Newbie question about anti theft alarm

Two days ago my 2003 LeSabre would not start. I seemed to have power, as the windows would work, radio, etc. I had a secrutiy key alarm indicator flashing on my dash indicator panel. I could not reset the alarm. I had the car towed to a dealer and he said the anti theft alarm was broken and I needed a new one. He installed a new one, then called and said my battery is no good.

My question is: can a bad or low battery cause a failure in the anit theft device. Might I have had a bad battery all the time, and this caused the anit theft device to malfunction?