All proofs
Project-declaredLean 4.31.0 · mathlib@fabf563a7c95

Verified Opening of verify eq true

ArkLib.Lattices.Ajtai.InnerOuter.WeakBinding.verifiedOpening_of_verify_eq_true

Plain-language statement

Extract reusable weak-opening facts from a successful verification (over 𝓜(q, α), where Lyubashevsky–Seiler invertibility applies).

Exact Lean statement

theorem verifiedOpening_of_verify_eq_true {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}
    {opening : Opening 𝓜(q, α) innerRows messageRows messageDigits blocks innerDigits}
    (hverify : verify_weak 𝓜(q, α) base βSq γ κ pp u opening = true) :
    VerifiedOpening 𝓜(q, α) base βSq γ κ pp u opening

Formal artifact

Lean source

Canonical source
Full Lean sourceLean 4
theorem verifiedOpening_of_verify_eq_true {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}    {opening : Opening 𝓜(q, α) innerRows messageRows messageDigits blocks innerDigits}    (hverify : verify_weak 𝓜(q, α) base βSq γ κ pp u opening = true) :    VerifiedOpening 𝓜(q, α) base βSq γ κ pp u opening := by  simp only [verify_weak, Bool.and_eq_true] at hverify  obtain ⟨⟨hall, hgamma, houter := hverify  refine (Simple.verify_eq_true_iff 𝓜(q, α) _ _ u ()).1 houter, by simpa using hgamma,    fun i => ?_  rw [List.all_eq_true] at hall  have hb := hall i (List.mem_finRange i)  simp only [Bool.and_eq_true, decide_eq_true_eq] at hb  obtain ⟨⟨⟨hpos, hshort, hscaled, hinner := hb  exact isUnit_of_l1Norm_le α hq5 hpos hshort hκ, hshort, hscaled,    (Simple.verify_eq_true_iff 𝓜(q, α) _ _ _ ()).1 hinner
Project
ArkLib
License
Apache-2.0
Commit
fad5cbf80877
Source
ArkLib/Commitments/Functional/Hachi/InnerOuter/Security.lean:183-200

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

Project-declaredLean 4.31.0

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...

cryptographyproof systemscoding theory

Source project: ArkLib

Person-level attribution pending.

View proof record
Project-declaredLean 4.31.0

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.

cryptographyproof systemscoding theory

Source project: ArkLib

Person-level attribution pending.

View proof record
Project-declaredLean 4.31.0

Gadget Decompose lawful

ArkLib.Lattices.Ajtai.gadgetDecompose_lawful

Plain-language statement

The base-b gadget decomposition is a lawful gadget decomposition.

cryptographyproof systemscoding theory

Source project: ArkLib

Person-level attribution pending.

View proof record