Private Computation of Systematically Encoded Data with Colluding Servers

Private Computation (PC), recently introduced by Sun and Jafar, is a generalization of Private Information Retrieval (PIR) in which a user wishes to privately compute an arbitrary function of data stored across several servers. We construct a PC scheme which accounts for server collusion, coded data, and non-linear functions. For data replicated over several possibly colluding servers, our scheme computes arbitrary functions of the data with rate equal to the asymptotic capacity of PIR for this setup. For systematically encoded data stored over colluding servers, we privately compute arbitrary functions of the columns of the data matrix and calculate the rate explicitly for polynomial functions. The scheme is a generalization of previously studied star-product PIR schemes.Comment: Submitted to IEEE International Symposium on Information Theory 2018. Version 2 fixes some typos and adds some clarifying remark

Hybrid Channel Pre-Inversion and Interference Alignment Strategies

In this paper we consider strategies for MIMO interference channels which combine the notions of interference alignment and channel pre-inversion. Users collaborate to form data-sharing groups, enabling them to clear interference within a group, while interference alignment is employed to clear interference between groups. To improve the capacity of our schemes at finite SNR, we propose that the groups of users invert their subchannel using a regularized Tikhonov inverse. We provide a new sleeker derivation of the optimal Tikhonov parameter, and use random matrix theory to provide an explicit formula for the SINR as the size of the system increases, which we believe is a new result. For every possible grouping of K = 4 users each with N = 5 antennas, we completely classify the degrees of freedom available to each user when using such hybrid schemes, and construct explicit interference alignment strategies which maximize the sum DoF. Lastly, we provide simulation results which compute the ergodic capacity of such schemes.Comment: Submitted to ICC 201

Node Repair for Distributed Storage Systems over Fading Channels

Distributed storage systems and associated storage codes can efficiently store a large amount of data while ensuring that data is retrievable in case of node failure. The study of such systems, particularly the design of storage codes over finite fields, assumes that the physical channel through which the nodes communicate is error-free. This is not always the case, for example, in a wireless storage system. We study the probability that a subpacket is repaired incorrectly during node repair in a distributed storage system, in which the nodes communicate over an AWGN or Rayleigh fading channels. The asymptotic probability (as SNR increases) that a node is repaired incorrectly is shown to be completely determined by the repair locality of the DSS and the symbol error rate of the wireless channel. Lastly, we propose some design criteria for physical layer coding in this scenario, and use it to compute optimally rotated QAM constellations for use in wireless distributed storage systems.Comment: To appear in ISITA 201

Weil-etale Cohomology over Local Fields

In a recent article, Lichtenbaum established the arithmetic utility of the Weil group of a finite field, by demonstrating a connection between certain Euler characteristics in Weil-etale cohomology and special values of zeta functions. In particular, the order of vanishing and leading coefficient of the zeta function of a smooth, projective variety over a finite field have a Weil-etale cohomological interpretation. These results rely on a duality theorem stated in terms of cup-product in Weil-etale cohomology. With Lichtenbaum's paradigm in mind, we establish results for the cohomology of the Weil group of a local field, analogous to, but more general than, results from Galois cohomology. We prove a duality theorem for discrete Weil group modules, which implies the main theorem of Local Class Field Theory. We define Weil- smooth cohomology for varieties over local fields, and prove a duality theorem for the cohomology of G_m on a smooth, proper curve with a rational point. This last theorem is analogous to, and implies, a classical duality theorem for such curves