Paper 2026/1646

General versus linear secret sharing under statistical security

Abstract

How few participants are needed before general secret-sharing schemes can outperform linear ones under a fixed security notion? Under statistical security, we show that the answer is five participants; under perfect security, the corresponding threshold remains unknown. It is known that the \(157\) connected access structures on five participants split into \(140\) Shannon-exact cases and seventeen exceptional cases. For the former, linear schemes attain the Shannon polymatroid region; for each of the latter, the exact linear contribution region is the all-pairs one-common-information region and is strictly smaller than the Shannon region. We investigate the statistical contribution regions of these seventeen exceptional structures. For fifteen of them, we construct a partial scheme whose contribution vector lies outside the exact linear region; Jafari--Khazaei's partial-to-statistical transfer then gives a statistically secure family with the same asymptotic vector. One dual pair remains open. For \(\Gamma_{30}\), we further show that the maximum information ratio under statistical security lies in \([14/9,1.6502)\), improving both previously established bounds; moreover, \(1.6502<5/3\), where \(5/3\) is the optimum for linear schemes.