A necessary and sufficient condition for the asymptotic idealness of the GRS threshold secret sharing scheme

• Published in: Information processing letters Vol. 114; no. 6; pp. 299 - 303
• Main Authors: Tiplea, Ferucio Laurentiu, Dragan, Constantin Catalin
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.06.2014; Elsevier Sequoia S.A
• Subjects: (Asymptotic) idealness; (Asymptotic) perfectness; Asymptotic methods; Asymptotic properties; Chinese Remainder Theorem; Cryptography; Data processing; Entropy; Information sharing; Probability distribution; Scientific papers; Secret sharing scheme; Studies; Thresholds; Entropy; (Asymptotic) idealness; Secret sharing scheme; Cryptography; Chinese Remainder Theorem; (Asymptotic) perfectness
• ISSN: 0020-0190; 1872-6119
• DOI: 10.1016/j.ipl.2014.01.008

The study of the asymptotic idealness of the Goldreich–Ron–Sudan (GRS, for short) threshold secret sharing scheme was the subject of several research papers, where sufficient conditions were provided. In this paper a necessary and sufficient condition is established; namely, it is shown that the GRS threshold secret sharing scheme is asymptotically ideal under the uniform distribution on the secret space if and only if it is based on 1-compact sequences of co-primes. •Previous results regarding the security of the GRS secret sharing scheme are improved.•The GRS scheme is asymptotically ideal iff it is based on 1-compact sequences of co-primes.•The asymptotic idealness issue of the GRS scheme is completely closed.•The importance of 1-compact sequences of co-primes is emphasized.