Output To Module SIS valid of verified
ArkLib.Lattices.Ajtai.InnerOuter.WeakBinding.outputToModuleSIS_valid_of_verified
Plain-language statement
Extractor validity (the reusable core). Two verified weak openings that differ yield a valid inner or outer Module-SIS witness (over 𝓜(q, α)). This is stated directly on the extracted VerifiedOpening facts, independent of how those facts were obtained , so it is reused both by the weak-binding bound below (where the VerifiedOpenings come fr...
Exact Lean statement
theorem outputToModuleSIS_valid_of_verified {base : ZMod q} {βSq γ κ : Nat}
{pp : PublicParams 𝓜(q, α)
innerRows messageRows messageDigits outerRows blocks innerDigits}
{u : Commitment 𝓜(q, α) outerRows}
{opening₁ opening₂ :
Opening 𝓜(q, α) innerRows messageRows messageDigits blocks innerDigits}
(hdiff : openingsDiffer 𝓜(q, α) opening₁ opening₂ = true)
(hv₁ : VerifiedOpening 𝓜(q, α) base βSq γ κ pp u opening₁)
(hv₂ : VerifiedOpening 𝓜(q, α) base βSq γ κ pp u opening₂) :
match outputToModuleSIS 𝓜(q, α) opening₁ opening₂ with
| Sum.inl z =>
ModuleSIS.relation 𝓜(q, α) (innerShort 𝓜(q, α) κ βSq) pp.innerMatrix z = true
| Sum.inr z =>
ModuleSIS.relation 𝓜(q, α) (outerShort 𝓜(q, α) γ)
pp.outerMatrix z = trueFormal artifact
Lean source
theorem outputToModuleSIS_valid_of_verified {base : ZMod q} {βSq γ κ : Nat} {pp : PublicParams 𝓜(q, α) innerRows messageRows messageDigits outerRows blocks innerDigits} {u : Commitment 𝓜(q, α) outerRows} {opening₁ opening₂ : Opening 𝓜(q, α) innerRows messageRows messageDigits blocks innerDigits} (hdiff : openingsDiffer 𝓜(q, α) opening₁ opening₂ = true) (hv₁ : VerifiedOpening 𝓜(q, α) base βSq γ κ pp u opening₁) (hv₂ : VerifiedOpening 𝓜(q, α) base βSq γ κ pp u opening₂) : match outputToModuleSIS 𝓜(q, α) opening₁ opening₂ with | Sum.inl z => ModuleSIS.relation 𝓜(q, α) (innerShort 𝓜(q, α) κ βSq) pp.innerMatrix z = true | Sum.inr z => ModuleSIS.relation 𝓜(q, α) (outerShort 𝓜(q, α) γ) pp.outerMatrix z = true := by unfold outputToModuleSIS by_cases hflat : PolyVec.flattenBlocks opening₁.innerDecomp = PolyVec.flattenBlocks opening₂.innerDecomp · obtain ⟨i, hfind⟩ := firstDiff?_some_of_differs hdiff have hmsgNe : opening₁.message i ≠ opening₂.message i := firstDiff?_eq_some_ne hfind simp only [hflat, if_true, hfind] exact inner_relation_of_verified α hv₁ hv₂ hflat hmsgNe · simp only [hflat, if_false] exact outer_relation_of_verified 𝓜(q, α) hv₁ hv₂ hflat- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Commitments/Functional/Hachi/InnerOuter/Security.lean:325-348
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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.