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Proof systems (provable security) are seen in many research papers related to cryptography and security. Where would I be able to find a comprehensive resource covering the purpose and the fundamentals of proof systems, and its common approaches such as the random oracle model method?

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First, as a heads up "approaches to provable security" or "approaches to security proofs" (which it sounds like you are interested in) means something different than "proof systems" in cryptography. Proof systems are a specific construct that are studied within cryptography (similar to encryption or signatures), of which zero-knowledge proofs are the best known example. The fact that there is a naming collision here is unfortunate, but its best to just keep it in mind and sometimes uses more awkward phrasing to avoid the collision.

There are a number of slightly different competing notions of "proof systems". I don't know a resource that collects them all into one document, but two popular ones are:

  1. Bellare and Rogaway's "Game-playing" model, which you can find exposition on here

  2. The "simulation" proof technique, which you can find exposition on here

As for the random oracle model, I believe chapter 6 from Katz's book Digital Signatures is probably a good pointer, although if you do not have institutional access it may not be freely available.

EDIT: Browsing eprint, I found this chapter on hybrid arguments, which also may be interesting.

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