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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.

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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.

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  • $\begingroup$ Thank you for your answer! What I don't understand is that the SIDH paper says that $\langle P_A,Q_A\rangle = E[l^{e_A}]$ and that both points generate the whole group $E[l^{e_A}]$. How is that possible if $P_A,Q_A$ only have order $l^{e_A}$ but $E[l^{e_A}]$ has cardinality $(l^{e_A})^2$? $\endgroup$ – jvdh Jun 3 at 16:53
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    $\begingroup$ Consider the group $G = \mathbb{Z}/2 \times \mathbb{Z}/2$, which is a group of cardinality $4 = 2^2$. The elements $P = (1,0) \in G$ and $Q = (0,1) \in G$ each have order $2 = 2^1$, but $P$ and $Q$ together generate $G$. $\endgroup$ – djao Jun 4 at 3:02
  • $\begingroup$ Thank you very much. One further question that comes to mind is why $E_0$ divides up neatly in one $l_A^{e_A}$ subgroup of cardinality $(l_A^{e_A})^2$ and a group of analogue cardinality for $l_B^{e_B}$. What prevents the points from having order $l_A$ or $l_B$? $\endgroup$ – jvdh Jul 10 at 13:55
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You appear to be under the impression that Elliptic Curve groups are always cyclic, and that there is only one subgroup of a given order.

That is not the case, and it is most definitely not the case in the groups we use for isogenies. There are a huge number of subgroups of the same order; hence just knowing the subgroup order doesn't tell you which subgroup we're talking about.

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  • $\begingroup$ However the authors state that $\langle P_A,Q_A\rangle = E_0[l_A^{e_A}]$. Doesn't that mean that $\langle [m_A]P_A + [n_A]Q_A\rangle = E_0[l_A^{e_A}]$ as well? $\endgroup$ – jvdh May 31 at 11:54
  • $\begingroup$ Okay, I think I got it. $⟨[m_A]P_A⟩ = E_0[l_A^{e_A}]$ but $\langle R \rangle= ⟨[m_A]P_A+[n_A]Q_A⟩$ might have a totally different order because of the addition. Is that correct? $\endgroup$ – jvdh May 31 at 12:06

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