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Title:
Improved Bounds on the Threshold Gap in Ramp Secret Sharing
Authors: Ignacio Cascudo, Jaron Skovsted Gundersen, Diego Ruano
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.
ePrint: https://eprint.iacr.org/2018/099
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