Paper 2019/174

Towards an Exponential Lower Bound for Secret Sharing

Kasper Green Larsen and Mark Simkin

Abstract

A secret sharing scheme allows a dealer to distribute shares of a secret among a set of $n$ parties $P=\{p_1,\dots,p_n\}$ such that any authorized subset of parties can reconstruct the secret, yet any unauthorized subset learns nothing about it. The family $\mathcal{A} \subseteq 2^P$ of all authorized subsets is called the access structure. Classic results show that if $\mathcal{A}$ contains precisely all subsets of cardinality at least $t$, then there exists a secret sharing scheme where the length of the shares is proportional to $\lg n$ bits plus the length of the secret. However, for general access structures, the best known upper bounds have shares of length exponential in $n$, whereas the strongest lower bound shows that the shares must have length at least $n/\log n$. Beimel conjectured that the exponential upper bound is tight, but proving it has so far resisted all attempts. In this paper we make progress towards proving the conjecture by showing that there exists an access structure $\mathcal{A}$, such that any secret sharing scheme for $\mathcal{A}$ must have either exponential share length, or the function used for reconstructing the secret by authorized parties must have an exponentially long description. As an example corollary, we conclude that if one insists that authorized parties can reconstruct the secret via a constant fan-in boolean circuit of size polynomial in the share length, then there exists an access structure that requires a share length that is exponential in $n$.