Federated　learning　emerged　to　solve　the　privacy　leakage　problem　of　traditional　centralized　machine　learning　methods.　Although　traditional　federated　learning　updates　the　global　model　by　updating　the　gradient,　an　attacker　may　still　infer　the　model　update　through　backward　inference,　which　may　lead　to　privacy　leakage　problems.　In　order　to　enhance　the　security　of　federated　learning,　we　propose　a　solution　to　this　challenge　by　presenting　a　multi-key　Cheon-Kim-Kim-Song　(CKKS)　scheme　for　privacy　protection　in　federated　learning.　Our　approach　can　enable　each　participant　to　use　local　datasets　for　federated　learning　while　maintaining　data　security　and　model　accuracy,　and　we　also　introduce　FedCMK,　a　more　efficient　and　secure　federated　learning　framework.　FedCMK　uses　an　improved　client　selection　strategy　to　improve　the　training　speed　of　the　framework,　redesigns　the　key　aggregation　process　according　to　the　improved　client　selection　strategy,　and　proposes　a　scheme　vMK-CKKS,　to　ensure　the　security　of　the　framework　within　a　certain　threshold.　In　particular,　the　vMK-CKKS　scheme　adds　a　secret　verification　mechanism　to　prevent　participants　from　malicious　attacks　through　false　information.　The　experiments　show　that　our　proposed　vMK-CKKS　schemes　significantly　improve　security　and　efficiency　compared　with　the　previous　encryption　schemes.　FedCMK　reduces　training　time　by　21%　on　average　while　guaranteeing　model　accuracy,　and　it　provides　robustness　by　allowing　participants　to　join　or　leave　during　the　process.　©　The　Author(s),　under　exclusive　license　to　Springer　Nature　Singapore　Pte　Ltd.　2024.