Const Rq mul coeff
ArkLib.Lattices.CyclotomicModulus.Rq.constRq_mul_coeff
Plain-language statement
Multiplying by the constant constRq Φ c scales coefficients by c.
Exact Lean statement
theorem constRq_mul_coeff (h1 : 1 ≤ Φ.φ.natDegree) (c : R) (x : Rq Φ) (k : ℕ) :
(constRq Φ c * x).1.coeff k = c * x.1.coeff kFormal artifact
Lean source
theorem constRq_mul_coeff (h1 : 1 ≤ Φ.φ.natDegree) (c : R) (x : Rq Φ) (k : ℕ) : (constRq Φ c * x).1.coeff k = c * x.1.coeff k := by have hmul : (constRq Φ c * x).1 = Φ.reduce ((constRq Φ c).1 * x.1) := rfl have hred : Φ.reduce (CompPoly.CPolynomial.C c * x.1) = CompPoly.CPolynomial.C c * x.1 := by apply Φ.reduce_eq_self_of_degree_lt rw [CompPoly.CPolynomial.toPoly_mul, CompPoly.CPolynomial.toPoly_C] have hx : x.1.toPoly.degree < Φ.φ.toPoly.degree := Φ.degree_toPoly_lt_of_reduced x.2 rcases eq_or_ne c 0 with hc | hc · simpa [hc] using lt_of_le_of_lt bot_le hx · rwa [Polynomial.degree_C_mul hc] rw [hmul, constRq_val Φ h1, hred] exact CompPoly.CPolynomial.coeff_C_mul x.1 c k- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/Lattices/CyclotomicRing/Rq.lean:347-358
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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...
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Person-level attribution pending.
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Plain-language statement
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Source project: ArkLib
Person-level attribution pending.
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Plain-language statement
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Source project: ArkLib
Person-level attribution pending.