In this paper we present a family of transforms that map existentially unforgeable signature schemes to signature schemes being strongly unforgeable. In spite of rising security, the transforms let us make a signature on a union of messages at once. The number of elements in this union depends on the signing algorithm of a scheme being transformed. In addition to that we define an existentially unforgeable signature scheme based on pairings, which satisfies all assumptions of the first part and is able to be subjected to transformation.