Algebraic geometric secret sharing schemes over large fields are asymptotically threshold

Fan Peng, Hao Chen and Chang-An Zhao

Vol. 319 (2022), No. 1, 213–232

DOI: 10.2140/pjm.2022.319.213

Abstract

Algebraic geometric secret sharing schemes were proposed by Chen and Cramer so that the fundamental theorem in information-theoretically secure multiparty computation can be established over constant-size base finite fields. These algebraic geometric secret sharing schemes defined by a curve of genus $g$ over a constant-size finite field ${\mathbb{𝔽}}_q$ are quasithreshold in the following sense: any subset of $u\le T-1$ players (nonqualified) has no information of the secret and any subset of $u\ge T+2g$ players (qualified) can reconstruct the secret. It is natural to ask how far from the threshold these quasithreshold secret sharing schemes are. How many subsets of $u\in [T,T+2g-1]$ players can recover the secret or have no knowledge of it?

We prove that if the size $q$ goes to infinity and $\lim <!--FUNCTION\; APPLICATION-->g∕q^{1∕2}=0$, then almost all subsets of $u\in [T,T+g-1]$ players have no information of the secret and almost all subsets of $u\in [T+g,T+2g-1]$ players can reconstruct the secret. Then algebraic geometric secret sharing schemes over large finite fields are asymptotically threshold in this case. We also analyze the case when the size  $q$ of the base field is fixed and the genus goes to infinity.

Keywords

algebraic geometric secret sharing, quasithreshold, threshold, algebraic-geometry codes

Mathematical Subject Classification

Primary: 14G50, 14H05, 14Q05, 94A60, 94B27

Milestones

Received: 11 September 2021

Revised: 21 January 2022

Accepted: 11 February 2022

Published: 28 August 2022