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What is the difference between Elliptic curves and Hyperelliptic curves in terms of security? I am relatively new to cryptography and have heard a great deal more about Elliptic curves than Hyperelliptic curves.

Why are Hyperelliptic curves not so prevalent in use?

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  • $\begingroup$ The main difference is that we don't yet understand the security of hyperelliptic curves as much as elliptic curves, partly because of less research on hyperelliptic curves. As to the question why are Hyperelliptic curves not so prevalent in use. My understanding is that elliptic curves are good enough for now, and hyperelliptic curves don't offer anything significantly better. $\endgroup$ – Changyu Dong Jun 6 '18 at 17:16
  • $\begingroup$ How about in pairing-based cryptography? $\endgroup$ – Fathima Abdur Rahman Jun 7 '18 at 8:48
  • $\begingroup$ A motivation of investigating hyperelliptic curves is actually hoping to find a better way of doing pairing-based cryptography. Security wise there is not much difference to my knowledge. As I said, we know relatively little about hyperelliptic curves, therefore there is no conclusive evidence yet to convince people to use hyperelliptic curves. $\endgroup$ – Changyu Dong Jun 7 '18 at 11:20