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$E(\mathbb{F}_{p^k})[r]$: all elements $P$ of $E(\mathbb{F}_{p^k})$ such that $rP = 0$. In other words, all points whose order divide $r$. In protocols $r$ is usually prime, so that means all points with order $r$ and the point at infinity. $\pi_p$: the Frobenius endomorphism. This is a function that takes an elliptic curve point such that $\pi_p((x, y)) = (...

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