With　the　advent　of　Cloud-storages,　secure　sharing　of　encrypted　data　with　fine-grained　access　control　has　become　an　important　challenge.　To　enforce　fine-grained　access　rights　on　encrypted　data　to　different　users,　Attribute-Base　Encryption　(ABE)　is　a　promising　cryptographic　tool.　But　in　many　situations,　some　pirate　users　sell　their　access　privileges　for　monetary　gain.　Hence,　an　efficient　Black-Box　Traitor　Tracing　(BB-TT)　system　is　vital　to　address　the　corresponding　Pirate　Decoders　(PDs).　But,　in　the　existing　BB-TTs,　the　tracing　complexity　is　directly　related　to　the　maximum　number　of　users　in　the　system.　This　article　proposes　an　ABE-based　access　control　system　that　provides　a　new　Relaxed　BB-TT.　Our　Relaxed　BB-TT　addresses　the　complexity　problem　by　exploring　the　pirate　user　with　an　optimal　number　of　decryption　queries　on　the　PD.　The　optimal　number　of　queries　leads　to　the　exciting　capability　of　traceability　of　stateful　decoders　(i.e.,　the　decoder　that　can　save　the　states　among　the　Tracer＇s　queries).　Finally,　our　system　provides　White-Box　traceability　and　reduces　the　amount　of　trust　in　the　authorities.　We　give　formal　security　proofs　and　extensive　experimental　results　on　different　classes　of　mobile　devices,　including　a　laptop　and　a　smartphone.