Outer relation of verified
ArkLib.Lattices.Ajtai.InnerOuter.WeakBinding.outer_relation_of_verified
Plain-language statement
Verified weak openings with different flattened witnesses give a valid outer relation.
Exact Lean statement
theorem outer_relation_of_verified {base : ZMod q} {βSq γ κ : Nat}
{pp : PublicParams Φ innerRows messageRows messageDigits outerRows blocks innerDigits}
{u : Commitment Φ outerRows}
{opening₁ opening₂ : Opening Φ innerRows messageRows messageDigits blocks innerDigits}
(hv₁ : VerifiedOpening Φ base βSq γ κ pp u opening₁)
(hv₂ : VerifiedOpening Φ base βSq γ κ pp u opening₂)
(hflat : PolyVec.flattenBlocks opening₁.innerDecomp ≠
PolyVec.flattenBlocks opening₂.innerDecomp) :
ModuleSIS.relation Φ (outerShort Φ γ)
pp.outerMatrix
(PolyVec.flattenBlocks opening₁.innerDecomp - PolyVec.flattenBlocks opening₂.innerDecomp)
= trueFormal artifact
Lean source
theorem outer_relation_of_verified {base : ZMod q} {βSq γ κ : Nat} {pp : PublicParams Φ innerRows messageRows messageDigits outerRows blocks innerDigits} {u : Commitment Φ outerRows} {opening₁ opening₂ : Opening Φ innerRows messageRows messageDigits blocks innerDigits} (hv₁ : VerifiedOpening Φ base βSq γ κ pp u opening₁) (hv₂ : VerifiedOpening Φ base βSq γ κ pp u opening₂) (hflat : PolyVec.flattenBlocks opening₁.innerDecomp ≠ PolyVec.flattenBlocks opening₂.innerDecomp) : ModuleSIS.relation Φ (outerShort Φ γ) pp.outerMatrix (PolyVec.flattenBlocks opening₁.innerDecomp - PolyVec.flattenBlocks opening₂.innerDecomp) = true := by have hne : PolyVec.flattenBlocks opening₁.innerDecomp - PolyVec.flattenBlocks opening₂.innerDecomp ≠ 0 := sub_ne_zero.mpr hflat have hshort : vecLInftyNorm Φ (PolyVec.flattenBlocks opening₁.innerDecomp - PolyVec.flattenBlocks opening₂.innerDecomp) ≤ subLInftyNormBound γ := sub_lInftyNorm_le Φ _ _ hv₁.outer_short hv₂.outer_short have heq : pp.outerMatrix *ᵥ PolyVec.flattenBlocks opening₁.innerDecomp = pp.outerMatrix *ᵥ PolyVec.flattenBlocks opening₂.innerDecomp := by simpa [Simple.commit] using hv₁.outer_eq.trans hv₂.outer_eq.symm have hker : pp.outerMatrix *ᵥ (PolyVec.flattenBlocks opening₁.innerDecomp - PolyVec.flattenBlocks opening₂.innerDecomp) = 0 := by rw [matVecMul_sub]; exact sub_eq_zero.mpr heq simp [ModuleSIS.relation, outerShort, hne, hshort, hker]- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Commitments/Functional/Hachi/InnerOuter/Security.lean:290-315
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Person-level attribution pending.
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Plain-language statement
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Source project: ArkLib
Person-level attribution pending.
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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.