1. Don’t tell Richard anything you don’t want everyone else to know. It’s impossible for him to keep anything under his hat
2. I’m not telling anyone yet, but Tom and I are getting married. Keep it under your hat, okay?

Antonyms: spill the beans; let the cat out of the bag. This phrase originates from the 1800s, when many men and women wore hats. The idea is to keep a secret in your head, underneath a hat.