The group structures of automorphism groups of elliptic function fields over finite fields and their applications to optimal locally repairable codes
The automorphism group of an elliptic curve over an algebraically closed field is well known. However, for various applications in coding theory and cryptography, we usually need to apply automorphisms defined over a finite field. Although we believe that the automorphism group of an elliptic curve...
Saved in:
Main Authors | , |
---|---|
Format | Journal Article |
Language | English |
Published |
26.08.2020
|
Subjects | |
Online Access | Get full text |
Cover
Loading…
Summary: | The automorphism group of an elliptic curve over an algebraically closed
field is well known. However, for various applications in coding theory and
cryptography, we usually need to apply automorphisms defined over a finite
field. Although we believe that the automorphism group of an elliptic curve
over a finite field is well known in the community, we could not find this in
the literature. Nevertheless, in this paper we show the group structure of the
automorphism group of an elliptic curve over a finite field. More importantly,
we characterize subgroups and abelian subgroups of the automorphism group of an
elliptic curve over a finite field.
Despite of theoretical interest on this topic, our research is largely
motivated by constructions of optimal locally repairable codes. The first
research to make use of automorphism group of function fields to construct
optimal locally repairable codes was given in a paper \cite{JMX20} where
automorphism group of a projective line was employed. The idea was further
generated to an elliptic curve in \cite{MX19} where only automorphisms fixing
the point at infinity were used. Because there are at most $24$ automorphisms
of an elliptic curve fixing the point at infinity, the locality of optimal
locally repairable codes from this construction is upper bounded by $23$. One
of the main motivation to study subgroups and abelian subgroups of the
automorphism group of an elliptic curve over a finite field is to remove the
constraints on locality. |
---|---|
DOI: | 10.48550/arxiv.2008.12119 |