Kriptografija temeljena na grafovima izogenija supersingularnih eliptičkih krivulja

master's thesis

• Title (english): Cryptography based on isogeny graphs of supersingular elliptic curves
• Author: Maja Marević
• Mentor: Matija Kazalicki (mentor)
• Committee member: Matija Kazalicki (predsjednik povjerenstva)
• Committee member: Filip Najman (član povjerenstva)
• Committee member: Marko Tadić (član povjerenstva)
• Committee member: Ivica Nakić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2020-09-29, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Algoritmi koji se u današnje vrijeme koriste u kriptografiji mogu se razbiti u polinomijalnom vremenu ako (kada) se kvantna računala počnu razvijati za civilne svrhe. Nesigurnost razmjene podataka na kvantnim računalima postala bi vrlo lako jedan od globalnih problema budućnosti. U ovom radu istražen je SIDH algoritam za koji se vjeruje da je kvantno otporan, s obzirom na to da na njega još uvijek nisu pronađeni ozbiljni napadi. Sadrži metodu razmjene ključeva temeljenu na Diffie-Hellmanovoj razmjeni te mu se sigurnost temelji na teškoći rješavanja problema vezanih uz supersingularne eliptičke krivulje. U prvom poglavlju napravljen je uvod za čitatelja, od osnovne definicije kriptografije do klasičnih i kvantnih algoritama. Drugo poglavlje daje jasne definicije eliptičke krivulje i njenih osnovnih svojstava. Također, definirano je ponašanje eliptičkih krivulja nad konačnim poljima. U trećem poglavlju uvodimo pojam izogenije, gdefiniciju grafa izogenije te opis razlike izogenija i izomorfizma. Četvrto poglavlje bazira se na SIDH protokolu. Od razmjene ključeva do računanja izogenija i traženja baze torzijskih podgrupa. Kako bi potvrdili kvantu otpornost, spomenuti su sigurnost i složenost SIDH algoritma.

Abstract (english)

The algorithms used in the present time for cryptographic purposes, can be broken in polynomial time if the development of a quantum computer becomes reality.Insecurity of data exchange on quantum computers would very easily become one of the global the problem of the future. In this master thesis, we are studying an algorithm that is believed to be quantum resistant, since no serious attacks against it have been found yet, and therefore can be used in post-quantum era, if this becomes a reality. It is called SIDH. It is a key exchange method based on Diffie-Hellmankey exchange, with security based on the dificulty of finding isogenies between supersingular elliptic curve. First chapter is defined as introduction to reader, from basic definition of cryptography to classic algorithms and why we should consider implementing a new ones. In second chapter one can find what are elliptic curves and how they behave on finite fields. Third chapter has focus on defining isogeny, isogeny graph and di erences between isomorphism and isogeny. In fourth chapter we are going through steps of SIDH algoritam, from key exchange to finding bases of torsion subgroups. There is subsection about complexity and security of SIDH algorithm.

• Keywords: grafovi izogenija; kvantna računala; SIDH algoritam
• Keywords (english): isogeny graphs; quantum computer; SIDH algorithm
• Language: croatian
• URN:NBN: urn:nbn:hr:217:583297
• Study programme: Title: Applied Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2021-03-01 12:14:41