We consider the problem of T -Private Information Retrieval with private side information (TPIR-PSI). In this problem, N replicated databases store K independent messages, and a user, equipped with a local cache that holds M messages as side information, wishes to retrieve one of the other K − M messages. The desired message index and the side information must remain jointly private even if any T of the N databases collude. We show that the capacityAs a special case obtained by setting T = 1, this result settles the capacity of PIR-PSI, an open problem previously noted by Kadhe et al. We also consider the problem of symmetric-TPIR with private side information (STPIR-PSI), where the answers from all N databases reveal no information about any other message besides the desired message. We show that the capacity of STPIR-PSI is 1 − T N if the databases have access to common randomness (not available to the user) that is independent of the messages, in an amount that is at least T N −T bits per desired message bit. Otherwise, the capacity of STPIR-PSI is zero. , W1, W2, Θ = 2, S = {1}) − I(W3; Q [Θ,S] 2 | Q [Θ,S] 1 , W1, W2, A [Θ,S] 1, Θ = 2, S = {1}). The first two mutual information terms on the right hand side are equal to zero because of (5) and (4), respectively, leaving only the negative mutual information term which must also be zero because mutual information cannot be negative.

A secure multi-party batch matrix multiplication problem (SMBMM) is considered, where the goal is to allow a master node to efficiently compute the pairwise products of two batches of massive matrices that originate at external source nodes, by distributing the computation across S honest but curious servers. Any group of up to X colluding servers should gain no information about the input matrices, and the master should gain no additional information about the input matrices beyond the product. A solution called Generalized Cross Subspace Alignment codes with Noise Alignment (GCSA-NA in short) is proposed in this work, based on cross-subspace alignment codes. These codes originated in secure private information retrieval, and have recently been applied to distributed batch computation problems where they generalize and improve upon the state of art schemes such as Entangled Polynomial Codes and Lagrange Coded Computing. The prior 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 (which can entirely take place beforehand, i.e., even before the input matrices are determined), flexible inter-server network topology, efficient batch processing, and tolerance to stragglers. The idea of noise-alignment can also be applied to construct schemes for N sources based on N -CSA codes, and to construct schemes for symmetric secure private information retrieval to achieve the asymptotic capacity.other server. This carries a high communication cost and requires the network topology among servers to be a complete graph (otherwise data security would be compromised), does not tolerate stragglers, and does not lend itself to batch processing. These aspects (batch processing, improved inter-server communication efficiency, various network topologies) are highlighted as open problems by Nodehi et al. in [40].Since GCSA codes are particularly efficient at batch processing and already encompass prior approaches to coded distributed computing, in this work we explore whether GCSA codes can also be applied to the problems identified by Nodehi et al. In particular, we will focus on the problem of multiplication of two matrices. As it turns out, in this context the answer is in the affirmative. Securing the data against any X colluding servers is already possible with GCSA codes as shown in [30]. The only remaining challenge is how to prevent the master node from learning anything about the inputs besides the result of the computation. Let us refer to the additional terms that are contained in the answers sent by the servers to the master node, which may collectively reveal information about the inputs beyond the result of the computation, as interference terms. To secure these interference terms, we use the idea of Noise Alignment (NA) -the workers communicate among themselves to share noise terms (unknown to the master node) that are structured in the same m...

The private search problem is introduced, where a dataset comprised of L i.i.d. records is replicated across N non-colluding servers, each record takes values uniformly from an alphabet of size K, and a user wishes to search for all records that match a privately chosen value, without revealing any information about the chosen value to any individual server. The capacity of private search is the maximum number of bits of desired information that can be retrieved per bit of download. The asymptotic (large K) capacity of private search is shown to be 1 − 1/N , even as the scope of private search is further generalized to allow approximate (OR) search over a number of realizations that grows with K. The results are based on the asymptotic behavior of a new converse bound for private information retrieval with arbitrarily dependent messages.

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The distributed matrix multiplication problem with an unknown number of stragglers is considered, where the goal is to efficiently and flexibly obtain the product of two massive matrices by distributing the computation across N servers. There are up to N − R stragglers but the exact number is not known a priori. Motivated by reducing the computation load of each server, a flexible solution is proposed to fully utilize the computation capability of available servers. The computing task for each server is separated into several subtasks, constructed based on Entangled Polynomial codes by Yu et al. The final results can be obtained from either a larger number of servers with a smaller amount of computation completed per server or a smaller number of servers with a larger amount of computation completed per server. The required finite field size of the proposed solution is less than 2N . Moreover, the optimal design parameters such as the partitioning of the input matrices are discussed. Our constructions can also be generalized to other settings such as batch distributed matrix multiplication and secure distributed matrix multiplication.