I am not a cygwin user so totally unfamiliar with what it provides, but you should be able to answer most questions about include files and linking from the man page.
If cygwin includes man pages see
man crypt, or find it
online here.
Usually the name of libs recognized by the linker are used without the 'lib' prefix so
-lcrypt would imply
libcrypt. Some like libmath may recogize abbreviations such as
-lm as well.
Good luck!
UPDATE: I see you found the answer while I was updating my reply!
If your issue is solved please use the Thread Tools link above top post to make this question as SOLVED! Thanks!