Geometric Shared Secret and/or Shared Control Schemes

A shared secret scheme is normally specified in terms of a desired security, P_d, and a concurrence scheme, Γ. The concurrence scheme (aka access structure) identifies subsets of participants (also called trustees or shareholders) each of which should be able to cooperatively recover the secret and/or initiate the controlled action. The security requirement is expressed as the maximum acceptable probability, P_d, that the secret can be exposed or the controlled action initiated by a collection of persons that doesn’t include at least one of the authorized subsets identified in the concurrence scheme. A concurrence scheme is said to be monotone if every set of participants that includes one or more sets from Γ is also able to recover the secret. The closure of Γ, denoted by $$ \hat \Gamma $$ is the collection of all supersets (not necessarily proper) of the sets in Γ, i.e., the collection of all sets of participants that can recover the secret and/or initiate the controlled action. A shared secret scheme implementing a concurrence scheme Γ is said to be perfect if the probability of recovering the secret is the same for every set, C, of participants: C$$ \hat \Gamma $$. Since, in particular, C could consist of only nonparticipants, i.e., of persons with no insider information about the secret, the probability, P, of an unauthorized recovery of the secret in a perfect scheme is just the probability of being able to “guess” the secret using only public information about Γ and the shared secret scheme implementing Γ: P ≤ P_td.