Research proof index

Pointwise weak-binding to Module-SIS bound for fixed samples (over 𝓜(q, α)).

Extract reusable weak-opening facts from a successful verification (over 𝓜(q, α), where Lyubashevsky–Seiler invertibility applies).

Binding reduces to Module-SIS for any commitment/Module-SIS shortness predicates closed under differences.

Core digit bound. Each base-b digit of zmodDigitDecomposition, viewed as a centered residue, has absolute value at most b - 1 , provided b - 1 ≤ q/2, so the digit (a natural number < b) does not wrap to a negative centered representative.

|R_q^H| ≤ q^k, from ψ : (R_q^H)^{d/k} ↪ R_q injective and |R_q| = q^{2^α}: |R_q^H|^{d/k} ≤ q^{2^α} = (q^k)^{d/k}.

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...