidirectional fashion in order to achieve coordination. A similar framework was adopted and improved in [8]. In [4], [6], [9], the authors addressed inner and outer bounds for the capacity region of a three-terminal network in the presence of a relay. The work of [4] was later extended in [5], [10] to derive a precise characterization of the strong coordination region for multi-hop networks. Starkly, the majority of the recent works on coordination have considered noisefree communication channels with the exception of two works:

We construct a joint coordination-channel polar coding scheme for strong coordination of actions between two agents X and Y, which communicate over a discrete memoryless channel (DMC) such that the joint distribution of actions follows a prescribed probability distribution. We show that polar codes are able to achieve our previously established inner bound to the strong noisy coordination capacity region and thus provide a constructive alternative to a random coding proof. Our polar coding scheme also offers a constructive solution to a channel simulation problem where a DMC and shared randomness are together employed to simulate another DMC. In particular, our proposed solution is able to utilize the randomness of the DMC to reduce the amount of local randomness required to generate the sequence of actions at agent Y. By leveraging our earlier random coding results for this problem, we conclude that the proposed joint coordinationchannel coding scheme strictly outperforms a separate scheme in terms of achievable communication rate for the same amount of injected randomness into both systems.