Ideal uniform multipartite secret sharing schemes

Multipartite secret sharing schemes are those having a multipartite access structure, in which the set of participants is divided into several parts and all participants in the same part play an equivalent role. Secret sharing schemes for these access structures have gained attention, as they generalize threshold secret sharing. This work focuses on the construction of linear secret sharing schemes for ideal uniform multipartite access structures, while avoiding the inefficiencies and randomness of known constructions. To achieve this, we have developed two approaches. The first approach combines polymatroid-based techniques with Gabidulin codes, which efficiently realize a particular family of ideal uniform multipartite access structures. The efficiency of this method is derived from the unique properties of Gabidulin codes, which allow for polynomial share and secret sizes relative to the number of participants. Our second approach applies linear algebraic techniques and polymatroid-based techniques to explicitly design linear secret sharing schemes for arbitrary ideal uniform multipartite access structures. The efficiency of this method stems from the properties of polynomials over finite fields. The schemes from this method are efficient for smaller numbers of parts relative to the number of participants. In summary, we present explicit constructions of linear secret sharing schemes for ideal uniform multipartite access structures, which have important applications in cryptography.