On Abelian and Homomorphic Secret Sharing Schemes

Homomorphic (resp. abelian) secret sharing is a generalization of ubiquitous linear secret sharing in which the secret value and the shares are taken from finite (resp. abelian) groups instead of vector spaces over a finite field. Homomorphic secret sharing was first defined by Benaloh and, later in the early nineties, Frankel and Desmedt presented some relevant results. Except for a few other related topics such as black-box secret sharing and secret sharing over rings, the subject has remained dormant for about three decades. The study of homomorphic secret sharing is resumed in this paper and three main results are presented: (1) mixed-linear schemes, a subclass of abelian schemes to be introduced in this paper, are more powerful than linear schemes in terms of the best achievable information ratio (the claim is proved for the port of a well-known almost entropic matroid), (2) the information ratios of dual access structures are equal for the class of abelian schemes and (3) every ideal homomorphic scheme can be transformed into an ideal linear scheme with the same access structure.