Eliptične krivulje nad različnimi obsegi

diplomsko delo

Petra Čačkov (Author), Marko Slapar (Mentor)

Abstract

Za boljše razumevnanje eliptičnih krivulj v uvodu definiramo projektivno ravnino in točke v neskončnosti, saj so te pomembne za njihovo obravnavo. Nato definiramo eliptične krivulje in predstavimo oblike v katerih jih lahko obravnavamo. Skozi celo diplomo jih v večini obravnavamo v Weierstrassovi obliki. Na eliptične krivulje lahko gledamo tudi kot množico na točk, ki rešijo enačbo za dano eliptično krivuljo. Ta množica točk, s točko v neskončnosti v kateri se sekajo vse premice vzporedne y osi, predstavlja abelovo grupo za seštevanje. V diplomi predstavimo grupno strukturo eliptičnih krivulj in definiramo seštevanje točk na njej. Ker pa se eliptične krivulje obnašajo različno, glede na to nad katerim obsegom jih obravnavamo, obravnavamo eliptične krivulje nad realnimi in racionalnimi števili ter nad končnim obsegom Z_p, kjer je p praštevilo. Obravnavamo jih tudi nad celimi števili, čeprav množica celih števil ni obseg, in množica točk, ki rešijo enačbo elliptične krivulje ni več grupa. Pri obravnavi eliptičnih krivulj nad realnimi števili se osredotočimo na iskanje ničel, med tem ko se v drugih primerih osredotočimo na iskanje in preštevanje točk, ki ležijo na dani eliptični krivulji.

Keywords

eliptične krivulje;Weierstrassova enačba;točka v neskončnosti;

Data

Language:	Slovenian
Year of publishing:	2016
Typology:	2.11 - Undergraduate Thesis
Organization:	UL PEF - Faculty of Education
Publisher:	[P. Čačkov]
UDC:	51(043.2)
COBISS:	11170377
Views:	870
Downloads:	162
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Elliptic curves over different fields
Secondary abstract:	For better understanding of elliptic curves, we first define projective plane and points at infinity. Then we define elliptic curves and show some different forms of equations that represent them. Throughout the thesis, mostly we use Weierstrass form for elliptic curves, since every elliptic curve, with at least one point lying on it, can be transformed into it. We can look on elliptic curves as a set of points that solve the given equation of elliptic curve. That set of points, with point at infinity in which all lines parallel to y axis meet, represent an abelian group for adding points. Furthermore, we define the group structure of elliptic curves and adding points on them. As elliptic curves act differently depending on the field they are studied in, we discuss elliptic curves over real numbers, rational numbers and over finite field Z_p where p is a prime number. We also consider elliptic curves over integer numbers, even though a set of integer numbers is not a field and we cannot define a group structure with adding points, like we did before. When dealing with elliptic curves over real numbers, we focus on finding zeroes of elliptic curves while in other cases the focus on finding and counting points on elliptic curve.
Secondary keywords:	mathematics;matematika;
File type:	application/pdf
Type (COBISS):	Bachelor thesis/paper
Thesis comment:	Univ. v Ljubljani, Pedagoška fak., Dvopredmetni učitelj: matematika-fizika
Pages:	34 str.
ID:	9171009