On the power of nonlinear secret-sharing

A secret-sharing scheme enables a dealer to distribute a secret among n parties such that only some predefined authorized sets of parties will be able to reconstruct the secret from their shares. The ( monotone) collection of authorized sets is called an access structure, and is freely identified with its characteristic monotone function f : {0, 1} n. {0, 1}. A family of secret-sharing schemes is called efficient if the total length of the n shares is polynomial in n. Most previously known secret-sharing schemes belonged to a class of linear schemes, whose complexity coincides with the monotone span program size of their access structure. Prior to this work there was no evidence that nonlinear schemes can be significantly more efficient than linear schemes, and in particular there were no candidates for schemes efficiently realizing access structures which do not lie in NC. The main contribution of this work is the construction of two efficient nonlinear schemes: ( 1) A scheme with perfect privacy whose access structure is conjectured not to lie in NC, and ( 2) a scheme with statistical privacy whose access structure is conjectured not to lie in P/poly. Another contribution is the study of a class of nonlinear schemes, termed quasi-linear schemes, obtained by composing linear schemes over different fields. While these schemes are (superpolynomially) more powerful than linear schemes, we show that they cannot efficiently realize access structures outside NC.