On asymptotically good ramp secret sharing schemes

Olav Geil (1) , Stefano Martin (1) , Umberto Martínez-Peñas (1) , Ryutaroh Matsumoto (2) , Diego Ruano (1)

1 Department of Mathematical Sciences [Aalborg]
2 TITECH - Tokyo Institute of Technology [Tokyo]

Asymptotically good sequences of ramp secret sharing schemes have been intensively studied by Cramer et al. in [1,2,3,4,5,6,7,8]. In those works the focus is on full privacy and full reconstruction. We propose an alternative definition of asymptotically good sequences of ramp secret sharing schemes where a small amount of information leakage is allowed (and possibly also non-full recovery). By a non-constructive proof we demonstrate the existence of sequences that – following our definition of goodness – have parameters arbitrary close to the optimal ones. Moreover – still using our definition – we demonstrate how to concretely construct asymptotically good sequences of schemes from sequences of algebraic geometric codes related to a tower of function fields. Our study involves a detailed treatment of the relative generalized Hamming weights of the involved codes.

Domaines: Cryptographie et sécurité [cs.CR]

Olav Geil, Stefano Martin, Umberto Martínez-Peñas, Ryutaroh Matsumoto, Diego Ruano. On asymptotically good ramp secret sharing schemes. WCC2015 - 9th International Workshop on Coding and Cryptography 2015, Anne Canteaut, Gaëtan Leurent, Maria Naya-Plasencia, Apr 2015, Paris, France. ⟨hal-01275753⟩