With　the　development　of　cloud　computing　and　big　data,　secure　multi-party　computation,　which　can　collaborate　with　multiple　parties　to　deal　with　a　large　number　of　transactions,　plays　an　important　role　in　protecting　privacy.　Private　set　intersection　(PSI),　a　form　of　multi-party　secure　computation,　is　a　formidable　cryptographic　technique　that　allows　the　sender　and　the　receiver　to　calculate　their　intersection　and　not　reveal　any　more　information.　As　the　data　volume　increases　and　more　application　scenarios　emerge,　PSI　with　multiple　participants　is　increasingly　needed.　Homomorphic　encryption　is　an　encryption　algorithm　designed　to　perform　a　mathematical-style　operation　on　encrypted　data,　where　the　decryption　result　of　the　operation　is　the　same　as　the　result　calculated　using　unencrypted　data.　In　this　paper,　we　present　a　cloud-assisted　multi-key　PSI　(CMPSI)　system　that　uses　fully　homomorphic　encryption　over　the　torus　(TFHE)　encryption　scheme　to　encrypt　the　data　of　the　participants　and　that　uses　a　cloud　server　to　assist　the　computation.　Specifically,　we　design　some　TFHE-based　secure　computation　protocols　and　build　a　single　cloud　server-based　private　set　intersection　system　that　can　support　multiple　users.　Moreover,　security　analysis　and　performance　evaluation　show　that　our　system　is　feasible.　The　scheme　has　a　smaller　communication　overhead　compared　to　existing　schemes.　©　2023　by　the　authors.