How to Share a Secret, Infinitely

Secret sharing schemes allow a dealer to distribute a secret piece of information among several parties so that any qualified subset of parties can reconstruct the secret, while every unqualified subset of parties learns nothing about the secret. The collection of qualified subsets is called an access structure. The best known access structure is the k-threshold one, where any subset of size k or more is qualified and all smaller subsets are not qualified. A major issue is the size of the shares needed to assure this property. For the threshold access structure, when k = 2 and there are n parties it is known that the share size must be log n even for secrets of 1 bit. In this work we consider the case where the set of parties is not known in advanced and could potentially be infinite. We would still like to give the t party arriving a small share as possible as a function of t. We show that for any access structure it is possible to do so by giving shares of size 2t−1. Our main result is a scheme for k-threshold access structure with shares of size (k− 1) · log t+ poly(k) · o(log t) bits. Finally, we prove that no secret sharing scheme for the 2-threshold access structure with share size at most log t+ log log t+O(1) exists. ∗Weizmann Institute of Science, Israel. Email: {ilan.komargodski,moni.naor,eylon.yogev}@weizmann.ac.il. Supported in part by a grant from the I-CORE Program of the Planning and Budgeting Committee, the Israel Science Foundation, BSF and the Israeli Ministry of Science and Technology. Moni Naor is the incumbent of the Judith Kleeman Professorial Chair. Ilan Komargodski is supported in part by a Levzion fellowship.