Galois Aut sum to Poly eq aeval
ArkLib.Lattices.CyclotomicModulus.galoisAut_sum_toPoly_eq_aeval
Plain-language statement
(S5) The core polynomial identity behind soundness: the monomial-remapped sum (before reduction) equals aeval (X^i) of the underlying polynomial. Both sides are ∑_{k<d} X^{ki}·a_k.
Exact Lean statement
theorem galoisAut_sum_toPoly_eq_aeval (α i : ℕ) (a : Rq (powTwoCyclotomic (R := R) α)) :
(∑ k ∈ range (powTwoCyclotomic (R := R) α).φ.natDegree,
CompPoly.CPolynomial.monomial (k * i) (a.1.coeff k)).toPoly
= Polynomial.aeval (Polynomial.X ^ i : Polynomial R) a.1.toPolyFormal artifact
Lean source
theorem galoisAut_sum_toPoly_eq_aeval (α i : ℕ) (a : Rq (powTwoCyclotomic (R := R) α)) : (∑ k ∈ range (powTwoCyclotomic (R := R) α).φ.natDegree, CompPoly.CPolynomial.monomial (k * i) (a.1.coeff k)).toPoly = Polynomial.aeval (Polynomial.X ^ i : Polynomial R) a.1.toPoly := by rw [toPoly_sum, show a.1.toPoly = ∑ k ∈ range (powTwoCyclotomic (R := R) α).φ.natDegree, Polynomial.monomial k (a.1.toPoly.coeff k) from a.1.toPoly.as_sum_range' _ (Rq.natDegree_val_toPoly_lt α a), map_sum] refine Finset.sum_congr rfl (fun k _ => ?_) rw [toPoly_monomial, aeval_X_pow_monomial, coeff_toPoly]- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/Lattices/CyclotomicRing/Galois/Automorphism.lean:170-180
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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.