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An asymmetric traitor tracing scheme based on Luc public key cryptosystem is proposed. The scheme uses Shamir's secret sharing scheme. Comparing with existed schemes, the new scheme has some advantages. Firstly, we use the Luc public key cryptosystem in the scheme to increase the security. Secondly, the length of user key and data header is short. The scheme is asymmetric which means data supplier does not have all user's key. The user's key is made by user and data supplier. The data supplier is unable to frame a legitimate user. Thirdly, the scheme supports black-box tracing. The user's key contains the data supplier's signature which makes the traitor tracing more persuasive. The user can choose public key and partial private key, which is more in line with the reality. Lastly, the user can join or withdrawal system at any time. When discovering the illegal user, the system can revoke the traitor but the legitimate user's key does not need to change. The user's key contains his own identity information that can make legitimate user not reluctantly to leak his key to other users.