Secret Sharing at the Shannon Ceiling

Abstract

For every $n\ge 9$ that is a multiple of 3, we construct an explicit access structure on $n$ participants. In every perfect secret-sharing scheme realising this access structure, if $S$ denotes the random secret, then the sum of the share entropies is at least $\left(\frac{n^2}{9}+\frac{2n}{3}\right)H(S)$, and some participant has share entropy at least $\left(\frac{n}{6}+\frac{1}{2}\right)H(S)$. After normalisation by $H(S)$, these are respectively $\Omega (n^2)$ and $\Omega (n)$ lower bounds and also give the same asymptotic lower bounds on the total and largest expected binary lengths of the shares. This improves by a logarithmic factor the longstanding general lower bounds of $\Omega (n^2/\log n)$ for total share size and $\Omega (n/\log n)$ for maximum share size due to Csirmaz. The proof uses only elementary Shannon inequalities, together with some averaging arguments. The Shannon-information method has universal $O(n^2)$ and $O(n)$ ceilings for the total and maximum normalised entropy lower bounds it can certify, so our construction reaches both ceilings up to constant factors.