T sdh denominator ne zero of opening equations
KZG.CommitmentScheme.t_sdh_denominator_ne_zero_of_opening_equations
Plain-language statement
If two accepted openings at the same query give different responses, the t-SDH denominator τ + (-query) cannot vanish. This is the small algebraic contradiction used to avoid a separate query = τ branch in the binding reduction.
Exact Lean statement
lemma t_sdh_denominator_ne_zero_of_opening_equations
(τ query resp₁ resp₂ cm prf₁ prf₂ : ZMod p) (hresp : resp₁ ≠ resp₂)
(hverifyEq₁ : cm - resp₁ = prf₁ * (τ - query))
(hverifyEq₂ : cm - resp₂ = prf₂ * (τ - query)) :
τ + -query ≠ 0Formal artifact
Lean source
lemma t_sdh_denominator_ne_zero_of_opening_equations (τ query resp₁ resp₂ cm prf₁ prf₂ : ZMod p) (hresp : resp₁ ≠ resp₂) (hverifyEq₁ : cm - resp₁ = prf₁ * (τ - query)) (hverifyEq₂ : cm - resp₂ = prf₂ * (τ - query)) : τ + -query ≠ 0 := by intro hzero have hτq : τ - query = 0 := by simpa [sub_eq_add_neg] using hzero have hcm₁ : cm = resp₁ := by simp only [hτq, MulZeroClass.mul_zero] at hverifyEq₁ exact sub_eq_zero.mp hverifyEq₁ have hcm₂ : cm = resp₂ := by simp only [hτq, MulZeroClass.mul_zero] at hverifyEq₂ exact sub_eq_zero.mp hverifyEq₂ exact hresp (hcm₁.symm.trans hcm₂)- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Commitments/Functional/KZG/Binding.lean:138-152
Reuse this declaration
Bring the exact result into your workflow
The import identifies the source module. Your project still needs the pinned package dependency shown on this page.
What this badge means
This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.
Continue in this project
Related declarations
Affine gaps lifted to interleaved codes
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.