Let $E(\mathbb{F_q})$ - elliptic curve.

$G_1 = E(\mathbb{F_q})[r]$. $|G_1| = r$.

$k$ is minimal such $r | q^k - 1$.

$\pi_q$ - $q$-power Frobenius endomorphism.

$G_2 = E(\mathbb{F_{q^k}})[r] \cap Ker([q] - \pi_q)$.

$\mu_r =$ {$z \in \mathbb{F_{q^k}}: z^r = 1$}.

$f_{s, P} \in \mathbb{F_q}(E)$: div$(f_{s, P}) =s(P) - ([s]P) - (s-1)(O)$.

Let $f_{s, P}^{(q^k - 1)/r}(Q)$ is non-degenerate bilinear pairing from $G_1 \times G_2 \to \mu_r$

Is it true that $r | s$ or $r | (s^k - 1)$ ?