Traitor　tracing　scheme　can　be　used　to　identify　a　decryption　key　is　illegally　used　in　public-key　encryption.　In　CCS’13,　Liu　et　al.　proposed　an　attribute-based　traitor　tracing　(ABTT)　scheme　with　blackbox　traceability　which　can　trace　decryption　keys　embedded　in　a　decryption　blackbox/device　rather　than　tracing　a　well-formed　decryption　key.　However,　the　existing　ABTT　schemes　with　blackbox　traceability　are　based　on　composite　order　group　and　the　size　of　the　decryption　key　depends　on　the　policies　and　the　number　of　system　users.　In　this　paper,　we　revisit　blackbox　ABTT　and　introduce　a　new　primitive　called　attribute-based　set　encryption　(ABSE)　based　on　key-policy　ABE　(KP-ABE)　and　identity-based　set　encryption　(IBSE),　which　allows　aggregation　of　multiple　related　policies　and　reduce　the　decryption　key　size　in　ABTT　to　be　irrelevant　to　the　number　of　system　users.　We　present　a　generic　construction　of　the　ABTT　scheme　from　our　proposed　ABSE　scheme　and　fingerprint　code　based　on　the　Boneh-Naor　paradigm　in　CCS’08.　We　then　give　a　concrete　construction　of　the　ABSE　scheme　which　can　be　proven　secure　in　the　random　oracle　model　under　the　decisional　BDH　assumption　and　a　variant　of　q-BDHE　assumption.　©　Springer　Nature　Switzerland　AG　2018.