Combinatorial constructions of repairable ramp schemes

A repairable ramp scheme is a ramp scheme in which a player can securely reconstruct a lost share with the help from a subset of players. This will take place without the participation of the dealer who set up the scheme. The repairing protocol should not compromise the security of the ramp scheme. Distribution designs were introduced by Stinson and Wei (Des. Codes Cryptogr. 86, 195–210 2018) and can be used to construct repairable ramp schemes. In this paper, we first give the definitions of a \(\varvec{(\tau _{1},\tau _{2},l_{1},l_{2})}\) $\mathbf{(}{\mathit{\tau }}_{\mathbf{1}}\mathbf{,}{\mathit{\tau }}_{\mathbf{2}}\mathbf{,}{\mathit{l}}_{\mathbf{1}}\mathbf{,}{\mathit{l}}_{\mathbf{2}}\mathbf{)}$ -distribution design and a repairable ramp scheme. And we use anti-Pasch Steiner triple systems as distribution designs to construct repairable ramp schemes. We determine the existence of an anti-Pasch Steiner triple system (QFSTS \(\varvec{(v)}\) $\mathbf{(}\mathit{v}\mathbf{)}$ ) with a minimum basic repairing set for \(\varvec{v\equiv 1,3\pmod 6}\) $\mathit{v}\mathbf{\equiv }\mathbf{1}\mathbf{,}\mathbf{3}\mathbf{(}\mathbf{mod}\mathbf{6}\mathbf{)}$ , \(\varvec{v\geqslant 9}\) $\mathit{v}\mathbf{⩾}\mathbf{9}$ and \(\varvec{v\ne 13}\) $\mathit{v}\mathbf{\ne }\mathbf{13}$ . Then we obtain a \(\varvec{(2,4,n,3)}\) $\mathbf{(}\mathbf{2}\mathbf{,}\mathbf{4}\mathbf{,}\mathit{n}\mathbf{,}\mathbf{3}\mathbf{)}$ -repairable ramp scheme containing \(\varvec{n}\) $\mathit{n}$ players with \(\varvec{\lceil \frac{2v}{3}\rceil \leqslant n\leqslant \frac{v(v-1)}{6}}\) $\mathbf{\lceil }\frac{\mathbf{2}\mathit{v}}{\mathbf{3}}\mathbf{\rceil }\mathbf{⩽}\mathit{n}\mathbf{⩽}\frac{\mathit{v}\mathbf{(}\mathit{v}\mathbf{-}\mathbf{1}\mathbf{)}}{\mathbf{6}}$ .