Systematic Derivation of Leakage in Massey Secret Sharing Schemes

Leakage attacks on secret sharing schemes can be based on the exploitation of partial information about individual shares to recover the underlying secret. In coding theory, linear exact repair schemes (LERSs) allow the recovery of a codeword symbol from limited information obtained from the remaining symbols. When applied to code-based secret sharing schemes, such repair schemes can be interpreted as recovering the secret from partial information about the shares, expressed in terms of subfield symbols. Guruswami and Wootters introduced the first explicit constructions of LERSs for low-rate Reed--Solomon codes that is optimal in terms of bandwidth. In this work, we first investigate the construction and algebraic properties of LERSs, building on the Guruswami-Wootters construction. We derive new explicit repair schemes from their construction and establish structural properties of the resulting schemes. Second, we further develop a systematic computational approach based on a Gr{ö}bner bases approach to identify leakage functions associated with LERSs for arbitrary linear codes. This approach enables us to investigate the existence and structural properties of LERSs across different code families and parameter ranges. We first applied our method to Reed-Solomon codes, which gave as further insights into the algebraic structure of LERSs. Moreover, by applying our method to generalized Reed-Muller and Tamo-Barg codes, we identify several LERSs that, to the best of our knowledge, have not previously been described in the literature.