Single-Server Multi-Message Private Information Retrieval with Side Information

We study the problem of single-server multi-message private information retrieval with side information. One user wants to recover $N$ out of $K$ independent messages which are stored at a single server. The user initially possesses a subset of $M$ messages as side information. The goal of the user is to download the $N$ demand messages while not leaking any information about the indices of these messages to the server. In this paper, we characterize the minimum number of required transmissions. We also present the optimal linear coding scheme which enables the user to download the demand messages and preserves the privacy of their indices. Moreover, we show that the trivial MDS coding scheme with $K-M$ transmissions is optimal if $N>M$ or $N^2+N \ge K-M$. This means if one wishes to privately download more than the square-root of the number of files in the database, then one must effectively download the full database (minus the side information), irrespective of the amount of side information one has available.Comment: 12 pages, submitted to the 56th Allerton conferenc

Higher Spin Chern-Simons Theory and the Super Boussinesq hierarchy

In this paper we construct a map between a solution of supersymmetric Chern-Simons higher spin gravity based on the superalgebra $sl(3|2)$ with Lifshitz scaling and the $N=2$ super Boussinesq hierarchy. We show that under this map the time evolution equations of both theories coincide. In addition, we identify the Poisson structure of the Chern-Simons theory induced by gauge transformation with the second Hamiltonian structure of the super Boussinesq hierarchy.Comment: 22 pages, no figure

Uniquely-Wilf classes

Two permutations in a class are Wilf-equivalent if, for every size, $n$, the number of permutations in the class of size $n$ containing each of them is the same. Those infinite classes that have only one equivalence class in each size for this relation are characterised provided either that they avoid at least one permutation of size 3, or at least three permutations of size 4.Comment: Updated to DMTCS styl

Cooperative Data Exchange based on MDS Codes

The cooperative data exchange problem is studied for the fully connected network. In this problem, each node initially only possesses a subset of the $K$ packets making up the file. Nodes make broadcast transmissions that are received by all other nodes. The goal is for each node to recover the full file. In this paper, we present a polynomial-time deterministic algorithm to compute the optimal (i.e., minimal) number of required broadcast transmissions and to determine the precise transmissions to be made by the nodes. A particular feature of our approach is that {\it each} of the $K-d$ transmissions is a linear combination of {\it exactly} $d+1$ packets, and we show how to optimally choose the value of $d.$ We also show how the coefficients of these linear combinations can be chosen by leveraging a connection to Maximum Distance Separable (MDS) codes. Moreover, we show that our method can be used to solve cooperative data exchange problems with weighted cost as well as the so-called successive local omniscience problem.Comment: 21 pages, 1 figur

On a Conjecture of EM Stein on the Hilbert Transform on Vector Fields

Let $v$ be a smooth vector field on the plane, that is a map from the plane to the unit circle. We study sufficient conditions for the boundedness of the Hilbert transform \operatorname H_{v, \epsilon}f(x) := \text{p.v.}\int_{-\epsilon}^ \epsilon f(x-yv(x)) \frac{dy}y where $\epsilon$ is a suitably chosen parameter, determined by the smoothness properties of the vector field. It is a conjecture, due to E.\thinspace M.\thinspace Stein, that if $v$ is Lipschitz, there is a positive $\epsilon$ for which the transform above is bounded on $L ^{2}$. Our principal result gives a sufficient condition in terms of the boundedness of a maximal function associated to $v$. This sufficient condition is that this new maximal function be bounded on some $L ^{p}$, for some $1<p<2$. We show that the maximal function is bounded from $L ^{2}$ to weak $L ^{2}$ for all Lipschitz maximal function. The relationship between our results and other known sufficient conditions is explored.Comment: 92 pages, 20+ figures. Final version of the paper. To appear in Memoirs AM