Applied Algebra (ΑΛ3)

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• Graduate Courses Outlines
• Department of Mathematics

General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΑΛ3
Semester	2
Course Title	Applied Algebra
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	General Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

The main purpose of the course is to introduce the student to the basic concepts, results, tools and methods of the theory of polynomials over finite fields and its applications to algebraic cryptography and coding theory, using tools from the theory of algebraic curves. In addition, the elementary theory of elliptic curves is developed and several applications are given to various areas of Mathematics and other sciences. At the end of the course, we expect the student to understand the basic concepts and the main theorems that are analysed in the course, to understand how these are applied to concrete examples arising in Mathematics and related sciences, to be able to apply them to derive new elementary consequences in various thematic fields, and finally to be able to perform some (not so obvious) calculations related to the construction and analysis of algebraic codes and encrypted messages.

General Competences

The course aims at enabling the graduate to acquire the ability to analyse and synthesize basic knowledge of the theory of polynomials over finite fields in connection with the basic elements of algebraic curves, in particular of elliptic curves, which is an important part of modern Mathematics with numerous applications in other sciences, with a view to applications in coding theory and algebraic cryptography. In particular, in the course are analyzed: the basic theory of codes (linear and cyclic codes), the elementary theory of elliptic curves and their applications to cryptography. When the graduate comes in for the first time in connection with the basic notions of coding theory and the central concepts of elliptic curves and their applications to contemporary cryptography, (s)he strengthens her/his creative, analytical and inductive thinking, and her/his ability to apply abstract knowledge in different areas of central interest with numerous applications in everyday life.

Syllabus

• Finite Fields and Polynomials: (1) Rings, ideals, homomorphisms, polynomials, fields, algebraic extensions, (2) Finite fields, irreducible polynomials over finite fields, factorization of polynomials over finite fields, and (3) Reminders from elementary number theory.
• The null-space of a matrix. Linear and cyclic codes.
• Algebraic cryptography.
• Basic theory of algebraic curves.
• Elliptic curves.
• Applications of elliptic curves to algebraic cryptography.

Teaching and Learning Methods - Evaluation

Delivery	Face to face
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Student's study Hours 78 Exercises: Problem Solving 70.5 Course total 187.5
Student Performance Evaluation	The evaluation is based on the combined performance of the graduate student in: Weekly homeworks, Presentations during the semester, Major Homework at the end of the course, Written examination at the end of the courses in Greek with questions and problems of development of theoretical topics and problem solving.

Attached Bibliography

• N. Koblitz: “Algebraic aspects of cryptography”, Springer-Verlag, (1998).
• Δ. Πουλάκης: “Κρυπτογραφία”, Εκδόσεις Ζήτη, (2004).
• Δ. Πουλάκης: “Γεωμετρία των Αλγεβρικών Καμπυλών”, Εκδόσεις Ζήτη, (2006).
• Ι. Αντωνιάδης και Α. Κοντογεώργης: “Πεπερασμένα Σώματα και Κρυπτογραφία”, Σύνδεσμος Ελληνικών Ακαδημαϊκών Βιβλιοθηκών, (2015).
• I.F. Blake, G. Seroussi, and N. Smart: “Elliptic Curves in Cryptography”, Lecture Note Series. Cambridge University Press, (1999).
• N. Koblitz: “A Course in Number Theory and Cryptography”, Springer-Verlag, (1994).