Improved Bounds on the Threshold Gap in Ramp Secret Sharing

Producción CientíficaAbstract: In this paper we consider linear secret sharing schemes over a finite field Fq, where the secret is a vector in Fℓq and each of the n shares is a single element of Fq. 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 ℓ≥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.Danish Council for Independent Research (grant DFF-4002- 00367)Ministerio de Economía, Industria y Competitividad (grants MTM2015-65764-C3-2-P / MTM2015-69138- REDT)RYC-2016-20208 (AEI/FSE/UE)Junta de Castilla y León (grant VA166G18