# Generating pairs of elliptic $\mathbb{F}_q$-curves isogenous over $\mathbb{F}_q$ such that nobody knows an $\mathbb{F}_q$-isogeny between them

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 ?

• Yes, it's interesting. Please do it. Jul 27 at 15:49