X–secure and T–private information retrieval (XSTPIR) is a form of private information retrieval where data security is guaranteed against collusion among up to X servers and the user’s privacy is guaranteed against collusion among up to T servers. The capacity of XSTPIR is characterized for arbitrary number of servers N, and arbitrary security and privacy thresholds X and T, in the limit as the number of messages K → ∞. Capacity is also characterized for any number of messages if either N = 3, X = T = 1 or if N ≤ X+T. Insights are drawn from these results, about aligning versus decoding noise, dependence of PIR rate on field size, and robustness to symmetric security constraints. In particular, the idea of cross subspace alignment, i.e., introducing a subspace dependence between Reed-Solomon code parameters, emerges as the optimal way to align undesired terms while keeping desired terms resolvable.

The problem of secure distributed batch matrix multiplication (SDBMM) studies the communication efficiency of retrieving a sequence of desired matrix products ${\mathbf{AB}} = ({\mathbf{A}}_{1}{\mathbf{B}}_{1},\,\,{\mathbf{A}}_{2}{\mathbf{B}}_{2},\,\,\cdots,\,\,{\mathbf{A}}_{S}{\mathbf{B}}_{S})$ from $N$ distributed servers where the constituent matrices ${\mathbf{A}}=({\mathbf{A}}_{1}, {\mathbf{A}}_{2}, \cdots, {\mathbf{A}}_{S})$ and ${\mathbf{B}}=({\mathbf{B}}_{1}, {\mathbf{B}}_{2},\cdots,{\mathbf{B}}_{S})$ are stored in $X$ -secure coded form, i.e., any group of up to $X$ colluding servers learn nothing about $\mathbf{ A, B}$ . It is assumed that ${\mathbf{A}}_{s}\in \mathbb {F}_{q}^{L\times K}, {\mathbf{B}}_{s}\in \mathbb {F}_{q}^{K\times M}, s\in \{1,2,\cdots, S\}$ are uniformly and independently distributed and $\mathbb {F}_{q}$ is a large finite field. The rate of an SDBMM scheme is defined as the ratio of the number of bits of desired information that is retrieved, to the total number of bits downloaded on average. The supremum of achievable rates is called the capacity of SDBMM. In this work we explore the capacity of SDBMM, as well as several of its variants, e.g., where the user may already have either ${\mathbf{A}}$ or ${\mathbf{B}}$ available as side-information, and/or where the security constraint for either ${\mathbf{A}}$ or ${\mathbf{B}}$ may be relaxed. We obtain converse bounds, as well as achievable schemes for various cases of SDBMM, depending on the $L, K, M, N, X$ parameters, and identify parameter regimes where these bounds match. In particular, the capacity for securely computing a batch of outer products of two vectors is $(1-X/N)^{+}$ , for a batch of inner products of two (long) vectors the capacity approaches $(1-2X/N)^{+}$ as the length of the vectors approaches infinity, and in general for sufficiently large $K$ (e.g., $K > 2\min (L,M)$ ), the capacity $C$ is bounded as $(1-2X/N)^{+}\leq C < (1-X/N)^{+}$ . A remarkable aspect of our upper bounds is a connection between SDBMM and a form of private information retrieval (PIR) problem, known as multi-message $X$ -secure $T$ -private information retrieval (MM-XSTPIR). Notable features of our achievable schemes include the use of cross-subspace alignment and a transformation argument that converts a scalar multiplication problem into a scalar addition problem, allowing a surprisingly efficient solution.

The problem of X-secure T-private information retrieval from MDS coded storage is studied in this paper, where the user wishes to privately retrieve one out of K independent messages that are distributed over N servers according to an MDS code. It is guaranteed that any group of up to X colluding servers learn nothing about the messages and that any group of up to T colluding servers learn nothing about the identity of desired message. A lower bound of achievable rates is proved by presenting a novel scheme based on cross-subspace alignment and a successive decoding with interference cancellation strategy. For large number of messages (K→∞) the achieved rate, which we conjecture to be optimal, improves upon the best known rates previously reported in the literature by Raviv and Karpuk, and generalizes an achievable rate for MDS-TPIR previously found by Freij-Hollanti et al. that is also conjectured to be asymptotically optimal. The setting is then expanded to allow unresponsive and Byzantine servers. Finally, the scheme is applied to find a new lower convex hull of (download, upload) pairs of secure and private distributed matrix multiplication that generalizes, and in certain asymptotic settings strictly improves upon the best known previous results.

