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.