The 26-year-old, who left Accrington Stanley in the summer, has been praised by Hibernian's manager Pat Fenlon.
"He has been training with the group for the past few weeks and he has impressed us with both his ability and his attitude," Fenlon told the club's official website.
"He will come in and automatically provide us with experienced cover for Ben Williams. We felt that it would be beneficial for Calum to get a regular run of first-team games with East Fife. His loan move will give him more experience and help his development."
Murdoch has also played for Dunfermline.