Research proof index

Lyubashevsky–Seiler: short elements are invertible (LS18, Cor. 1.2; Hachi, Lemma 3). Over the power-of-two cyclotomic modulus powTwoCyclotomic α (φ = X^{2^α}+1) with a prime q ≡ 5 (mod 8), a nonzero element of Rq (powTwoCyclotomic α) with centered ℓ₁ norm ≤ κ and κ² < q is a unit: by the algebraic core a non-unit forces q ∣ ‖c‖₂², whil...

X^e + σ_{-1}(X^e) ∈ R_q^H when d/2k ∣ e: X^e is then σ_{4k+1}-fixed (its exponent is a multiple of d/2k), and the sum is symmetric under σ_{-1} (which has order 2).

(C-3 helper) aeval (X^n) and aeval (X^{n mod 2^{α+1}}) agree in the quotient.

Full coefficient of a (possibly high-degree) scaled monomial. For p < d, the p-th coefficient of the reduced c·X^m is (-1)^{m/d}·c at the folded position p = m mod d, and 0 elsewhere , the sign records how many times the exponent wrapped past X^d = -1.

The j-th coefficient of the reduced c·X^i (for i < d) is c·[j = i].

The scaled monomial c·X^j ∈ Rq Φ factors as (constant c)·X^j.