Vol. 319, No. 1, 2022

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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

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 𝔽q are quasithreshold in the following sense: any subset of u T 1 players (nonqualified) has no information of the secret and any subset of u 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 [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 gq12 = 0, then almost all subsets of u [T,T + g 1] players have no information of the secret and almost all subsets of u [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.

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algebraic geometric secret sharing, quasithreshold, threshold, algebraic-geometry codes
Mathematical Subject Classification
Primary: 14G50, 14H05, 14Q05, 94A60, 94B27
Received: 11 September 2021
Revised: 21 January 2022
Accepted: 11 February 2022
Published: 28 August 2022
Fan Peng
College of Mathematics and Statistics
Guangxi Normal University
Yucai Campus
Hao Chen
College of Information Science and Technology / Collage of Cyber Security
Jinan University
Chang-An Zhao
Department of Mathematics
Sun Yat-sen University
Guangdong Key Laboratory of Information Security