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On asymptotically good ramp secret sharing schemes

Abstract : 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.
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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⟩

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