Federated　Learning　is　a　distributed　machine　learning　framework,　which　mainly　adopts　cloud-edge　collaborative　computing　mode　and　supports　multiple　participants　to　train　models　without　directly　sharing　local　data.　However,　participants＇　sensitive　information　may　still　be　leaked　through　their　gradients.　Besides,　incorrect　aggregated　results　returned　by　the　aggregation　server　may　reduce　the　effect　of　joint　modeling.　This　article　proposes　a　privacy-preserving　and　verifiable　federated　learning　method　called　PPVerifier　to　support　privacy　protection　and　verification　of　aggregated　results　in　the　cloud-edge　collaborative　computing　environment.　By　the　integrating　Paillier　homomorphic　encryption　and　random　number　generation　technique,　all　gradients　and　their　ciphertexts　can　be　protected.　Meanwhile,　an　additive　secret-sharing　scheme　is　introduced　to　resist　potential　collusion　attacks　among　the　aggregation　server,　malicious　participants,　and　edge　nodes.　Moreover,　a　verification　scheme　based　on　discrete　logarithm　is　proposed,　which　can　not　only　verify　the　correctness　of　aggregated　results,　but　also　discover　lazy　aggregation　servers,　and　the　verification　overhead　can　be　reduced　by　over　half　compared　with　a　bilinear　aggregate　signature　method.　Finally,　theoretical　analysis　and　experiments　conducted　on　the　MNIST　data　set　prove　that　our　proposed　method　can　achieve　gradient　protection　and　correctness　verification　of　the　aggregated　results　with　higher　efficiency.