The　Industrial　Internet　of　Things　(IIoT)　has　revolutionized　industrial　processes,　but　the　adoption　of　edge　computing　brings　significant　security　challenges,　particularly　in　authentication　and　key　agreement　between　resource-constrained　edge　gateways　and　cloud　application　servers.　This　article　presents　a　symmetric-key　authentication　and　key　agreement　scheme　(SKAS)　tailored　for　the　IIoT.　The　proposed　scheme　integrates　symmetric　encryption　and　elliptic　curve　Diffie-Hellman　(ECDH)　key　exchange　to　facilitate　efficient　mutual　authentication　and　secure　communication　channel　establishment　in　untrusted　edge　computing　environments.　A　specialized　symmetric-key　authentication　mechanism　is　designed　for　edge　gateways,　making　it　compatible　with　scenarios　where　legacy　infrastructure　functions　as　the　edge　gateway.　The　security　of　SKAS　is　formally　verified　using　ProVerif,　confirming　its　robustness　against　known　IIoT-specific　attacks.　Performance　evaluations　demonstrate　that　SKAS　achieves　notable　efficiency　in　computation,　communication,　and　storage　overhead,　making　it　a　lightweight　and　practical　solution　for　IIoT　applications.　©　The　Author(s),　under　exclusive　license　to　Springer　Nature　Singapore　Pte　Ltd.　2025.