SIDH cryptosystem question I'm trying to understand the SIDH cryptosystem and got confused at this point:

Alice fixes base $\{P_A,Q_A\}$ so that it generates $E_0[l_A^{e_A}]$. Then she chooses secret parameters $m_A,n_A$ and computes the secret isogeny $\phi_A: E_0 \to E_A$  with kernel $\langle [m_A]P_A +[n_A]Q_A\rangle$. Further section 4 of the 2011 SIDH paper states that both $P_A$ and $Q_A$ have order $l_A^{e_A}$ and are independent of each other. 

Doesn't that mean that $[m_A]P_A$ and $[n_A]Q_A$ also have order $l_A^{e_A}$ and therefore $ker(\phi_A) = E_0[l_A^{e_A}]$? That would mean that $\phi_A$ is not secret.
I invented SIDH. $E_0[\ell_A^{e_A}]$ has cardinality $(\ell_A^{e_A})^2$. Each of $P_A$, $Q_A$, and $R$ has order $\ell_A^{e_A}$ and they all generate different subgroups. This is possible because $E_0[\ell_A^{e_A}]$ is a non-cyclic group.