**Improved Bounds on the Threshold Gap in Ramp Secret Sharing**

*Ignacio Cascudo and Jaron Skovsted Gundersen and 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.

**Category / Keywords: **Secret Sharing

**Original Publication**** (in the same form): **IEEE Transactions on Information Theory
**DOI: **10.1109/TIT.2019.2902151

**Date: **received 15 Jan 2018, last revised 4 Mar 2019

**Contact author: **jaron at math aau dk

**Available format(s): **PDF | BibTeX Citation

**Note: **Accepted at IEEE Transactions on Information Theory. IEEE early access version available at https://ieeexplore.ieee.org/document/8654006

**Version: **20190304:103345 (All versions of this report)

**Short URL: **ia.cr/2018/099

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