Paper 2018/099

Improved Bounds on the Threshold Gap in Ramp Secret Sharing

Ignacio Cascudo, Jaron Skovsted Gundersen, and Diego Ruano


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.

Note: Accepted at IEEE Transactions on Information Theory. IEEE early access version available at

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Published elsewhere. IEEE Transactions on Information Theory
Secret Sharing
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jaron @ math aau dk
2019-03-04: revised
2018-01-29: received
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      author = {Ignacio Cascudo and Jaron Skovsted Gundersen and Diego Ruano},
      title = {Improved Bounds on the Threshold Gap in Ramp Secret Sharing},
      howpublished = {Cryptology ePrint Archive, Paper 2018/099},
      year = {2018},
      doi = {10.1109/TIT.2019.2902151},
      note = {\url{}},
      url = {}
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