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Stand-alone simulation proof: Ideal functions and leakage collection

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Are ideal functions and leakage collections the terms of UC security? Why I often see them in a simulation-based proof under stand-alone model? Some papers use the definition in Lindell's How to simulate it? while defining ideal functions (sometimes with leakage collections), which really confuse me a lot!

I sit in a Tesla and translated this thread with Ai:

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