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Secret sharing schemes with optimal and universal communication overheadshave been obtained independently by Bitar et al. and Huang et al. However,their constructions require a finite field of size q > n, where n is the numberof shares, and do not provide strong security. In this work, we give a generalframework to construct communication efficient secret sharing schemes based onsequences of nested linear codes, which allows to use in particular algebraicgeometry codes and allows to obtain strongly secure and communication efficientschemes. Using this framework, we obtain: 1) schemes with universal and closeto optimal communication overheads for arbitrarily large lengths n and a fixedfinite field, 2) the first construction of schemes with universal and optimalcommunication overheads and optimal strong security (for restricted lengths),having in particular the component-wise security advantages of perfect schemesand the security and storage efficiency of ramp schemes, and 3) schemes withuniversal and close to optimal communication overheads and close to optimalstrong security defined for arbitrarily large lengths n and a fixed finitefield.

Communication efficient and strongly secure secret sharing schemes based on algebraic geometry codes