Prof. Amos Beimel

Evolving secret-sharing schemes, introduced by Komargodski, Naor, and Yogev (TCC 2016b), are secret-sharing schemes in which the dealer does not know the number of parties that will participate. The parties arrive one by one and when a party arrives the dealer gives it a share; the dealer cannot update this share when other parties arrive. Komargodski and Paskin-Cherniavsky (TCC 2017) constructed evolving a· i -threshold secret-sharing schemes (for every 0 < a< 1), where any set of parties whose maximum party is the i-th party and contains at least ai parties can reconstruct the secret; any set such that all its prefixes are not an a-fraction of the parties should not get any information on the secret. The length of the share of the i-th party in their scheme is O(i⁴log i). As the number of parties is unbounded, this share size can be quite large. In this work we suggest studying a relaxation of evolving threshold secret-sharing schemes; we consider evolving (a, b)-ramp secret-sharing schemes for 0 < b< a< 1. Again, we require that any set of parties whose maximum party is the i-th party and contains at least ai parties can reconstruct the secret; however, we only require that any set such that all its prefixes are not a b-fraction of the parties should not get any information on the secret. For all constants 0 < b< a< 1, we construct an evolving (a, b)-ramp secret-sharing scheme where the length of the share of the i-th party is O(1). Thus, we show that evolving ramp secret-sharing schemes offer a big improvement compared to the known constructions of evolving a· i -threshold secret-sharing schemes.