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Let $\mathbb{F}_q$ be a large finite field. What if I invent how to efficiently construct pairs of elliptic "cryptographically strong" $\mathbb{F}_q$-curves $E_1$, $E_2$ isogenous over $\mathbb{F}_q$, that is $\#E_1(\mathbb{F}_q) = \#E_2(\mathbb{F}_q)$, in such a way that nobody (including me) knows an $\mathbb{F}_q$-isogeny between them ? Is this useful in discrete logarithm or isogeny-based cryptography ?

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    $\begingroup$ Yes, it's interesting. Please do it. $\endgroup$ Jul 27 at 15:49

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