Access structures of hyperelliptic secret sharing schemes

Abstract

In CRYPTO 2006 Chen and Cramer proposed secret sharing schemes (SSS) from algebraic–geometric (AG) codes. The schemes are ramp schemes with gap bounded by 2g, where g is the genus of the underlying curve. Subsequently, Chen, Ling and Xing explicitly gave a complete characterization of the access structures for one special and important instance-elliptic secret sharing schemes (the ones from algebraic–geometric codes associated with elliptic curves), and additionally constructed weighted threshold secret sharing schemes from algebraic curves. In elliptic SSS case, one single point on an elliptic curve was computed to determine whether a set, with size in the gap mentioned above, is qualified. In this paper, we generalize Chen, Ling and Xing's idea and method to the case where the underlying curve is a hyperelliptic curve of arbitrary genus. By the means of Cantor's algorithm, we compute a reduced divisor to determine whether a set is qualified. Moreover, we construct a weighted hyperelliptic secret sharing schemes. Thus we reduce the gap size from 2g to g−1 in both ideal and weighted hyperelliptic SSS cases. One explicit example is provided.