Number Theory and Applications

Code:

M3015

Acronym:

M3015

Level:

300

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2022/2023 - 2S

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	0	Official Study Plan	3	-	6	56	162
L:CC	4	study plan from 2021/22	2	-	6	56	162
			3
L:F	2	Official Study Plan	2	-	6	56	162
			3
L:G	0	study plan from 2017/18	2	-	6	56	162
			3
L:Q	0	study plan from 2016/17	3	-	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

 

Program

1. Introduction: Numbers (Peano axioms, induction, well-ordering)
2. Unique factorization domains (Euclidean Algorithm in Z and K[x], Euclidean domains, PID, UFD).
3. Some arithmetic functions (perfect numbers, Mersene primes, Möbius function, Dirichlet product, Euler's phi-function, prime counting function).
4. Modular arithmetic (Euler's Theorem, Fermat's little Theorem).
5. Primitive roots and the group of units U(Z/nZ)
6. Quadratic residuies and the law of quadratic reciprocity
7  Cryptography: classical and public key ciphers, Deffie-Hellman RSA, fast algorithm to calculate powers mod n, digital signatures, ElGamal, eliptic curves
8. Primality Tests( Miller-Rabin)

Mandatory literature

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

Complementary Bibliography

Shoup Victor; A computational introduction to number theory and algebra. ISBN: 0-521-85154-8
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.

Software

SageMath

Evaluation Type

Distributed evaluation without final exam

Assessment Components

designation	Weight (%)
Teste	50,00
Exame	50,00
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

The final grade is obtained through the sum of the scores of the midterm quiz and the final exam.

The make-up exam will have two parts that correspond accordingly to the midterm quiz and the final exam. The students can choose which part they want to hand in and which score they would like to keep.

Examinations or Special Assignments

1 midterm quiz and 1 final exam