# Game-based proofs and semantic security

After reading this question, I understood that "game-based" and "simulation-based" have to do with the way that security properties are postulated. In particular, IND-CPA is a canonical example of game-based security property and semantic security is a canonical example of simulation-based security property. Then, I guess that the "game-based proof method" refers to doing proofs for this kind of security statements. Did I get this right?

By looking back at Shoup's tutorial on game-based proofs, I see that "semantic security" of ElGamal is his first example. How?! Isn't that the canonical example of the simulation-based proof method? Is it because he postulates "semantic security" as "polynomial indistinguishability" which is game-based? Where does my confusion comes from? Is that my intuition (a "simulation-based proof" is a proof of a simulation-based security definition) is wrong?

• Shoup literally states "this definition comes from [GM84], where is called polynomial indistinguishability, and semantic security is actually the name of a syntactically different, but equivalent, characterization". – Maeher Feb 21 '20 at 8:44
• @Maeher this means that he does a game-based proof for polynomial indistinguishably, which is a game-based definition, and this is why he follows the game-based approach? – CTN Feb 21 '20 at 13:02