Number Theory and Applications

Code:

M3015

Acronym:

M3015

Level:

300

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2018/2019 - 1S

Active?	Yes
Web Page:	https://moodle.up.pt/course/view.php?id=372
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Biology

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:B	2	Official Study Plan	3	-	6	56	162
L:CC	9	Plano de estudos a partir de 2014	2	-	6	56	162
			3
L:F	1	Official Study Plan	2	-	6	56	162
			3
L:G	1	study plan from 2017/18	3	-	6	56	162
L:M	47	Official Study Plan	2	-	6	56	162
			3
L:Q	0	study plan from 2016/17	3	-	6	56	162
MI:ERS	3	Plano Oficial desde ano letivo 2014	2	-	6	56	162

Teaching language

Suitable for English-speaking students

Objectives

To introduce the basic concepts and results of Number Theory, together with some of its computational aspects. To give some of its cryptographical applications. 

Learning outcomes and competences

To know the basic concepts and results of Number Theory, as well as some of its computational aspects and some of its cryptographical applications. 

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

M141 Álgebra Linear I

M142 Álgebra Linear II

CC101 Introdução à Programação

Program

1. Introduction: Numbers (Peano axioms, induction, well-ordering)
2. Unique factorization in Z
3. Unique factorization in k[x]
4. Gauss integers and applications to unique factorization
5. Arithmetic functions
6. Congruences
7. Primitive roots and the group of units U(Z/nZ)
8. Artin conjecture and nth power residues
9. Quadratic residuies
10. The Law of quadratic reciprocity
11.  Legendre symbol and proof of the law of quadratic reciprocity
12. Algebraic numbers and algebraic integers
13. Quadratic Gauss sum
14. Introduction to algebraic number theory
15. Unique factorization in field of algebraic numbers
16. Cryptography: classical ciphers
19. Public Key Cryptography: Deffie-Hellman RSA
20. Fast algorithm to calculate powers mod n. Cryptographic system ElGamal
21. Finite fields
22. Proof of the Theorem that the multiplicative group of a finite field is cyclic.
24. Elliptic curves

Mandatory literature

Ireland Kenneth; A classical introduction to modern number theory. ISBN: 0-387-90625-8

Complementary Bibliography

Vinogradov I. M.; Elements of number theory. ISBN: 0-486-60259-1
Shoup Victor; A computational introduction to number theory and algebra. ISBN: 0-521-85154-8
Menezes Alfred J.; Handbook of applied cryptography. ISBN: 0-8493-8523-7
Endler O.; Teoria dos Números Algébricos

Teaching methods and learning activities

Lectures on the concepts and results of the subject matter, with many examples, and exercise solving classes.

Evaluation Type

Distributed evaluation without final exam

Assessment Components

designation	Weight (%)
Teste	100,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	106,00
Frequência das aulas	56,00
Total:	162,00

Eligibility for exams

There are no rules concerning the attendance frequency.

Calculation formula of final grade

1. During the regular time ("época normal") the final grade is obtained by the sum of the grades of two quizzes ("testes"):

Quiz 1: in the first quiz, which will take place during class time and which is still to be scheduled, students can obtain up to 10 points. A student who does not obtain 2 points will automatically obtain a "failed" in the first exam period.

Teste 2: in the second quiz, which will take place in January, students can obtain up to 10 points. A student who does not obtain 2 points will automatically obtain a "failed" in the first exam period.

The final grade is the sum of the scores obtained in both quizzes or "failed" in case one of the scores is below 2 points.

2. The make-up exam consists of only one exam which contains two parts corresponding to Quiz 1 + 2. Students have to have at least 2 points in each of the parts.

Examinations or Special Assignments

2 quizzes + make up exam