Linear Exact Repair Schemes for Distributed Storage and Secure Distributed Matrix Multiplication

In this thesis we develop exact repair schemes capable of repairing or circumventing unavailable servers of a distributed network in the context of distributed storage and secure distributed matrix multiplication. We develop the (Λ, Γ, W, ⊙)-exact repair scheme framework for discussing both of these contexts and develop a multitude of explicit exact repair schemes utilizing decreasing monomial-Cartesian codes (DMC codes). Specifically, we construct novel DMC codes in the form of augmented Cartesian codes and rectangular monomial-Cartesian codes, as well as design exact repair schemes utilizing these constructions inspired by the schemes from Guruswami and Wootters [16] and Chen and Zhang [6]. In the context of distributed storage we demonstrate the existence of both high rate and low bandwidth systems based on these schemes, and we develop two methods to extend them to the l-erasure case. Additionally, we develop a family of hybrid schemes capable of attaining high rates, low bandwidths, and a balance in between which proves to be competitive compared to existing schemes. In the context of secure distributed matrix multiplication we develop similarly impactful schemes which have very competitive communication costs. We also construct an encoding algorithm based on multivariate interpolation and prove it is T-secure.