Historic Authentication Schemes Before Computers
As explained by Cort Ammon, they mostly relied on physical, rather than mathematical processes.

Here is a video on Youtube titled "How to launch a nuclear missile", taken in the Titan Missile Museum mentioned by Cort Ammon.

The message contains a seven character string. 
The personnel in the shelter have a set of envelopes with the first two letters on the outside. 
If the inside of the envelope matches the message, they launch.

Here is another one from a British sub, starting at 23:00. Decryption is done behind a curtain.

It is all very manual, no computer is visible.
Note the wooden club used to protect the codes (24:17).