Project-declaredLean 4.32.0 · mathlib@81a5d257

Extractability

InductiveMerkleTree.extractability

program verification separation logic cryptography extractability

Project documentation

The extractability theorem for Merkle trees. Adapting from the SNARGs book Lemma 18.5.1: For any adversary 𝒜 whose committing and opening phases together obey the two-phase total query bound qb, if the game runs 𝒜.commit and 𝒜.opening, and the extractor algorithm is run on the resulting cache and root, then with probability at most κ does `𝒜...

Exact Lean statement

theorem extractability [DecidableEq α] [Fintype α] [Inhabited α]
    {s : Skeleton} (𝒜 : Adversary α s) (qb : ℕ)
    (h_IsQueryBound_qb : 𝒜.IsTwoPhaseTotalQueryBound qb) :
    Pr[AdversaryWinsExtractabilityGame |
        extractabilityGame 𝒜] ≤
        ((qb + s.depth) ^ 2 : ENNReal) / (2 * Fintype.card α)
        + 1 / (Fintype.card α)

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

theorem extractability [DecidableEq α] [Fintype α] [Inhabited α]    {s : Skeleton} (𝒜 : Adversary α s) (qb : ℕ)    (h_IsQueryBound_qb : 𝒜.IsTwoPhaseTotalQueryBound qb) :    Pr[AdversaryWinsExtractabilityGame |        extractabilityGame 𝒜] ≤        ((qb + s.depth) ^ 2 : ENNReal) / (2 * Fintype.card α)        + 1 / (Fintype.card α) := by  calc    _ = Pr[AdversaryWinsExtractabilityGame ∘ Prod.fst |          (extractabilityGame 𝒜).withQueryLog] :=      (probEvent_withQueryLog _ _).symm    _ ≤ Pr[fun (vals, log) =>            LogHasCollision log ∨            (¬ LogHasCollision log ∧ AdversaryWinsExtractabilityGame vals) |          (extractabilityGame 𝒜).withQueryLog] :=      probEvent_mono'' fun ⟨_, _⟩ => by tauto    _ ≤ Pr[fun (vals, log) => LogHasCollision log |            (extractabilityGame 𝒜).withQueryLog] +        Pr[fun (vals, log) =>            ¬ LogHasCollision log ∧ AdversaryWinsExtractabilityGame vals |          (extractabilityGame 𝒜).withQueryLog] :=      probEvent_or_le ..    _ ≤ ((qb + s.depth) ^ 2 : ENNReal) / (2 * Fintype.card α) +        Pr[fun (vals, log) =>            ¬ LogHasCollision log ∧ AdversaryWinsExtractabilityGame vals |          (extractabilityGame 𝒜).withQueryLog] := by      gcongr      convert OracleComp.probEvent_logCollision_le_birthday_total (spec := spec α)        (extractabilityGame 𝒜) (qb + s.depth)        (extractabilityGame_isTotalQueryBound 𝒜 qb h_IsQueryBound_qb)        (fun _ => le_rfl) using 2      · rfl      all_goals norm_cast    _ ≤ ((qb + s.depth) ^ 2 : ENNReal) / (2 * Fintype.card α) +        1 / (Fintype.card α) := by      have h' := extractabilityGame_not_logHasCollision_wins_le_inv_card 𝒜      gcongr; norm_cast

Project: VCVio
License: Apache-2.0
Commit: 2ceb2d825ee3
Source: VCVio/CryptoFoundations/MerkleTree/Inductive/Extractability.lean:678-714

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