In this paper, we propose a novel construction for secure distributed matrix multiplication (SDMM) based on algebraic geometry (AG) codes, which we call the PoleGap SDMM scheme. The proposed construction is inspired by the Gap Additive Secure Polynomial (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.

Abstract A perfect matroid design (PMD) is a matroid whose flats of the same rank all have the same size. In this paper we introduce the q -analogue of a PMD and its properties. In order to do so, we first establish a new cryptomorphic definition for q -matroids. We show that q -Steiner systems are examples of q -PMD’s and we use this q -matroid structure to construct subspace designs from q -Steiner systems. We apply this construction to the only known q -Steiner system, which has parameters S (2, 3, 13; 2), and hence establish the existence of a new subspace design with parameters 2-(13, 4, 5115; 2).