Broadcast enforced threshold schemes with disenrollment

Blakley, Blakley, Chan and Massey conjectured a lower bound on the entropy of broadcast messages in threshold schemes with disenrollment. In an effort to examine the conjecture, we identify their original scheme definition has a limitation: a coalition of participants can reconstruct all shared secrets without broadcast from the dealer, and hence render the dealer no control over disenrollment. We introduce a constraint that delays this lack of control of the dealer over disenrollment. We also establish the lower bounds on the entropy of broadcast messages in such a model. We demonstrate the need for new models by presenting a construction under open problems