Number Theory and Applications
Keywords |
Classification |
Keyword |
OFICIAL |
Mathematics |
Instance: 2019/2020 - 1S
Cycles of Study/Courses
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
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
PARI/GP
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 |
106,00 |
Frequência das aulas |
56,00 |
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 tests of one hour each (on dates to be determined). Each test will have a weight of 30% of the final mark. The exam will consist of:
1) a one-hour test weighing 40% of the final grade for students wishing to use the sum of the scores obtained in the two tests,
or (disjunctive):
2) a 3 hour test with a weight of 100%.
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.