Private　set　intersection　(PSI)　is　a　fundamental　cryptographic　protocol　which　has　a　wide　range　of　applications.　It　enables　two　clients　to　compute　the　intersection　of　their　private　datasets　without　revealing　non-matching　elements.　The　advent　of　cloud　computing　drives　the　ambition　to　reduce　computation　and　data　management　overhead　by　outsourcing　such　computations.　However,　since　the　cloud　is　not　trustworthy,　some　cryptographic　methods　should　be　applied　to　maintain　the　confidentiality　of　datasets.　But,　in　doing　so,　data　owners　may　be　excluded　from　access　control　on　their　outsourced　datasets.　Therefore,　to　control　access　rights　and　to　interact　with　authorized　users,　they　have　to　be　online　during　the　protocol.　On　the　other　hand,　none　of　the　existing　cloud-based　PSI　schemes　support　fine-grained　access　control　over　outsourced　datasets.　This　paper,　for　the　first　time,　proposes　an　attribute-based　private　set　intersection　(AB-PSI)　scheme　providing　fine-grained　access　control.　AB-PSI　allows　a　data　owner　to　control　intersection　computations　on　its　outsourced　dataset　by　defining　an　access　control　policy.　We　also　provide　security　definitions　for　an　AB-PSI　scheme　and　prove　the　security　of　our　scheme　in　the　standard　model.　We　implement　our　scheme　and　report　performance　evaluation　results.　(C)　2020　Elsevier　Inc.　All　rights　reserved.