Improved Polynomial Secret-Sharing Schemes

Despite active research on secret-sharing schemes for arbitrary access structures for more than 35 years, we do not understand their share size – the best known upper bound for an arbitrary n-party access structure is 2^{O ( n )}, while the best known lower bound is Ω(n/ log (n) ). Consistent with our knowledge, the share size can be anywhere between these bounds. To better understand this question, one can study specific families of secret-sharing schemes. For example, linear secret-sharing schemes, in which the sharing and reconstruction functions are linear mappings, have been studied in many papers, e.g., it is known that they require shares of size at least 2^{0.5 n}. Secret-sharing schemes in which the sharing and/or reconstruction are computed by low-degree polynomials have been recently studied by Paskin-Cherniavsky and Radune [ITC 2020] and by Beimel, Othman, and Peter [CRYPTO 2021]. It was shown that secret-sharing schemes with sharing and reconstruction computed by polynomials of degree 2 are more efficient than linear schemes (i.e., schemes in which the sharing and reconstruction are computed by polynomials of degree one). Prior to our work, it was not known if using polynomials of higher degree can reduce the share size. We show that this is indeed the case, i.e., we construct secret-sharing schemes for arbitrary access structures with reconstruction by degree-d polynomials, where as the reconstruction degree d increases, the share size decreases. As a step in our construction, we construct conditional disclosure of secrets (CDS) protocols. For example, we construct 2-server CDS protocols for functions f: [ N] × [ N] → { 0, 1 } with reconstruction computed by degree-d polynomials with message size N^{O ( log log d / log d )}. Combining our results with a lower bound of Beimel et al. [CRYPTO 2021], we show that increasing the degree of the reconstruction function in CDS protocols provably reduces the message size. To construct our schemes, we define sparse matching vectors, show constructions of such vectors, and design CDS protocols and secret-sharing schemes with degree-d reconstruction from sparse matching vectors.