Improved Bounds on the Threshold Gap in Ramp Secret Sharing.

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. [ABSTRACT FROM AUTHOR]