14,678 research outputs found
Index Calculus in Class Groups of Plane Curves of Small Degree
We present a novel index calculus algorithm for the discrete logarithm problem (DLP) in degree 0 class groups of curves over finite fields. A heuristic analysis of our algorithm indicates that asymptotically for varying q, ``essentially all\u27\u27 instances of the DLP in degree 0 class groups of curves represented by plane models of a fixed degree d over can be solved in an expected time of .
A particular application is that heuristically, ``essentially all\u27\u27 instances of the DLP in degree 0 class groups of non-hyperelliptic curves of genus 3 (represented by plane curves of degree 4) can be solved in an expected time of .
We also provide a method to represent ``sufficiently general\u27\u27 (non-hyperelliptic) curves of genus by plane models of degree . We conclude that on heuristic grounds the DLP in degree 0 class groups of ``sufficiently general\u27\u27 curves of genus (represented initially by plane models of bounded degree) can be solved in an expected time of
Discrete logarithms in curves over finite fields
A survey on algorithms for computing discrete logarithms in Jacobians of
curves over finite fields
On rationality of the intersection points of a line with a plane quartic
We study the rationality of the intersection points of certain lines and
smooth plane quartics C defined over F_q. For q \geq 127, we prove the
existence of a line such that the intersection points with C are all rational.
Using another approach, we further prove the existence of a tangent line with
the same property as soon as the characteristic of F_q is different from 2 and
q \geq 66^2+1. Finally, we study the probability of the existence of a rational
flex on C and exhibit a curious behavior when the characteristic of F_q is
equal to 3.Comment: 17 pages. Theorem 2 now includes the characteristic 2 case;
Conjecture 1 from the previous version is proved wron
The type numbers of closed geodesics
A short survey on the type numbers of closed geodesics, on applications of
the Morse theory to proving the existence of closed geodesics and on the recent
progress in applying variational methods to the periodic problem for Finsler
and magnetic geodesicsComment: 29 pages, an appendix to the Russian translation of "The calculus of
variations in the large" by M. Mors
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