How to Collaborate between Threshold Schemes

Threshold schemes have been used to protect secrets by distributing shares to participants. To protect two secrets, we can use two separate traditional schemes, say, a (t1, n1) scheme and a (t2, n2) scheme. If there are u (<=min(t1, t2)) participants involved in both schemes, each of these u participants must keep two different shares. This paper proposes a method that allows each common participant to keep only one share. Our method constructs two polynomials with u common crossover points. We give theoretical details and two demonstrative examples. This algorithm can also handle the collaboration between more than two schemes.