Research proof index

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...

Main combinatorial bound (at most two contributions per position). For p < d, the p-th coefficient of ψ(a) is ± c₁ ± c₂ where c₁, c₂ are coefficients of the (at most two) entries of a whose index matches p's residue class mod d/2k (j₁ := p mod d/2k from the first half, j₂ := d/2k + p mod d/2k from the second); all other summands...

Vanishing outside the matching residue classes. If j's index does not match p's residue class mod d/2k, the p-th coefficient of the j-th packed summand a_j · X^{packExp j} is 0. Combined with packExp_mod_eq and the support of fixed elements (fixedSubring_coeff_eq_zero), only summands whose index agrees with p mod d/2k can contrib...