Patent ID: 7818570

Claim:
A computer-implemented method for generating data representing an exclusive set system over a set U of entities such that each element of the exclusive set system is associated with cryptographic data, the method comprising: obtaining, by a computer system comprising circuitry, one or more coordinate systems for the set U, wherein each coordinate system associates each entity in U with a plurality of coordinates; determining, by the computer system, functions ƒ(u) each of which is defined on the set U, each function being a polynomial in one or more of the coordinates of u in at least one of the coordinate systems; determining by the computer system, for each said function ƒ, a corresponding subset S ƒ ⊂U such that ƒ is not equal to a predefined value on S ƒ but is equal to the predefined value on U\S ƒ , wherein said exclusive set system comprises the subsets S ƒ , and wherein each subset S ƒ is associated with first cryptographic data for use in a cryptographic operation which is for: (a) receiving a first input associated with an entity; (b) receiving a second input associated with a validity property of the entity and/or with an entitlement operable to be provided to the entity; and (c) combining the first input and the second input to prove the validity property and/or provide the entitlement; wherein at least one subset S ƒ has a plurality of entities, and the associated first cryptographic data are for providing the first input for each entity in the subset S ƒ ; wherein said one or more coordinate systems comprise a first coordinate system which comprises s coordinates where s is a predefined integer greater than one, each coordinate having values in a finite field having fewer elements than the number of entities in the set U; wherein for the first coordinate system, for at least one pair (i,j) of the s coordinates in the first coordinate system, for at least one subset R of U, the following property (A) holds: (A) said functions ƒ(u) comprise a first function ƒƒ(u) which is a polynomial in the ith and jth coordinates, wherein for each entity u in U the value of the jth coordinate of u is completely defined by the value of the ith coordinate of u and the value of the first function ƒƒ(u) on the entity u, wherein the first function ƒƒ(u) is equal to said predefined value on the subset R; wherein the method further comprises outputting said first cryptographic data for use by devices as the first input in said cryptographic operation, wherein the devices are for obtaining second cryptographic data for use as the second input in the cryptographic operation to prove the validity property and/or provide the entitlement for each entity of one or more of the subsets S ƒ .