software downloads from the internet "is damaged and can’t be opened. You should move it to the Trash"

I am unable to open download software from the internet (chrome or safari). The Error message is filename.dmg is damaged and can’t be opened. You should move it to the Trash. I am attempting to download legal and trusted software from known sites. I am currently under the impression that this is a glitch in the gatekeeper, since the function of allow downloads from "anywhere" feature was removed in past software releases.