We propose three private information retrieval (PIR) protocols for distributed storage systems (DSSs) where data is stored using an arbitrary linear code. The first two protocols, named Protocol 1 and Protocol 2, achieve privacy for the scenario with noncolluding nodes. Protocol 1 requires a file size that is exponential in the number of files in the system, while Protocol 2 requires a file size that is independent of the number of files and is hence simpler. We prove that, for certain linear codes, Protocol 1 achieves the maximum distance separable (MDS) PIR capacity, i.e., the maximum PIR rate (the ratio of the amount of retrieved stored data per unit of downloaded data) for a DSS that uses an MDS code to store any given (finite and infinite) number of files, and Protocol 2 achieves the asymptotic MDS-PIR capacity (with infinitely large number of files in the DSS). In particular, we provide a necessary and a sufficient condition for a code to achieve the MDS-PIR capacity with Protocols 1 and 2 and prove that cyclic codes, Reed-Muller (RM) codes, and a class of distance-optimal local reconstruction codes achieve both the finite MDS-PIR capacity (i.e., with any given number of files) and the asymptotic MDS-PIR capacity with Protocols 1 and 2, respectively. Furthermore, we present a third protocol, Protocol 3, for the scenario with multiple colluding nodes, which can be seen as an improvement of a protocol recently introduced by Freij-Hollanti et al.. Similar to the noncolluding case, we provide a necessary and a sufficient condition to achieve the maximum possible PIR rate of Protocol 3. Moreover, we provide a particular class of codes that is suitable for this protocol and show that RM codes achieve the maximum possible PIR rate for the protocol. For all three protocols, we present an algorithm to optimize their PIR rates.

We propose an information-theoretic private information retrieval (PIR) scheme for distributed storage systems where data is stored using a linear systematic code of rate R > 1/2. The proposed scheme generalizes the PIR scheme for data stored using maximum distance separable codes recently proposed by Tajeddine and El Rouayheb for the scenario of a single spy node. We further propose an algorithm to optimize the communication price of privacy (cPoP) using the structure of the underlying linear code. As an example, we apply the proposed algorithm to several distributed storage codes, showing that the cPoP can be significantly reduced by exploiting the structure of the distributed storage code.

Synergistic-cooperative combination of enamine catalysis with transition metal catalysis is an emerging and exciting field aiming to achieve organic transformations that cannot be accomplished by individual catalysis. The biggest obstacle in this field lies in the catalyst incompatibility arising from Lewis acid-Lewis base interactions. Several strategies including soft/hard combination of a Lewis acid and a Lewis base, the utilization of a chelating ligand, and mixing an ammonium salt with a Lewis acid have been developed to solve the incompatibility problem. A number of new reactions and new reaction modes have been discovered using these strategies. In this review article, we aim to highlight these new discoveries found in the literature.

We propose a private information retrieval (PIR) protocol for distributed storage systems with noncolluding nodes where data is stored using an arbitrary linear code. An expression for the PIR rate, i.e., the ratio of the amount of retrieved data per unit of downloaded data, is derived, and a necessary and a sufficient condition for codes to achieve the maximum distance separable (MDS) PIR capacity are given. The necessary condition is based on the generalized Hamming weights of the storage code, while the sufficient condition is based on code automorphisms. We show that cyclic codes and Reed-Muller codes satisfy the sufficient condition and are thus MDS-PIR capacity-achieving.

Private information retrieval (PIR) protocols make it possible to retrieve a file from a database without disclosing any information about the identity of the file being retrieved. These protocols have been rigorously explored from an informationtheoretic perspective in recent years. While existing protocols strictly impose that no information is leaked on the file's identity, this work initiates the study of the tradeoffs that can be achieved by relaxing the requirement of perfect privacy. In case the user is willing to leak some information on the identity of the retrieved file, we study how the PIR rate, as well as the upload cost and access complexity, can be improved. For the particular case of replicated servers, we propose two weakly-private information retrieval schemes based on two recent PIR protocols and a family of schemes based on partitioning. Lastly, we compare the performance of the proposed schemes.