On polynomial secret sharing schemes

Nearly all secret sharing schemes studied so far are linear or multi-linear schemes. Although these schemes allow to implement any monotone access structure, the share complexity, SC, may be suboptimal – there are access structures for which the gap between the best known lower bounds and best known multi-linear schemes is exponential. There is growing evidence in the literature, that non-linear schemes can improve share complexity for some access structures, with the work of Beimel and Ishai (CCC’01) being among the first to demonstrate it. This motivates further study of non linear schemes. We initiate a systematic study of polynomial secret sharing schemes (PSSS), where shares are (multi-variate) polynomials of secret and randomness vectors ~s,~r respectively over some finite field Fq. Our main hope is that the algebraic structure of polynomials would help obtain better lower bounds than those known for the general secret sharing. Some of the initial results we prove in this work are as follows.