[curves] The genus 3 setting

Kim Laine laine at math.berkeley.edu
Thu Apr 10 09:32:59 PDT 2014

Dear all,

All curves of genus <= 2 are hyperelliptic so the fast non-hyperelliptic
index calculus methods do not apply. Really the increase in speed comes
from some simple geometry tricks that work for plane curves, but
hyperelliptic curves are never plane curves (whereas for instance genus 3
non-HE are plane quartics).

Ben Smith's isogeny was the first demonstration of isogeny attacks.
Recently there has been interesting advance in computing more general
explicit isogenies using completely different methods (work of Robert,
Bisson, Cosset and some others) so these isogeny attacks are very relevant
now and seem hard to avoid in genus 3.


On Thu, Apr 10, 2014 at 7:37 AM, Schmidt, Jörn-Marc <
Joern-Marc.Schmidt at secunet.com> wrote:

> Dear Watson Ladd,
> >However, there is a problem in that there may be an isogeny to a
> non-hyperelliptic curve.
> I was looking just a bit into it: [1] presents such an isogeny for curves
> defined over fields with characteristic > 3.
> Do you know of a similar result for smaller characteristics?
> Best,
> Jörn
> [1] Benjamin Smith, Isogenies and the discrete logarithm problem in
> jacobians of genus 3 hyperelliptic curves
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