Ciphertext-policy　attribute-based　keyword　search　(CP-ABKS)　schemes　facilitate　the　fine-grained　keyword　search　over　encrypted　data,　such　as　those　sensed/collected　from　Industrial　Internet　of　Things　(IIoT)　devices　and　stored　in　the　cloud.　However,　existing　CP-ABKS　schemes　generally　have　significant　computation　and　storage　requirements,　which　are　beyond　those　of　resource-constrained　IIoT　devices.　Therefore,　in　this　paper,　we　design　a　secure　online/offline　data　sharing　framework　(DSF),　which　supports　online/offline　encryption　and　outsourced　decryption.　Using　the　healthcare　setting　as　a　case　study,　we　demonstrate　how　DSF　can　be　deployed　in　the　cloud-assisted　Healthcare　IIoT　(HealthIIoT)　system.　We　not　only　prove　that　the　DSF　is　selectively　secure　in　the　chosen　access　structure　security　model　but　also　demonstrate　its　efficiency　and　feasibility　in　practical　scenarios　using　experiments.