Algebraic Geometry Codes for Secure Distributed Matrix Multiplication

In this paper, we propose a novel construction for secure distributed matrix multiplication (SDMM) based on algebraic geometry (AG) codes. The proposed construction is inspired by the GASP code, where so-called gaps in a certain polynomial are utilized to achieve higher communication rates. Our construction considers the gaps in a Weierstrass semigroup of a rational place in an algebraic function field to achieve a similar increase in the rate. This construction shows that there is potential in utilizing AG codes and their subcodes in SDMM since we demonstrate a better performance compared to state-of-the-art schemes in some parameter regimes.Comment: 14 pages, 1 figur

Efficient Recovery of a Shared Secret via Cooperation: Applications to SDMM and PIR

This work considers the problem of privately outsourcing the computation of a matrix product over a finite field $\mathbb{F}_q$ to $N$ helper servers. These servers are considered to be honest but curious, i.e., they behave according to the protocol but will try to deduce information about the user's data. Furthermore, any set of up to $X$ servers is allowed to share their data. Previous works considered this collusion a hindrance and the download cost of the schemes increases with growing $X$. We propose to utilize such linkage between servers to the user's advantage by allowing servers to cooperate in the computational task. This leads to a significant gain in the download cost for the proposed schemes. The gain naturally comes at the cost of increased communication load between the servers. Hence, the proposed cooperative scheme can be understood as outsourcing both computational cost and communication cost. While the present work exemplifies the proposed server cooperation in the case of a specific secure distributed matrix multiplication (SDMM) scheme, the same idea applies to many other use cases as well. For instance, other SDMM schemes as well as linear private information retrieval (PIR) as a special case of SDMM are instantly covered.Comment: 10 pages, 2 figure