Function binding
KZG.CommitmentScheme.function_binding
Plain-language statement
The KZG scheme satisfies function binding provided ARSDH holds.
Exact Lean statement
theorem function_binding {g₁ : G₁} {g₂ : G₂}
(L : ℕ) (hn : 1 ≤ n) (hp : p ≥ n + 2) (hg₁ : g₁ ≠ 1)
(hpair : pairing g₁ g₂ ≠ 0)
[SampleableType G₁] (arsdhError : ℝ≥0)
(hArsdh : Groups.arsdhAssumption (G₁ := G₁) (G₂ := G₂) (g₁ := g₁) (g₂ := g₂)
n arsdhError) :
Commitment.functionBinding (L := L) (init := pure ∅) (impl := randomOracle)
(hn := rfl)
(kzg (n := n) (g₁ := g₁) (g₂ := g₂) (pairing := pairing)) arsdhErrorFormal artifact
Lean source
theorem function_binding {g₁ : G₁} {g₂ : G₂} (L : ℕ) (hn : 1 ≤ n) (hp : p ≥ n + 2) (hg₁ : g₁ ≠ 1) (hpair : pairing g₁ g₂ ≠ 0) [SampleableType G₁] (arsdhError : ℝ≥0) (hArsdh : Groups.arsdhAssumption (G₁ := G₁) (G₂ := G₂) (g₁ := g₁) (g₂ := g₂) n arsdhError) : Commitment.functionBinding (L := L) (init := pure ∅) (impl := randomOracle) (hn := rfl) (kzg (n := n) (g₁ := g₁) (g₂ := g₂) (pairing := pairing)) arsdhError := by letI := Classical.decEq G₁ letI scheme := kzg (n := n) (g₁ := g₁) (g₂ := g₂) (pairing := pairing) simp only [Commitment.functionBinding] intro AuxState adversary letI game := Commitment.functionBindingGame (init := pure ∅) (impl := randomOracle) (hn := rfl) (AuxState := AuxState) (scheme := scheme) (adversary := adversary) letI game_ext := functionBindingGameExt (g₁ := g₁) (g₂ := g₂) AuxState adversary scheme change Pr[Commitment.functionBindingCondition (Data := Fin (n + 1) → ZMod p) | game] ≤ arsdhError exact calc Pr[Commitment.functionBindingCondition (Data := Fin (n + 1) → ZMod p) | game] _ = Pr[functionBindingCondExt n L | game_ext] := function_binding_game_ext_eq_function_binding_game (pairing := pairing) adversary _ ≤ Pr[(Groups.arsdhCondition n) ∘ mapFunctionBindingToArsdh hn | game_ext] := function_binding_cond_le_arsdh_cond (pairing := pairing) hn hp hg₁ hpair adversary _ = Pr[(Groups.arsdhCondition n) | mapFunctionBindingToArsdh hn <$> game_ext] := map_instance_drag hn adversary scheme _ = Groups.arsdhExperiment (g₁ := g₁) (g₂ := g₂) n (reduction (g₁ := g₁) (g₂ := g₂) (pairing := pairing) L hn AuxState adversary) := arsdh_game_eq (g₁ := g₁) (g₂ := g₂) (pairing := pairing) hn adversary _ ≤ arsdhError := arsdh_error_bound (g₁ := g₁) (g₂ := g₂) (pairing := pairing) hn arsdhError hArsdh adversary- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Commitments/Functional/KZG/FunctionBinding/Basic.lean:635-665
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affine_gaps_lifted_to_interleaved_codes
Project documentation
This lemma proves the final algebraic step in the DG25 Theorem 3.1 proof. It shows that if R > e + 1, then e * (R / (R - 1)) < e + 1. The intuition is that the fraction R / (R - 1) is always greater than 1, but as R gets larger, it gets closer to 1. The hypothesis R > e + 1 provides a strong enough bound to ensure the product e * (fraction) do...
Source project: ArkLib
Person-level attribution pending.
Gadget Decompose coeff
ArkLib.Lattices.Ajtai.gadgetDecompose_coeff
Plain-language statement
The k-th coefficient (k < deg φ) of a gadget-decomposition block is exactly the corresponding digit of the corresponding input coefficient.
Source project: ArkLib
Person-level attribution pending.
Gadget Decompose lawful
ArkLib.Lattices.Ajtai.gadgetDecompose_lawful
Plain-language statement
The base-b gadget decomposition is a lawful gadget decomposition.
Source project: ArkLib
Person-level attribution pending.