Rang eliptičkih krivulja

master's thesis

• Title (english): Rank of elliptic curves
• Author: Luka Žunić
• Mentor: Andrej Dujella (mentor)
• Committee member: Andrej Dujella (predsjednik povjerenstva)
• Committee member: Robert Manger (član povjerenstva)
• Committee member: Matija Kazalicki (član povjerenstva)
• Committee member: Boris Muha (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2024-02-28, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Kroz ovaj rad su uvedeni osnovni pojmovi o eliptičkim krivuljama nad poljem racionalnih brojeva \(\mathbb{Q}\) i nad konačnim poljima \(\mathbb{F}_p\). Definiran je rang eliptičke krivulje, pojam koji ima veliku važnost u istraživanju eliptičkih krivulja te u njihovoj primjeni u kriptografiji. Opisane su neke od metoda za pronalazak ranga eliptičke krivulje te je pomoću Mestre-Nagaovih suma konstruirana eliptička krivulja ranga \(\geq 8\). Kroz proučavanje eliptičkih krivulja koristimo pojmove algebarskih struktura, koje navodimo u prvom poglavlju. Primjene eliptičkih krivulja u kriptografiji su danas jako istraživana tema, opisujemo problem diskretnog logaritma i težinu rješavanja tog problema u kontekstu eliptičkih krivulja. Opisujemo neke od kriptosustava koji koriste eliptičke krivulje te povezujemo težinu pronalaska eliptičkih krivulja visokog ranga s njihovom primjenom.

Abstract (english)

In this thesis, we introduce some of the fundamental concepts of elliptic curves over the field of rational numbers \(\mathbb{Q}\) and finite fields \(\mathbb{F}_p\). We define the rank of an elliptic curve, a notion of significant importance in the study of elliptic curves and their application in cryptography. Various methods for determining the rank of an elliptic curve are described, and an elliptic curve of rank \(\geq 8\) is constructed using the Mestre-Nagao sums. Through the study of elliptic curves, we utilize concepts of algebraic structures, which are outlined in the first chapter. The applications of elliptic curves in cryptography are a highly researched topic. We discuss the discrete logarithm problem and the difficulty of solving this problem in the context of elliptic curves. We describe some of the cryptosystems utilizing elliptic curves, and the connection between the challenge of finding elliptic curves with high rank and their practical application.

• Keywords: polje racionalnih brojeva; konačna polja; kriptografija; diskretni logaritam; kriptosustavi
• Language: croatian
• URN:NBN: urn:nbn:hr:217:536904
• Study programme: Title: Computer Science and Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: sveučilišni magistar računarstva i matematike (sveučilišni magistar računarstva i matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2024-03-14 13:02:57