2
$\begingroup$

enter image description here

How do you know if an isogeny is surjective or not, and how do you tell how many points on E maps to E'? Does the answer lie in the degree of the isogeny function?

$\endgroup$
3
$\begingroup$

Isogenies are always surjective, but there's a nuance. They are surjective over the algebraic closure. The correct statement would be "for every $\mathbb{F}_{17}$-point on the green curve there are three $\bar{\mathbb{F}}_{17}$-points on the blue curve which map to it".

To find the preimages of a point, write down a polynomial system and find its solutions. A computer algebra system (e.g., SageMath) may help. The number of solutions is obviously related to the degree of the polynomial system.

$\endgroup$
  • $\begingroup$ thank you! I have apparently forgot the original definition of algebraic varieties and I’m also thankful for your paper ‘mathematics on isogeny based crypto’ $\endgroup$ – edlothia Aug 23 '18 at 0:39

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.