As you do not wan't to use 3rd party libraries and no .NET over 3.5 you have some problems...
System.IO.Compression[
^] namespace have a DeflateStream and GZipStream class - since .NET 2.0 - to compressing files. As
deflate[
^] is the original algorithm for PKZIP (the grandfather of modern zip) so any tool capable to read zip will read it happily...It's not exactly a modern zip format in the meaning of ability, but it is compatible with, so you may go with it...