Number Theory and Criptography

Code:

M3032

Acronym:

M3032

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2024/2025 - 2S

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

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	48	162
L:CC	16	study plan from 2021/22	2	-	6	48	162
			3
L:F	1	Official Study Plan	3	-	6	48	162
L:G	0	study plan from 2017/18	2	-	6	48	162
			3
L:M	58	Official Study Plan	2	-	6	48	162
			3
L:MA	25	Official Study Plan	3	-	6	48	162
L:Q	0	study plan from 2016/17	3	-	6	48	162

Teaching language

Suitable for English-speaking students

Objectives

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

Learning outcomes and competences

Upon completing this curricular unit, the student should:

(1) master basic concepts, methods, and results of number theory;

(2) be able to analyze and solve problems within number theory, using the methods and results that best apply to the problems under study;

(3) to appreciate the computational aspects of number theory and some of its cryptographic applications;

(4) be able to communicate efficiently and clearly their problem solving and understanding of the subject.

Working method

Presencial

Program

• Some notes on the history of Number Theory.


• Divisibility in commutative rings, namely in the ring of integers and in rings of polynomials over fields: irreducible elements, primes and units.


• Euclidean domains and unique factorization.


• Modular arithmetic: some computational aspects; review of the theorems of Fermat, Euler and the so-called "Chinese Remainder Theorem".


• The RSA cipher and some of its applications.


• Rudiments on primality tests and factorization test; numbers of Mersenne and Fermat.


• Primitive roots and the Diffie-Hellman key-exchange protocol.


• Quadratic residues, law of quadratic reciprocity, and the resolution of modular equations of the second degree; Pépin test for Fermat numbers.


• Some aspects of analytical number theory.


• Brief introduction to algebraic number theory.

Mandatory literature

Manuel Delgado e António Machiavelo; Teoria dos Números: uma introdução com aplicações
Vinogradov I. M.; Elements of number theory. ISBN: 0-486-60259-1
Ireland Kenneth; A classical introduction to modern number theory. ISBN: 0-387-90625-8

Complementary Bibliography

Harold M. Edwards; Fermat.s last theorem. ISBN: 0-387-90230-9
Oystein Ore; Number theory and its history. ISBN: 0-486-65620-9
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
José Plínio de Oliveira Santos; Introdução à Teoria dos Números, IMPA, 2000

Teaching methods and learning activities

The contact hours are distributed in theoretical and theoretical-practical classes. In the first ones, the contents of the program are studied, often using examples to illustrate the concepts treated and to guide the students in the resolution of exercises and problems. In the theoretical-practical classes, exercises and problems are solved, which are indicated in advance for each week. List of exercises and other course materials are available on the course page at Sigarra. In addition to the classes, there are weekly attendance periods where students have the opportunity to clarify their doubts.

Software

Sage (https://www.sagemath.org)
PARI/GP (https://pari.math.u-bordeaux.fr)

keywords

Physical sciences > Mathematics > Number theory

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Teste	60,00
Exame	40,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	114,00
Frequência das aulas	48,00
Total:	162,00

Eligibility for exams

There are no rules concerning attendance frequency.

Calculation formula of final grade

Approval of the course unit is obtained in the final exam.

There will be two one-hour tests (on dates to be announced). Each test will be worth 30% of the final grade.

The exam will consist of 3 parts, two of which will correspond to the two tests. The student may choose not to complete one or both of these parts of the exam, in which case each part not completed will be awarded the grade obtained by the student in the corresponding test.

The examination of the "appeal" period will be made in the same way as the one of "normal" period.

Special assessment (TE, DA, ...)

Examinations required under special statutes shall consist of a written test that may be preceded by an oral test, to assess if the student satisfies minimum conditions to attempt to obtain approval at the discipline in the written test.