Plain-language statement
The KZG scheme satisfies evaluation binding provided t-SDH holds.
Exact Lean statement
theorem binding {g₁ : G₁} {g₂ : G₂} (hg₁ : g₁ ≠ 1)
(hpair : pairing g₁ g₂ ≠ 0) [SampleableType G₁] (tSdhError : ℝ≥0)
(htSdh : Groups.tSdhAssumption (p := p) (G₁ := G₁) (G₂ := G₂) (g₁ := g₁)
(g₂ := g₂) n tSdhError) :
Commitment.binding (init := pure ∅) (impl := randomOracle)
(kzg (n := n) (g₁ := g₁) (g₂ := g₂) (pairing := pairing)) tSdhErrorFormal artifact
Lean source
theorem binding {g₁ : G₁} {g₂ : G₂} (hg₁ : g₁ ≠ 1) (hpair : pairing g₁ g₂ ≠ 0) [SampleableType G₁] (tSdhError : ℝ≥0) (htSdh : Groups.tSdhAssumption (p := p) (G₁ := G₁) (G₂ := G₂) (g₁ := g₁) (g₂ := g₂) n tSdhError) : Commitment.binding (init := pure ∅) (impl := randomOracle) (kzg (n := n) (g₁ := g₁) (g₂ := g₂) (pairing := pairing)) tSdhError := by letI scheme := kzg (n := n) (g₁ := g₁) (g₂ := g₂) (pairing := pairing) simp only [Commitment.binding] intro AuxState adversary letI game := Commitment.bindingGame (init := pure ∅) (impl := randomOracle) (AuxState := AuxState) (scheme := scheme) (adversary := adversary) letI game_ext := bindingGameExt (g₁ := g₁) (g₂ := g₂) AuxState adversary scheme change Pr[Commitment.bindingCondition (Data := Fin (n + 1) → ZMod p) | game] ≤ tSdhError exact calc Pr[Commitment.bindingCondition (Data := Fin (n + 1) → ZMod p) | game] _ = Pr[bindingCondExt (p := p) (n := n) | game_ext] := binding_game_ext_eq_binding_game (pairing := pairing) adversary _ ≤ Pr[(Groups.tSdhCondition (p := p) (g₁ := g₁)) ∘ mapBindingToTsdh (p := p) (n := n) | game_ext] := binding_cond_le_t_sdh_cond (pairing := pairing) hg₁ hpair adversary _ = Pr[Groups.tSdhCondition (p := p) (g₁ := g₁) | mapBindingToTsdh (p := p) (n := n) <$> game_ext] := map_binding_instance_drag adversary scheme _ = Groups.tSdhExperiment (g₁ := g₁) (g₂ := g₂) n (bindingReduction (g₁ := g₁) (g₂ := g₂) (pairing := pairing) AuxState adversary) := t_sdh_game_eq (g₁ := g₁) (g₂ := g₂) (pairing := pairing) adversary _ ≤ tSdhError := t_sdh_error_bound (g₁ := g₁) (g₂ := g₂) (pairing := pairing) tSdhError htSdh adversary- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Commitments/Functional/KZG/Binding.lean:743-770
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affine_gaps_lifted_to_interleaved_codes
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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.