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Let an elliptic curve $E$, and 2 points on such curve $P$ and $O$ the methods I’m talking about consist in creating a weaker elliptic curve $F$ and mapping $P$ and $O$ to $F$ while successfully preserving the discrete logarithm $s$ between the two such as $F=s×O$.

The most well know example is moving elliptic curves into hyperelliptic curves but as far I’m aware this only work on extension fields of medium sized degree (and not over prime field). But of course, I’m more interested in curve over prime fields.
As an example outside hyperelliptic curve, is there a case that consider building a different curve defined on a field from different characteristic while preserving the same order in a subgroup ?

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  • $\begingroup$ "successfully preserving their discrete logarithm relation"? you need to explain that for clarity $\endgroup$
    – kodlu
    Commented Nov 14 at 0:43
  • $\begingroup$ @kodlu Simple. I remember about a paper where an elliptic curve is constructed easely but computing 1 that doesn t change the discrete logarithm between $O$ and $F$ rarely happen. $\endgroup$
    – user2284570
    Commented Nov 14 at 4:33
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    $\begingroup$ I'm sorry that's still unclear to me. If you found the paper or wrote what you are suggesting explicitly it would help the question. What is the discrete logarithm between $F$ and $G$ for example? Define it. $\endgroup$
    – kodlu
    Commented Nov 14 at 4:45
  • $\begingroup$ @kodlu Ok, I rewrote it more formally. Essentially I don t want about papers that fails to build to keep the same scalar. In my case I m thinking about cases that don t necessarily involve the Generator $\endgroup$
    – user2284570
    Commented Nov 14 at 4:50

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