The Construction of Geometric Threshold Schemes with Projective Geometry

In a secret sharing scheme, a dealer has a secret and distributes shares of the secret to the participants. The shares are distributed in such a way that only certain subsets of the participants can combine their shares to recover the secret. We use projective geometry to construct a secret sharing scheme for which the shares are represented by points in a projective 4-space. A sharply focused set is a set of k points, no three collinear, in a nite projective plane in which the k 2 distinct secants formed by this set intersect a given line in exactly k points. We use sharply focused sets to design a scheme so that certain subsets of participants can pool their shares to calculate the secret. Several examples show how sharply focused sets can be used for diiering sizes of the subsets of the participants that are allowed to calculate the secret. We provide a proof that sharply focused sets obtained from a certain construction correspond to cosets of a group. This implies that the possible sizes of these sharply focused sets are divisors of the group order. Finally, we provide a new construction that uses subplanes in planes of prime power order. The previous construction method works only on planes of odd order, but the subplane construction method works in planes of both even and odd order. iii This abstract accurately represents the content of the candidate's thesis. I recommend its publication.