Communication Efficient and Strongly Secure Secret Sharing Schemes Based on Algebraic Geometry Codes

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Secret-sharing schemes with optimal and universal communication overheads have 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 number of shares, and do not provide strong security. In this paper, we give a general framework to construct communication efficient secret-sharing schemes based on sequences of nested linear codes, which allows to use, in particular, algebraic geometry codes and allows to obtain strongly secure and communication-efficient schemes. Using this framework, we obtain: 1) schemes with universal and close to optimal communication overheads for arbitrarily large lengths $n$ and a fixed finite field; 2) the first construction of schemes with universal and optimal communication overheads and optimal strong security (for restricted lengths), having, in particular, the component-wise security advantages of perfect schemes and the security and storage efficiency of ramp schemes; and 3) schemes with universal and close to optimal communication overheads and close to optimal strong security defined for arbitrarily large lengths $n$ and a fixed finite field.