Faktorizacija velikih prirodnih brojeva

master's thesis

• Author: Marija Laća
• Mentor: Andrej Dujella (mentor)
• Committee member: Andrej Dujella (predsjednik povjerenstva)
• Committee member: Dražen Adamović (član povjerenstva)
• Committee member: Marcela Hanzer (član povjerenstva)
• Committee member: Pavle Pandžić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2019-07-16, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Ovaj rad fokusira se na proste brojeve i faktorizacijske algoritme velikih prirodnih brojeva. Prosti ili prim brojevi su jedan od fundamentalnih pojmova u matematici. To su brojevi koji su djeljivi samo s brojem 1 i sa samim sobom. Algoritam je konačan slijed dobro definiranih naredbi za ostvarenje zadatka. Faktorizacija velikih prirodnih brojeva i pronalazak velikih prostih brojeva iznimno je bitan u kriptografiji. Jos u 3. stoljeću pr. Kr. Euklid je dokazao da ne postoji najveći prosti broj te da prostih brojeva ima beskonačno mnogo. Kako se skup prirodnih brojeva nastavlja, prosti brojevi postaju sve manje učestali, te je pronalazak velikih prostih brojeva postao zanimacija mnogih. U prvom poglavlju komentirat ćemo učestalost prostih brojeva i što su to Mersennovi prosti brojevi. U drugom poglavlju upoznat ćemo se s raznim testovima prostosti. Naivni način za provjeru prostosti broja je pokusno dijeljenje, a obradeni su i Fermatov i Miller-Rabinov test. Ako prirodan broj \(n\) ne prođe neki od testova prostosti, onda znamo da je \(n\) sigurno složen broj. Iako su korisni za provjeru prostosti, takvi testovi nam najčešće ne daju niti jedan netrivijalni faktor od \(n\). Postavlja se pitanje kako naći netrivijalni faktor velikog složenog broja i koji algoritam je najbrži za pronalazak takvih faktora. To se smatra teškim problemom i na njegovoj težini su zasnovani neki od najvažnijih kriptosustava s javnim ključem. U trećem poglavlju obradit ćemo 6 algoritama za faktorizaciju prirodnih brojeva.

Abstract (english)

This paper focuses on prime numbers and factorisation algorithms of large numbers. Prime numbers have only two factors, number 1 and itself, and are one of the fundamental terms in mathematics. Algorithm is a process or set of rules to be followed in calculations or other problem-solving operations. Factorization of large numbers and finding large prime numbers is extremely important in cryptography. In 3rd century BC, Euclid proved that the largest prime number doesn’t exist, and that there are infinite number of primes. As the set of natural numbers continues, they become less frequent, so finding new prime numbers has become a mission for many people. In chapter 1, we will discuss frequency of prime numbers and what are Mersenne’s prime numbers. In chapter 2, we will talk about primality testing. Most basic way to check if a number is prime is by trial division. Fermat’s primality test and Miller-Rabin’s test is also discussed in this paper. If a number \(n\) doesn’t pass one of the primality tests, then we know for sure that \(n\) is a composite number. Even though they are useful for primality testing, these tests rarely give us a non-trivial factor of \(n\). How can we find factors of large numbers, and what is the fastest algorithm to do that? That is considered a hard problem and therefore some of the most important cryptosystems with public key are based on the complexity of factorization. In chapter 3 we will discuss six factorisation algorithms.

• Keywords: prosti brojevi; faktorizacijski algoritmi velikih prirodnih brojeva; Euklid; Mersennovi prosti brojevi; Fermatov test; Miller-Rabinov test
• Keywords (english): prime numbers; factorisation algorithms of large numbers; Euclid; Mersenne’s prime numbers; Fermat’s primality test; Miller-Rabin’s test
• Language: croatian
• URN:NBN: urn:nbn:hr:217:942151
• Study programme: Title: Mathematics and Computer Science Education; specializations in: Mathematics and Computer Science Education; Course: Mathematics and Computer Science Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike i informatike (magistar/magistra edukacije matematike i informatike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-09-30 12:02:15