Perfectly Correct
ArkLib.Lattices.Ajtai.InnerOuter.perfectlyCorrect
Plain-language statement
Unconditional perfect correctness with the concrete binary decomposition. Both message and inner decompositions are the genuine base-b digit decomposition of ZMod q (zmodDigitDecomposition, the Hachi gadget inverse G⁻¹). All weak-verifier side conditions are discharged automatically: the trivial challenge cᵢ = 1 is short (Rq.l1Norm_one), a...
Exact Lean statement
theorem perfectlyCorrect (b κ : ℕ) (hb : 1 < b) (hκ : 1 ≤ κ) (hbq : b - 1 ≤ q / 2)
(hdeg : 1 ≤ Φ.φ.natDegree) (hmsg : 0 < messageDigits) (hinner : 0 < innerDigits)
(hqm : q ≤ b ^ messageDigits) (hqi : q ≤ b ^ innerDigits) :
(commitmentScheme Φ (messageRows := messageRows) (innerRows := innerRows)
(outerRows := outerRows) (blocks := blocks) (b : ZMod q)
(messageRows * messageDigits * (Φ.φ.natDegree * (b - 1) ^ 2))
(b - 1) κ
(Decomposition.ofDigits Φ (zmodDigitDecomposition b messageDigits hb hqm)
(zmodDigitDecomposition b innerDigits hb hqi))).PerfectlyCorrectFormal artifact
Lean source
theorem perfectlyCorrect (b κ : ℕ) (hb : 1 < b) (hκ : 1 ≤ κ) (hbq : b - 1 ≤ q / 2) (hdeg : 1 ≤ Φ.φ.natDegree) (hmsg : 0 < messageDigits) (hinner : 0 < innerDigits) (hqm : q ≤ b ^ messageDigits) (hqi : q ≤ b ^ innerDigits) : (commitmentScheme Φ (messageRows := messageRows) (innerRows := innerRows) (outerRows := outerRows) (blocks := blocks) (b : ZMod q) (messageRows * messageDigits * (Φ.φ.natDegree * (b - 1) ^ 2)) (b - 1) κ (Decomposition.ofDigits Φ (zmodDigitDecomposition b messageDigits hb hqm) (zmodDigitDecomposition b innerDigits hb hqi))).PerfectlyCorrect := by refine perfectlyCorrect_of_digits Φ (b : ZMod q) _ _ κ hdeg hmsg hinner _ _ ?_ ?_ ?_ ?_ · rw [Rq.l1Norm_one Φ hdeg]; norm_num · rw [Rq.l1Norm_one Φ hdeg]; exact hκ · intro pp m i exact gadgetDecompose_zmod_vecL2NormSq_le Φ hb hqm hbq (m i) · intro pp m exact vecLInftyNorm_flattenBlocks_le Φ _ (fun i => gadgetDecompose_zmod_vecLInftyNorm_le Φ hb hqi hbq _)- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Commitments/Functional/Hachi/InnerOuter/Correctness.lean:222-238
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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.