Secret Sharing, Slice Formulas, and Monotone Real Circuits

A secret-sharing scheme allows to distribute a secret s among n parties such that only some pre-defined “authorized” sets of parties can reconstruct the secret, and all other “unauthorized” sets learn nothing about s . For over 30 years, it was known that any (monotone) collection of authorized sets can be realized by a secret-sharing scheme whose shares are of size 2 n − o ( n ) and until recently no better scheme was known. In a recent breakthrough, Liu and Vaikuntanathan (STOC 2018) have reduced the share size to 2 0 . 994 n + o ( n ) , and this was further improved by several follow-ups accumulating in an upper bound of 1 . 5 n + o ( n ) (Applebaum and Nir, CRYPTO 2021). Following these advances, it is natural to ask whether these new approaches can lead to a truly sub-exponential upper-bound of 2 n 1 − ε for some constant ε > 0 , or even all the way down to polynomial upper-bounds. In this paper, we relate this question to the complexity of computing monotone Boolean functions by monotone real circuits (MRCs) – a computational model that was introduced by Pudlák (J. Symb. Log., 1997) in the context of proof complexity. We introduce a new notion of “separable” MRCs that lies between monotone real circuits and monotone real formulas (MRFs). As our main results, we show that recent constructions of general secret-sharing schemes implicitly give rise to separable MRCs for general monotone functions of similar complexity, and that some monotone functions (in monotone NP) cannot be computed by sub-exponential size separable MRCs. Interestingly, it seems that proving similar lower-bounds for general MRCs is beyond the reach of current techniques. We use this connection to obtain lower-bounds against a natural family of secret-sharing schemes, as well as new non-trivial upper-bounds for MRCs. Specifically, we conclude that recent approaches for secret-sharing schemes cannot achieve sub-exponential share size and that every monotone function can be realized by an MRC (or even MRF) of complexity 1 . 5 n + o ( n ) . To the best of our knowledge, this is the first improvement over the trivial 2 n − o ( n ) upper-bound. Along the way, we show that the recent constructions of general secret-sharing schemes implicitly give rise to Boolean formulas over slice functions and prove that such formulas can be simulated by separable MRCs of similar size. On a conceptual level, our paper continues the rich line of study that relates the share size of secret-sharing schemes to monotone complexity measures.