Traitor　tracing　scheme　is　a　paradigm　to　classify　the　users　who　illegal　use　of　their　decryption　keys　in　cryptosystems.　In　the　ciphertext-policy　attribute-based　cryptosystem,　the　decryption　key　usually　contains　the　users＇　attributes,　while　the　real　identities　are　hidden.　The　decryption　key　with　hidden　identities　enables　malicious　users　to　intentionally　leak　decryption　keys　or　embed　the　decryption　keys　in　the　decryption　device　to　gain　illegal　profits　with　a　little　risk　of　being　discovered.　To　mitigate　this　problem,　the　concept　of　blackbox　traceability　in　the　ciphertext-policy　attribute-based　scheme　was　proposed　to　identify　the　malicious　user　via　observing　the　I/O　streams　of　the　decryption　device.　However,　current　solutions　with　blackbox　traceability　are　impractical　since　either　the　composite-order　group　or　the　linear　complexity　of　system　users　is　required.　In　this　article,　we　proposed　a　secure　ciphertext-policy　attribute-based　set　encryption　scheme　with　the　short　decryption　key.　The　proposed　scheme　bases　on　the　prime-order　group　to　improve　computational　performances　and　aggregates　multiple　attributes　into　a　constant-size　attribute　set　to　reduce　the　costs　of　communication　overheads.　By　applying　our　proposed　scheme　with　fingerprint　codes,　we　then　give　an　instantiation　of　the　ciphertext-policy　attribute-based　scheme　with　blackbox　traceability.　Our　scheme　is　provably　secure　under　various　q-type　assumptions.　(C)　2020　Elsevier　Inc.　All　rights　reserved.