Question on cryptographic advantage
If you're totally lost, you might not have the needed background to get much use out of the paper.  The paper draws on many concepts and foundational ideas from theoretical cryptography (including how to formalize notions of security for various cryptographic tasks), and it assumes familiarity with those ideas.  If the definition looks confusing, it's probably because you don't have that familiarity; the paper's intended audience was cryptographers who already have that familiarity.

You might want to spend a few weeks studying theoretical cryptography (foundations of cryptography), e.g., from Lindell-Katz, the Goldreich book, and/or many lecture notes online on the foundations of cryptography.  You'll find that this kind of definition is routine, and studying that material will prepare you to be able to understand this material and work through the definitions on your own.