Attribute-based　encryption　(ABE)　and　attribute-based　keyword　search　(ABKS)　facilitate　fine-grained　access　and　search　control　for　cloud-assisted　Industrial　Internet　of　Things　(IIoT).　However,　existing　schemes　suffer　from　the　following　drawbacks:　(1)　their　computational　overhead　in　data　outsourcing　and　retrieval　is　exceptionally　high;　(2)　they　obtain　wrong　search　results　if　one　or　more　of　the　queried　keywords　are　wrongly　selected;　(3)　in　most　existing　ABKS,　the　untrustworthiness　of　the　cloud　server　is　not　taken　into　account,　and　the　correctness　of　the　search　results　is　not　verified;　(4)　existing　verifiable　ABKS　schemes　do　not　support　search　results　verification　without　the　main　data,　and　verifiers　have　to　first　download　the　search　results　and　then　check　their　correctness.　In　this　paper,　we　design　a　verifiable　online/offline　multi-keyword　search　(VMKS)　scheme　providing　high-level　solutions　to　the　aforementioned　problems.　We　use　the　healthcare　setting　as　a　case　study,　and　we　demonstrate　how　our　VMKS　can　be　deployed　in　cloud-assisted　Healthcare　IIoT　(HealthIIoT).　We　prove　the　security　of　our　VMKS　in　the　standard　model　and　under　the　hardness　assumption　of　the　decisional　bilinear　Diffie-Hellman　(DBDH)　problem.　Empirical　results　demonstrate　that　our　VMKS　speeds　up　encryption　and　verification　processes　by　more　than　70%　and　90%,　respectively.　Moreover,　our　scheme　reduces　the　communication　overhead　in　the　verification　phase　by　more　than　80%.　©　2022　Elsevier　Ltd