Talk by visiting PhD Student Mark Abspoel

2018.09.07 | Malene Bisgaard Blaabjerg Andersen

Date Tue 18 Sep
Time 14:00 15:00
Location Building 5335, Room 295

Title: Shamir's scheme is the only strongly multiplicative LSSS with maximal adversary

Abstract: We consider linear secret sharing schemes (LSSS) over a finite field K with the shares in K. An LSSS with t-adversary and n players is strongly multiplicative if it has (n-t)-product reconstruction. It is well-known that for strongly multiplicative LSSS with the secret in K it holds that t ≤ (n-1)/3. This bound is sharp, as equality can be attained using Shamir's scheme. We show that in fact Shamir's scheme is the only strongly multiplicative LSSS with maximal adversary t. In particular, this implies that the number of players in this case is bounded by q+1, where q is the order of K.  

 

We generalize this result to strongly multiplicative LSSS with the secret in an extension field L over K of finite degree k. We show that it holds that t ≤ (n-2k+1)/3, and that equality can be attained using an extension of Shamir's scheme, where we take the evaluation point of the secret in L. We also show that this scheme is the only one that attains maximal t.

About the Speaker: Mark Abspoel is a PhD student at CWI since January 2017, under the supervision of Ronald Cramer. Before this, he completed a Masters degree in Mathematics at Leiden University, with his thesis written at CWI on the topic of secret sharing. Mark's research interests are multi-party computation and secret sharing. Part of his PhD is funded by Philips Research, through the SODA Horizon 2020 project.

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