feat(Crypto): define PRGs and their security definition - #876
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| abbrev Family (Seed Output : ℕ → Type*) := ∀ n, Generator (Seed n) (Output n) | ||
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| namespace Family | ||
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| variable {Seed Output : ℕ → Type*} | ||
| variable [∀ n, Fintype (Seed n)] [∀ n, Nonempty (Seed n)] | ||
| variable [∀ n, Fintype (Output n)] [∀ n, Nonempty (Output n)] |
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Why are Generators arbitrary types? Couldn't they just be instances of a structure. Then if the structure got a Fintype or Finite instance, there would be no need to declare [forall n, Fintype (Seed n)] etc. This is just a design comment.
| /-- An unbounded adversary tests whether its input is in the generator's range. | ||
| It is not assumed to be admissible for a computationally restricted class. -/ | ||
| noncomputable def rangeAdversary (G : Generator Seed Output) : Adversary Output := by | ||
| classical |
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Is this actually needed here? To me it sounds like
:= fun output => PMF.pure (decide (output ∈ Set.range G)) might work
| /-- Zero-error security against arbitrary tests is equivalent to exactly uniform output. -/ | ||
| theorem secure_zero_iff_outputDist_eq_uniform (G : Generator Seed Output) : | ||
| G.Secure (fun _ => True) 0 ↔ G.outputDist = PMF.uniformOfFintype Output := by | ||
| classical |
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It seems better to explicitly declare what Props need to be Decidable than use classical.
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Formalizes Boneh-Shoup's definitions of PRGs and proves the impossibility of making an extending PRG against an arbitrary adversary. Takes the approach of #766 by defining the construction and proving the impossibility in certain contexts, motivating the future computational security definitions.
Composed with Astra using Codex based on some code that I had written by hand.