The goal of coded distributed computation is to efficiently distribute a computation task, such as matrix multiplication, N-linear computation, or multivariate polynomial evaluation, across S servers through a coding scheme, such that the response from any R servers ( R is called the recovery threshold) is sufficient for the user to recover the desired computed value. Current state-of-art approaches are based on either exclusively matrix-partitioning (Entangled Polynomial (EP) Codes for matrix multiplication), or exclusively batch processing (Lagrange Coded Computing (LCC) for N-linear computations or multivariate polynomial evaluations). We present three related classes of codes, based on the idea of Cross-Subspace Alignment (CSA) which was introduced originally in the context of secure and private information retrieval. CSA codes are characterized by a Cauchy-Vandermonde matrix structure that facilitates interference alignment along Vandermonde terms, while the desired computations remain resolvable along the Cauchy terms. These codes are shown to unify, generalize and improve upon the state-of-art codes for distributed computing. First we introduce CSA codes for matrix multiplication, which yield LCC codes as a special case, and are shown to outperform LCC codes in general in download-limited settings. While matrix-partitioning approaches (EP codes) for distributed matrix multiplication have the advantage of flexible server computation latency, batch processing approaches (CSA, LCC) have significant advantages in communication costs as well as encoding and decoding complexity per matrix multiplication. In order to combine the benefits of these approaches, we introduce Generalized CSA (GCSA) codes for matrix multiplication that bridge the extremes of matrix-partitioning and batch processing approaches and demonstrate synergistic gains due to cross subspace alignment. Finally, we introduce N-CSA codes for N-linear distributed batch computations and multivariate batch polynomial evaluations. N-CSA codes include LCC codes as a special case, and are in general capable of outperforming LCC codes in download-constrained settings by upto a factor of N. Generalizations of N-CSA codes to include X-secure data and B-byzantine servers are also provided.

A secure multi-party batch matrix multiplication problem (SMBMM) is considered, where the goal is to allow a master to efficiently compute the pairwise products of two batches of massive matrices, by distributing the computation across S servers. Any X colluding servers gain no information about the input, and the master gains no additional information about the input beyond the product. A solution called Generalized Cross Subspace Alignment codes with Noise Alignment (GCSA- NA) is proposed in this work, based on cross-subspace alignment codes. The state of art solution to SMBMM is a coding scheme called polynomial sharing (PS) that was proposed by Nodehi and Maddah-Ali. GCSA-NA outperforms PS codes in several key aspects-more efficient and secure inter-server communication, lower latency, flexible inter-server network topology, efficient batch processing, and tolerance to stragglers. The idea of noise alignment can also be combined with N-source Cross Subspace Alignment (N-CSA) codes and fast matrix multiplication algorithms like Strassen's construction. Moreover, noise alignment can be applied to symmetric secure private information retrieval to achieve the asymptotic capacity.

The problem of private information retrieval with graph-based replicated storage was recently introduced by Raviv, Tamo and Yaakobi. Its capacity remains open in almost all cases. In this work the asymptotic (large number of messages) capacity of this problem is studied along with its generalizations to include arbitrary $T$ -privacy and $X$ -security constraints, where the privacy of the user must be protected against any set of up to $T$ colluding servers and the security of the stored data must be protected against any set of up to $X$ colluding servers. A general achievable scheme for arbitrary storage patterns is presented that achieves the rate $(\rho _{\min }-X-T)/N$ , where $N$ is the total number of servers, and each message is replicated at least $\rho _{\min }$ times. Notably, the scheme makes use of a special structure inspired by dual Generalized Reed Solomon (GRS) codes. A general converse is also presented. The two bounds are shown to match for many settings, including symmetric storage patterns. Finally, the asymptotic capacity is fully characterized for the case without security constraints $(X=0)$ for arbitrary storage patterns provided that each message is replicated no more than $T+2$ times. As an example of this result, consider PIR with arbitrary graph based storage ( $T=1, X=0$ ) where every message is replicated at exactly 3 servers. For this 3-replicated storage setting, the asymptotic capacity is equal to $2/\nu _{2}(G)$ where $\nu _{2}(G)$ is the maximum size of a 2-matching in a storage graph $G[V,E]$ . In this undirected graph, the vertices $V$ correspond to the set of servers, and there is an edge $uv\in E$ between vertices $u,v$ only if a subset of messages is replicated at both servers $u$ and $v$ .