# 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.

No, this is not equivalent. To see why, take the example with $$ℓ_A = 2$$ and $$e_A = 2$$, then the torsion group is isomorphic to the additive group $$ℤ/4ℤ × ℤ/4ℤ$$. Set $$P=(1,3)$$ and $$Q=(1,1)$$, then neither is a multiple of the other, however they do not generate the whole group because $$2P = 2Q$$, and indeed there is no way to write $$(1,0)$$ (and many other vectors) as a combination of $$P$$ and $$Q$$.
Another way to see that $$P$$ and $$Q$$ are not independent is to form the matrix $$\begin{pmatrix}1 & 3\\1 & 1\end{pmatrix}$$ and compute its discriminant in $$ℤ/4ℤ$$, which is $$2$$. So the matrix is not invertible mod $$4$$, and thus the linear combinations of its rows do not generate the whole $$ℤ/4ℤ×ℤ/4ℤ$$. This is in a sense the analogue of computing the Weil pairing.
Note that $$e_A>1$$ was the key to this counter-example. If $$e_A=1$$, then it is indeed sufficient to check that $$[i]P≠Q$$ for all $$0.
What is true in general is that $$P_A$$ and $$Q_A$$ are independent if and only if $$P'=[ℓ_A^{e_A-1}]P_A$$ and $$Q'=[ℓ_A^{e_A-1}]Q_A$$ are, which you can check by enumerating all multiples of $$P'$$, as above. This is how the pairing check is usually done in implementations of SIDH.