Attribute Based Group Signatures were first introduced in [12]. It was proposed

to serve the purpose of including attributes in a group signature scheme. Group

Signatures allow a member of a group to sign on behalf of the others while in

ABGS schemes the aim is to allow a member of the group only possessing certain

attributes to sign on behalf of the rest.

Chaum and van Heist [9] proposed the first group signature in order to implement e-cash systems. The two underlying security notions of group signatures

are anonymity and traceability. An anonymous scheme is a scheme that does not

reveal the signers identity. A traceable scheme is such that a group of colluding

members cannot forge a signature that will not be traced to at least one of them.

Since Group Signatures was introduced, different security notions were defined;

some examples are unlinkability, unforgeablitity, collusion resistance, exculpability, and framing resistance. In Bellare et al. [4] strong definitions of the core

requirements were formulated and defined. The authors in that paper proposed

two security notions that implies all the rest and they defined them as full traceability and full anonymity. Since the scheme introduced in our paper could be

considered as a type of group signature scheme we adopt the security notions

presented in Bellare et al.s work and modify them as done in [12]( See Section

3.2). The reader is referred to Bellare et al.s work in [4] for further details about