Better Secret-Sharing via Robust Conditional Disclosure of Secrets.

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 2n−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 20.994n+o(n), which was later improved to 20.892n+o(n) by Applebaum et al. (EUROCRYPT 2019). In this paper we improve the exponent of general secret-sharing down to 0.637. For the special case of linear secret-sharing schemes, we get an exponent of 0.762 (compared to 0.942 of Applebaum et al.). As our main building block, we introduce a new robust variant of conditional disclosure of secrets (robust CDS) that achieves unconditional security even under bounded form of re-usability. We show that the problem of general secret-sharing reduces to robust CDS with sub-exponential overhead and derive our main result by implementing robust CDS with a non-trivial exponent. The latter construction follows by presenting a general immunization procedure that turns standard CDS into a robust CDS.