Give out x Give up: Quando utilizar

If your company doesn't wish to reveal its customer names, it won't:

a) give out their names; or
b) give up their names?

Please provide a rationale behind your choice if you have one.

My copy of Webster's Third New International Dictionary on CD gives one of the senses of b) as meaning "to make public, reveal." And no, I won't throw the dictionary out the window because worthless.