Abstract: A method of managing an axiom includes determining if an H in an A?H axiom in existing axioms is a predetermined expression; and if the H is the predetermined expression, collecting, from the existing axioms, a B?I axiom in which the B includes the H, deleting the A?H axiom from the existing axioms, and generating a new B?I axiom from the collected B?I axiom by replacing the H in the B with the A.

Abstract: A method and apparatus are provided for broadcast encryption using a bilinear map, defined on elliptic curves. The method for the broadcast encryption using the bilinear map includes generating a first random number for all nodes except for a plurality of leaf nodes of an a-ary tree structure, configured in a plurality of depths, generating ‘a’ pieces of a second random number to allocate the generated second random number to all nodes except for a root node of the a-ary tree structure, generating public key information by applying the second random number to a second cyclic group, and generating a secret key group by applying the first and the second random numbers to a first cyclic group.