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),which has the security advantages of perfect schemes and the storage efficiencyof ramp schemes, and 3) schemes with universal and close to optimalcommunication overheads and close to optimal strong security defined forarbitrarily large lengths n and a fixed finite field.
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