Constructing ideal (t,n) threshold secret sharing schemes leads to some limitations on the maximum number of users, that are able to join the secret sharing scheme. We aim to remove these limitations by reducing the information rate of the constructed threshold secret sharing schemes. In this paper we propose recursive construction algorithms of (t,n) threshold secret sharing schemes, based on the generalized vector space construction. Using these algorithms we are able to construct a (t,n) threshold secret sharing scheme for any arbitrary n.

We present a recursive algorithm for constructing linear discrete-time systems which interpolate the desired 1st-and 2nd-order information. The recursive algorithm constructs a new system and connects it to the previous system in the cascade form every time new information is added. These procedures yield a practical realization of all the interpolants.