Leakage-Resilient Secret Sharing with Constant Share Size

Abstract

In this work, we consider the leakage resilience of algebraic-geometric (AG for short) codes based ramp secret sharing schemes extending the analysis on the leakage resilience of linear threshold secret sharing schemes over prime fields that is done by Benhamouda et al. Since there does not exist any explicit efficient construction of AG codes over prime fields, we consider constructions over prime fields with the help of concatenation method and constructions of codes over field extensions. Extending the Fourier analysis done by Benhamouda et al., one can show that concatenated algebraic geometric codes over prime fields do produce some nice leakage-resilient secret sharing schemes. One natural and curious question is whether AG codes over extension fields produce better leakage-resilient secret sharing schemes than the construction based on concatenated AG codes. Such construction provides another advantage compared to the construction over prime fields using concatenation method. It is clear that AG codes over extension fields give secret sharing schemes with smaller reconstruction for a fixed privacy parameter t. In this work, it is also confirmed that indeed AG codes over extension fields have stronger leakage-resilience under some reasonable assumptions. These advantages strongly motivate the study of secret sharing schemes from AG codes over extension fields. The current paper has two main contributions: (i) we obtain leakage-resilient secret sharing schemes with constant share sizes and unbounded numbers of players. (ii) via a sophisticated Fourier Analysis, we analyze the leakage-resilience of secret sharing schemes from codes over extension fields. This is of its own theoretical interest independent of its application to secret sharing schemes from algebraic geometric codes over extension fields.