Linearne kongruencije, linearne diofantske jednadžbe i programska podrška

undergraduate thesis

• Title (english): Linear Congruences, Linear Diophantine Equations and Software Support
• Author: Nikola Seletković
• Mentor: Mario Krnić (mentor)
• Committee member: Mario Krnić (predsjednik povjerenstva)
• Committee member: Lenka Mihoković (član povjerenstva)
• Committee member: Anamari Nakić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Electrical Engineering and Computing; Zagreb
• Defense date and country: 2020-07-03, Croatia
• Scientific / art field, discipline and subdiscipline: TECHNICAL SCIENCES; Computing

Abstract

Ključni pojmovi iz elementarne teorije brojeva koji su obrađeni u ovom radu su djeljivost, kongruencije i diofantske jednadžbe. U kontekstu djeljivosti, bitno je spomenuti pojam najvećeg zajedničkog djelitelja koji se računa pomoću Euklidovog algoritma te osnovni teorem aritmetike. Kineski teorem o ostatcima, Gaussova definicija kongruencije i diofantske jednadžbe pokazale su se ključnima za razvoj kriptografije. Njen temelj čine veliki prosti brojevi koji predstavljaju ključeve pomoću kojih se kodira i dekodira poruka, a to je u praksi rješavanje linearnih kongruencija te sustava linearnih kongruencija pomoću Kineskog teorema o ostatcima. Diofantske se jednadžbe koriste za traženje rješenja polinoma u skupu cijelih brojeva.

Abstract (english)

The main terms from the elemental number theory that have been mentioned in this thesis are divisibility, congruence and diophantine equations. Within the context of divisibility, it is important to mention the term of the greatest common divisor that is being calculated by the Euclidian algorithm, as well as the Fundamental Theorem of Arithmetic. The Chinese Remainder Theorem, Gauss' definition of congruence and diophantine equations have proved to be key for the development of cryptography. Its foundation consists of large prime numbers that represent the keys by which the messages are coded and decoded. In practice, this means solving linear congruences and the systems of linear congruences by applying the Chinese Remainder Theorem. Diophantine equations are used to determine whether or not polynomial equation has integer solutions.

• Keywords: elementarna teorija brojeva; djeljivost; najveći zajednički višekratnik; Euklidov algoritam; prosti brojevi; linearne kongruencije; sustav linearnih kongruencija; Kineski teorem o ostatcima; linearne diofantske jednadžbe
• Keywords (english): elemental number theory; divisibility; greatest common divisor; Euclidian algorithm; prime numbers; linear congruences; system of linear congruences; Chinese Remainder Theorem; linear diophantine equations
• Language: croatian
• URN:NBN: urn:nbn:hr:168:299664
• Study programme: Title: Computing; Study programme type: university; Study level: undergraduate; Academic / professional title: sveučilišni/a prvostupnik/ prvostupnica (baccalaureus/baccalaurea) inženjer/inženjerka računarstva (sveučilišni/a prvostupnik/ prvostupnica (baccalaureus/baccalaurea) inženjer/inženjerka računarstva)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Closed access
• Created on: 2023-01-14 09:54:51