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Paper:

TR22-006 | 12th January 2022 13:02

Secret Sharing, Slice Formulas, and Monotone Real Circuits

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TR22-006
Authors: Benny Applebaum, Amos Beimel, Oded Nir, Naty Peter, Toniann Pitassi
Publication: 12th January 2022 17:39
Downloads: 103
Keywords: 


Abstract:

A secret-sharing scheme allows to distribute a secret $s$ among $n$ parties such that only some predefined ``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 $2^{0.994n+o(n)}$, 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 $2^{n^{1-\epsilon}}$ for some constant $\epsilon>0$, 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 Pudl\'{a}k (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.



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