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Gradient coding is a coding theoretic framework to provide robustness against slow or unresponsive machines, known as stragglers, in distributed machine learning applications. Recently, Kadhe et al. (2019) proposed a gradient code based on a combinatorial design, called balanced incomplete block design (BIBD), which is shown to outperform many existing gradient codes in worst-case adversarial straggling scenarios. However, parameters for which such BIBD constructions exist are very limited (Colbourn and Dinitz, 2006).

Coded caching aims to minimize the network’s peak-time communication load by leveraging the information pre-stored in the local caches at the users. The original setting by Maddah-Ali and Niesen, which considered single file retrieval, has been recently extended to general Scalar Linear Function Retrieval (SLFR) by Wan et al., who proposed a linear scheme that surprisingly achieves the same optimal load under the constraint of uncoded cache placement as in single file retrieval.

In distributed storage systems, cooperative regenerating codes tradeoff storage for repair bandwidth in the case of multiple node failures. In rack-aware distributed storage systems, there is no cost associated with transferring symbols within a rack. Hence, the repair bandwidth will only take into account cross-rack transfer. Rack-aware regenerating codes for the case of single node failures have been studied and their repair bandwidth tradeoff characterized.

We consider the problem of designing secure and private codes for distributed matrix-matrix multiplication. A master server owns two private matrices and hires worker nodes to help compute their product. The matrices should remain information-theoretically private from the workers. Some of the workers are malicious and return corrupted results to the master. We design a framework for security against malicious workers in private matrix-matrix multiplication. The main idea is a careful use of Freivalds’ algorithm to detect erroneous matrix multiplications.

In the conventional robust $T$ -colluding private information retrieval (PIR) system, the user needs to retrieve one of the possible messages while keeping the identity of the requested message private from any $T$ colluding servers. Motivated by the possible heterogeneous privacy requirements for different messages, we consider the $(N, T_{1}~:~K_{1}, T_{2}~:~K_{2})$ two-level PIR system with a total of $K_{2}$ messages in the system, where $T_{1}\geq T_{2}$ and $K_{1}\leq K_{2}$ .

The compound secure groupcast problem is considered, where the key variables at $K$ receivers are designed so that a transmitter can securely groupcast a message to any $N$ out of the $K$ receivers through a noiseless broadcast channel. The metric is the information theoretic tradeoff between key storage $\alpha $ , i.e., the number of bits of the key variable stored at each receiver per message bit, and broadcast bandwidth $\beta $ , i.e., the number of bits of the broadcast information sent by the transmitter per message bit. We have three main results.

We consider the problems of Private and Secure Matrix Multiplication (PSMM) and Fully Private Matrix Multiplication (FPMM), for which matrices privately selected by a master node are multiplied at distributed worker nodes without revealing the indices of the selected matrices, even when a certain number of workers collude with each other.

A majority of coded matrix-matrix computation literature has broadly focused in two directions: matrix partitioning for computing a single computation task and batch processing of multiple distinct computation tasks. While these works provide codes with good straggler resilience and fast decoding for their problem spaces, these codes would not be able to take advantage of the natural redundancy of re-using matrices across batch jobs.

Although blockchain, the supporting technology of various cryptocurrencies, has offered a potentially effective framework for numerous decentralized trust management systems, its performance is still sub-optimal in real-world networks. With limited bandwidth, the communication complexity for nodes to process a block scales with the growing network size and hence becomes the limiting factor of blockchain’s performance. In this paper, we suggest a re-design of existing blockchain systems, which addresses the issue of the communication burden.

We consider over-the-air convex optimization on a $d$ dimensional space where coded gradients are sent over an additive Gaussian noise channel with variance $\sigma ^{2}$ . The codewords satisfy an average power constraint $P$ , resulting in the signal-to-noise ratio (SNR) of $P/\sigma ^{2}$ . We derive bounds for the convergence rates for over-the-air optimization. Our first result is a lower bound for the convergence rate showing that any code must slowdown the convergence rate by a factor of roughly $\sqrt {d/\log (1+ \mathtt {SNR})}$ .