Independent parameters basis for torsion-groups in SIDH: Is the Weil-pairing necessary? In the original SIDH paper by De Feo, Jao and Plût, the basis points $P_A$ and $Q_A$ are supposed to be independent points in $E(\mathbb{F}_{p^2})$ of order $\ell_A^{e_A}$ for some small prime $\ell_A$ on some supersingular curve $E$. To check independence, they suggest computing the Weil-pairing $e(P_A, Q_A)$ and check that the result still has order $\ell_A^{e_A}$.
My question is this: Is another valid approach to checking independence to check if some discrete logarithm exists from one point to the other? Of course, $|E[\ell_A^{e_A}]|$ is $\ell_A$-power smooth, so taking discrete logarithms is easy by Pohlig-Hellman, but I am sure if this approach is valid/equivalent with calculating the Weil-pairing.
As you mentioned, computing the discrete logarithm is an easy way to check for linear dependency. The usual way to compute discrete logs over elliptic curves relies on some pairing, but there are ways to do so without any pairing. See for instance &quot;Isogeny-based key compression without pairings&quot; at PKC'21 (https://eprint.iacr.org/2021/272.pdf), where pairing-less discrete logs are used for key compression.