Summary: In the generalized group-oriented cryptosystem, the sender can send a conditional message to a group of users such that only the specified sets of users in this group can cooperate to decrypt this message. We use an ElGamal cryptosystem and an elliptic curve ElGamal cryptosystem to achieve the purposes of generalization and group-orientation, respectively. Both of our schemes are more efficient than Tsai et al.'s scheme in terms of sender's computational complexity.