C Inf Norm psi le
ArkLib.Lattices.CyclotomicModulus.cInfNorm_psi_le
Plain-language statement
Hachi [NOZ26, §3, Lemma 6]: ‖ψ(a)‖∞ ≤ 2β. If every entry of the vector a : (R_q^H)^{d/k} has centered coefficient ℓ∞-norm at most β (i.e. ‖a‖∞ ≤ β in the mod± q convention of [NOZ26, §2.1]), then the packed ring element ψ(a) ∈ R_q satisfies ‖ψ(a)‖∞ ≤ 2β. The hypotheses are the standing assumptions of [NOZ26, §3]: k = 2^κ divides `d/2...
Exact Lean statement
theorem cInfNorm_psi_le (α κ : ℕ) (h2 : (2 : ZMod q) ≠ 0) (hk : 2 * 2 ^ κ ∣ 2 ^ α) {β : ℕ}
(a : Fin (2 ^ α / 2 ^ κ) → fixedSubring (R := ZMod q) α (2 ^ κ))
(ha : (zmodCenteredView q).vecCInfNorm
(fun i => ((a i : Rq (powTwoCyclotomic (R := ZMod q) α))).1) ≤ β) :
(zmodCenteredView q).cInfNorm (psi α (2 ^ κ) a).1 ≤ 2 * βFormal artifact
Lean source
theorem cInfNorm_psi_le (α κ : ℕ) (h2 : (2 : ZMod q) ≠ 0) (hk : 2 * 2 ^ κ ∣ 2 ^ α) {β : ℕ} (a : Fin (2 ^ α / 2 ^ κ) → fixedSubring (R := ZMod q) α (2 ^ κ)) (ha : (zmodCenteredView q).vecCInfNorm (fun i => ((a i : Rq (powTwoCyclotomic (R := ZMod q) α))).1) ≤ β) : (zmodCenteredView q).cInfNorm (psi α (2 ^ κ) a).1 ≤ 2 * β := by rw [CenteredCoeffView.cInfNorm] apply Finset.sup_le intro p _ change (ZMod.valMinAbs ((psi α (2 ^ κ) a).1.coeff p)).natAbs ≤ 2 * β by_cases hp : p < 2 ^ α · exact coeff_psi_abs_le_two_vecCInfNorm q α κ h2 hk a ha hp · rw [coeff_eq_zero_of_le α (psi α (2 ^ κ) a) (by omega), ZMod.valMinAbs_zero] exact Nat.zero_le _- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/Lattices/CyclotomicRing/Subfield/NormBound.lean:165-177
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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.