Transparent PDFs using Illustrator

I have the trial version of Illustrator installed in Windows 2000. I have created a custom stamp in Adobe Acrobat 5.0, and it works well except the file is very large. Since I do not understand what I'm doing, here is what I have done.

We scanned a signature, saved it as a GIF with transparent background, dropped it into Illustrator, and saved is as a PDF. That file works as it should for a custom stamp in Acrobat, but it is 400KB.

The stamp files that are supplied with Acrobat are very small. I was told that the reason they are so small is that the graphics are vector instead of bitmap. So, I had a company digitize the signature in AutoCAD, save it as an EPS file, and drop it into Illustrator. They saved it as a PDF and got the same result -- a large file.

Can anyone educate me on a way to create a compact PDF file with a transparent background, whether vector or bitmap? The file only needs to be the size of a signature. I'm frustrated and otherwise lost on this project.