Paper 2018/099 Original: eprint.iacr.org/2018/099 ↗

Improved Bounds on the Threshold Gap in Ramp Secret Sharing

Abstract

In this paper we consider linear secret sharing schemes over a finite field $\mathbb{F}_q$, where the secret is a vector in $\mathbb{F}_q^\ell$ and each of the $n$ shares is a single element of $\mathbb{F}_q$. We obtain lower bounds on the so-called threshold gap $g$ of such schemes, defined as the quantity $r-t$ where $r$ is the smallest number such that any subset of $r$ shares uniquely determines the secret and $t$ is the largest number such that any subset of $t$ shares provides no information about the secret. Our main result establishes a family of bounds which are tighter than previously known bounds for $\ell\geq 2$. Furthermore, we also provide bounds, in terms of $n$ and $q$, on the partial reconstruction and privacy thresholds, a more fine-grained notion that considers the amount of information about the secret that can be contained in a set of shares of a given size. Finally, we compare our lower bounds with known upper bounds in the asymptotic setting.