10 research outputs found
Arithmetic using compression on elliptic curves in Huff's form and its applications
In this paper for elliptic curves provided by Huff's equation and general Huff's equation and degree 2 compression function on these curves, herein we provide formulas for doubling and differential addition after compression, which for Huff's curves are as efficient as Montgomery's formulas for Montgomery's curves . For these curves we also provided point recovery formulas after compression, which for a point on these curves allows to compute after compression using the Montgomery ladder algorithm, and then recover . Using formulas of Moody and Shumow for computing odd degree isogenies on general Huff's curves, we have also provide formulas for computing odd degree isogenies after compression for these curves.Moreover, it is shown herein how to apply obtained formulas using compression to the ECM algorithm. In the appendix, we present examples of Huff's curves convenient for the isogeny-based cryptography, where compression can be used
Exceptional elliptic curves over quartic fields
We study the number of elliptic curves, up to isomorphism, over a fixed
quartic field having a prescribed torsion group as a subgroup. Let
, where , be a torsion group such that the modular
curve is an elliptic curve. Let be a number field such that
there is a positive and finite number of elliptic curves over having
as a subgroup. We call such pairs \emph{exceptional}. It is
known that there are only finitely many exceptional pairs when varies
through all quadratic or cubic fields. We prove that when varies through
all quartic fields, there exist infinitely many exceptional pairs when
or and finitely many otherwise
Torsion points on elliptic curves over number fields of small degree
We determine the set of possible prime orders of -rational points
on elliptic curves over number fields of degree , for and
Torsion points on elliptic curves over number fields of small degree
Barry Mazur famously classified the finitely many groups that can occur as a torsion subgroup of an elliptic curve over the rationals. This thesis deals with generalizations of this to higher degree number fields. Merel proved that for all integers d one has that the number of isomorphsim classes of torsion groups of elliptic curves over number fields of degree d is finite. This thesis consists of 4 chapters, the first is introductory and the other tree are research articles. Chapter two deals with the computation of gonalities of modular curves, and the application of these computations to the question which cyclic subgroups can occur as the torsion subgroup of infinitely many non-isomorphic elliptic curves over number fields of degree <7. In the second chapter a general theory for finding rational points on symmetric powers of curves is developed that is similar to symmetric power Chabauty. Application of this theory to symmetric powers of modular curves allows us to determine which primes can divide the order of the torsion subgroup of an elliptic curve over a number field of degree <7. The last chapter studies elliptic curve with a point of order 17 over a number field of degree 4