Secure　group　communication　in　Vehicle　Ad　hoc　Networks　(VANETs)　over　open　channels　remains　a　challenging　task.　To　enable　secure　group　communications　with　conditional　privacy,　it　is　necessary　to　establish　a　secure　session　using　Authenticated　Key　Agreement　(AKA).　However,　existing　AKAs　suffer　from　problems　such　as　cross-domain　dynamic　group　session　key　negotiation　and　heavy　computational　burdens　on　the　Trusted　Authority　(TA)　and　vehicles.　To　address　these　challenges,　we　propose　a　dynamic　privacy-preserving　anonymous　authentication　scheme　for　condition　matching　in　fog-cloud-based　VANETs.　The　scheme　employs　general　Elliptic　Curve　Cryptosystem　(ECC)　technology　and　fog-cloud　computing　methods　to　decrease　computational　overhead　for　On-Board　Units　(OBUs)　and　supports　multiple　TAs　for　improved　service　quality　and　robustness.　Furthermore,　certificateless　technology　alleviates　TAs　of　key　management　burdens.　The　security　analysis　indicates　that　our　solution　satisfies　the　communication　security　and　privacy　requirements.　Experimental　simulations　verify　that　our　method　achieves　optimal　overall　performance　with　lower　computational　costs　and　smaller　communication　overhead　compared　to　state-of-the-art　solutions.　©　2024　by　the　authors.