|laziness, impatience, and hubris|
I knew that my algorithm couldn't have been as spiffy as it seemed, and I knew that if I posted it, someone would be able to point me towards what I'm missing. :) Thank you for that.
If nothing else, though, this has certainly been an interesting and educational exercise. And, I've found out the correct term to google on for this class of software: "Key Sharing". I had googled all over the place to find out if someone had already done what I was trying to do, or found an even better scheme. Just didn't know the right terminology. That blasted self-taught thing bites me sometimes. ;-)