The　well-known　open　problem　in　public　key　encryption　with　keyword　search　is　how　to　avoid　internal　adversaries　as　the　server.　Implicitly,　the　internal　attack　is　implemented　as　follows.　Upon　receiving　a　trapdoor,　the　probability　polynomial　time　internal　adversary　can　always　act　as　a　sender　to　produce　each　ciphertext　for　each　keyword　if　keyword　space　is　bounded　by　a　polynomial　of　the　security　parameter.　Then,　the　adversary　runs　the　test　algorithm　for　the　trapdoor　and　all　produced　ciphertext,　and　then　infer　the　correct　keyword.　To　overcome　this　problem,　the　original　framework　must　be　changed　slightly.　A　fundamental　goal　is　creates　a　secure　bridge　between　the　sender　and　receiver.　It　not　only　keeps　testability　of　the　server,　but　also　avoids　imitating　a　sender.　Witness-based　searchable　encryption　(WBSE)　is　a　manner　to　realize　the　design　goal.　In　this　paper,　we　formalize　an　abstracted　notion,　witness-based　searchable　encryption　with　aggregative　trapdoor.　Under　the　notion,　we　present　a　nearly　optimal　solution　for　WBSE　under　the　barrier　with　trapdoor　size　proportional　to　n　(the　number　of　senders).　Comparing　with　the　existing　scheme　with　trapdoor　size　O(n),　the　proposed　scheme　is　based　on　bilinear　map,　and　offers　size　only　in　n.　©　2020,　Springer　Nature　Switzerland　AG.