I was wondering what "securely realizes" means. I see this in some cryptographic papers but I don't know what it means for a protocol to "securely realize" a function $F$. Is it just a fancy way of saying "uses" or does it mean something else?
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A standard approach in cryptography is to separate out the ideal specification (of what a scheme is supposed to achieve) from the particular instantiation (the implementation of the scheme).
To specify the security and functionality goals, sometimes cryptographers specify an "ideal functionality", which is an idealization of what we are hoping to achieve. Then, one designs a specific instantiation -- a particular scheme -- and proves that the scheme meets the goals set out in the "ideal functionality". We describe this by saying that the scheme securely realizes the ideal functionality. How to prove that is another can of worms entirely... but that's what is meant by the terminology.
This kind of approach and terminology is common in, for instance, theoretical work that adopts an "ideal world/real world" formalization (including, e.g., simulation-based security; indifferentiability; universal composability; and other lines of work in cryptography). See Wikipedia on universal composability for a high-level introduction.