Mem rel Poly Eval of rel In
ArkLib.Lattices.Ajtai.InnerOuter.mem_relPolyEval_of_relIn
Project documentation
Pull-back lemma (the hRel for the bridge's CWSS): a QuadEvalWitness accepted by QuadEval's relIn at the reinterpreted statement toQuadEvalStatement Φ s is accepted by relPolyEval at the polynomial-level statement s. The MSIS disjuncts are preserved verbatim (toQuadEvalStatement keeps pp); the opening disjunct converts the matrix-leve...
Exact Lean statement
theorem mem_relPolyEval_of_relIn (base : ZMod q) (βSq γ κ : ℕ)
(s : PolyEvalStatement Φ innerRows messageDigits outerRows innerDigits dRows m r)
(w : QuadEvalWitness Φ innerRows (2 ^ m) messageDigits (2 ^ r) innerDigits)
(h : (toQuadEvalStatement Φ s, w) ∈ relIn Φ base βSq γ κ) :
(s, w) ∈ relPolyEval Φ base βSq γ κFormal artifact
Lean source
theorem mem_relPolyEval_of_relIn (base : ZMod q) (βSq γ κ : ℕ) (s : PolyEvalStatement Φ innerRows messageDigits outerRows innerDigits dRows m r) (w : QuadEvalWitness Φ innerRows (2 ^ m) messageDigits (2 ^ r) innerDigits) (h : (toQuadEvalStatement Φ s, w) ∈ relIn Φ base βSq γ κ) : (s, w) ∈ relPolyEval Φ base βSq γ κ := by cases w with | opening o => obtain ⟨hvo, hec⟩ := h refine ⟨hvo, ?_⟩ change CMlPolynomial.eval (Hachi.toPolynomial (derivedMsgMatrix Φ base o)) (s.xl ++ s.xh) = s.y rw [← Hachi.splitForm_monomialBasis_eq_eval (derivedMsgMatrix Φ base o) s.xl s.xh] exact hec | msisB z => exact h | msisD z => exact h- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Commitments/Functional/Hachi/QuadEval/Bridge.lean:186-200
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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.