Output To Module SIS valid
ArkLib.Lattices.Ajtai.InnerOuter.WeakBinding.outputToModuleSIS_valid
Plain-language statement
A successful pair of weak openings yields a valid inner or outer Module-SIS witness (over 𝓜(q, α)). The boolean-verification wrapper around outputToModuleSIS_valid_of_verified: it extracts the VerifiedOpening facts from verify_weak and applies the core.
Exact Lean statement
theorem outputToModuleSIS_valid (base : ZMod q)
(hq5 : q % 8 = 5) (βSq γ κ : Nat) (hκ : κ ^ 2 < q)
(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)
(hwin : (openingsDiffer 𝓜(q, α) opening₁ opening₂ &&
verify_weak 𝓜(q, α) base βSq γ κ pp u opening₁ &&
verify_weak 𝓜(q, α) base βSq γ κ pp u opening₂) = true) :
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 (base : ZMod q) (hq5 : q % 8 = 5) (βSq γ κ : Nat) (hκ : κ ^ 2 < q) (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) (hwin : (openingsDiffer 𝓜(q, α) opening₁ opening₂ && verify_weak 𝓜(q, α) base βSq γ κ pp u opening₁ && verify_weak 𝓜(q, α) base βSq γ κ pp u opening₂) = true) : 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 simp only [Bool.and_eq_true] at hwin obtain ⟨⟨hdiff, hverify₁⟩, hverify₂⟩ := hwin exact outputToModuleSIS_valid_of_verified α hdiff (verifiedOpening_of_verify_eq_true α hq5 hκ hverify₁) (verifiedOpening_of_verify_eq_true α hq5 hκ hverify₂)- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Commitments/Functional/Hachi/InnerOuter/Security.lean:353-373
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Plain-language statement
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Person-level attribution pending.