Kvadratni recipročnostni zakon : delo diplomskega seminarja

Perić, Dejan (author), Vavpetič, Aleš (mentor)

V delu je predstavljen problem rešljivosti kvadratne enačbe v kolobarju ostankov. Definiran je Legendrov simbol, ki poda rešljivost določene kvadratne enačbe. S tem simbolom je formuliran kvadratni recipročnostni zakon in njuna dodatka. Ti izreki so dokazani z uporabo primitivnih n-tih korenov enote in Gaussovimi vsotami ter služijo ugotavljanju vrednosti Legendrovega simbola. Iz teh rezultatov je prikazana uporaba Goldwasser-Micalijevega kriptosistema.

• Language: Slovenian
• Keywords: kvadratni recipročnostni zakon, kvadratni ostanek po modulu p, Legendrov simbol, Gaussova vsota, primitivni koren enote, Goldwasser-Micalijev kriptosistem
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2022
• PID: 20.500.12556/RUL-139403
• UDC: 511
• COBISS.SI-ID: 120685827
• Publication date in RUL: 02.09.2022

Secondary language

• Language: English
• Title: Quadratic reciprocity law

Abstract

In this thesis, the problem of a solvability of quadratic equation in a residue ring is presented. Legendre symbol is defined, which indices the solvability of a certain quadratic equation. This symbol is used in the formulation of the quadratic reciprocity law and its two supplements. These theorems are proved with the use of primitive nth roots of unity and the Gauss sums, and are used for specifying the value of the Legendre symbol. With the use of the results in this thesis the use of Goldwasser-Micali cryptosystem is shown.

• Keywords: quadratic reciprocity law, quadratic residue modulo p, Legendre symbol, Gauss sum, primitive root of unity, Goldwasser-Micali cryptosystem