1 | Course Title: | THEORY OF ELLIPTIC CURVES AND ITS APPLICALTIONS I |
2 | Course Code: | MAT6111 |
3 | Type of Course: | Optional |
4 | Level of Course: | Third Cycle |
5 | Year of Study: | 1 |
6 | Semester: | 1 |
7 | ECTS Credits Allocated: | 5 |
8 | Theoretical (hour/week): | 3 |
9 | Practice (hour/week) : | 0 |
10 | Laboratory (hour/week) : | 0 |
11 | Prerequisites: | none |
12 | Recommended optional programme components: | None |
13 | Language: | Turkish |
14 | Mode of Delivery: | Face to face |
15 | Course Coordinator: | Prof. Dr. İSMAİL NACİ CANGÜL |
16 | Course Lecturers: | Prof. Dr. Osman Bizim |
17 | Contactinformation of the Course Coordinator: |
Uludağ Üniversitesi, Fen-Edebiyat Fakültesi Matematik Bölümü, Görükle Bursa-TÜRKİYE 0 224 294 17 57 / obizim@uludag.edu.tr |
18 | Website: | |
19 | Objective of the Course: | The theory of elliptic curves brings important areas of mathematics such as abstract algebra, number theory and related fields. The aim of this course is to make the students get all connections among all these areas. The goal is to teach the elementary theory of elliptic curves. So students can bring new ideas the theory of elliptic curves and have the ability conduct original research and independent publication. |
20 | Contribution of the Course to Professional Development |
21 | Learning Outcomes: |
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22 | Course Content: |
Week | Theoretical | Practical |
1 | Basic concepts of elliptic curves, the group law on the elliptic curves and proof of associativity. | |
2 | Other equations for elliptic curves, Legendre equation, cubic equations and quartic equations. The j-invariant of an elliptic curve and isomorphisms and endomorphisms of the curves. | |
3 | Torsion points of elliptic curves and division polynomials of an elliptic curve. Weil paring, Tate-Licthenbaum pairing. | |
4 | Elliptic curves over finite fields, counting the number of the points on these curves and the theorem of Hasse, The frobenius endomorphism, Schoof’s Algorithm. | |
5 | Determining the group structure of the points on the elliptic curves over finite fields and the group order. Some family of elliptic curves over finite fields, singular and supersingular curves. | |
6 | The discrete logarithm problem, general attacks on discerete logs, baby step, giant step, Pollard’s method, the Pohling-Hellman method. | |
7 | MOV attack, Frey-Rück attack and other attacks. | |
8 | The basic concepts in the elliptic curve cryptography, Diffie-Hellman key Exchange, Massey-Omura encryption. | |
9 | ElGamal public key encryption and the digital signatures, the digital signature algorithm. | |
10 | The elliptic curves over Q and the torsion subgroup and the Lutz-Nagell theorem, the method of descent, the Mordell- Weil theorem. | |
11 | 2-Selmer groups, Shafarevich-Tate grous, a nontrivial Shafarevich-Tate groups, Galois cohomology. | |
12 | The elliptic curves over C, doubly periodic functions, tori are elliptic curves, the arithmetic and geometric mean. | |
13 | Division poliynomials and the torsion subgroups, Doud’s method, complex multiplication and numerical examples, the integrality of j-invariants. | |
14 | Hyperelliptic curves, Cantor’s algorithm, zeta functions, Fermat’s last theorem, sketch of Wiles’s proof. |
23 | Textbooks, References and/or Other Materials: |
[1] Rational Points on Elliptic Curves, J. H. Silverman ve J. Tate, [2]The Arithmetic of Elliptic Curves, J. H. Silverman, [3]Elliptic Curves, L. C. Washington. [4] Introduction to Elliptic Curves and Modular Forms, N. Koblitz. |
24 | Assesment |
TERM LEARNING ACTIVITIES | NUMBER | PERCENT |
Midterm Exam | 0 | 0 |
Quiz | 0 | 0 |
Homeworks, Performances | 0 | 0 |
Final Exam | 1 | 100 |
Total | 1 | 100 |
Contribution of Term (Year) Learning Activities to Success Grade | 0 | |
Contribution of Final Exam to Success Grade | 100 | |
Total | 100 | |
Measurement and Evaluation Techniques Used in the Course | ||
Information |
25 | ECTS / WORK LOAD TABLE |
Activites | NUMBER | TIME [Hour] | Total WorkLoad [Hour] |
Theoretical | 14 | 3 | 42 |
Practicals/Labs | 0 | 0 | 0 |
Self Study and Preparation | 14 | 5 | 70 |
Homeworks, Performances | 0 | 0 | 0 |
Projects | 0 | 0 | 0 |
Field Studies | 0 | 0 | 0 |
Midtermexams | 0 | 0 | 0 |
Others | 14 | 5 | 70 |
Final Exams | 1 | 13 | 13 |
Total WorkLoad | 195 | ||
Total workload/ 30 hr | 6,5 | ||
ECTS Credit of the Course | 6,5 |
26 | CONTRIBUTION OF LEARNING OUTCOMES TO PROGRAMME QUALIFICATIONS | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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LO: Learning Objectives | PQ: Program Qualifications |
Contribution Level: | 1 Very Low | 2 Low | 3 Medium | 4 High | 5 Very High |