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I'm reading Mathematics of Isogeny Based Cryptography by Luca De Feo. At some point (pg. 32), he says

"A walk of length $e_A$ in the $l_A$-isogeny graph corresponds to a kernel of size $l_A^{e_A};$ and this kernel is cyclic if and only if the walk does not backtrack."

I don't understand why the kernel is cyclic iff the walk does not backtrack.

Here we have a graph where the vertices are j-invariants of supersingular elliptic curves and the edges are $l_a-$isogenies.

Can someone help me?

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    $\begingroup$ If you backtrack then you are using the dual isogeny of the previous step; the composition is $[\ell_A]$, which has non-cyclic kernel. Similarly, if you have a non-cyclic kernel of order $\ell_A^{e_A}$ that is not cyclic, then it contains a subgroup isomorphic to $(\mathbb{Z}/(\ell_A))^2$, so you can factor $[\ell_A]$ out of the isogeny - and this represents a backtracking step. $\endgroup$
    – Ben Smith
    Commented Jan 9, 2023 at 9:08

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