Syllabus ( MATH 662 )
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Basic information
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| Course title: |
Elliptic Curves |
| Course code: |
MATH 662 |
| Lecturer: |
Assoc. Prof. Dr. Fatma KARAOĞLU CEYHAN
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| ECTS credits: |
7.5 |
| GTU credits: |
3 (3+0+0) |
| Year, Semester: |
1/2, Fall and Spring |
| Level of course: |
Third Cycle (Doctoral) |
| Type of course: |
Area Elective
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| Language of instruction: |
English
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| Mode of delivery: |
Face to face , Group study
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| Pre- and co-requisites: |
none |
| Professional practice: |
No |
| Purpose of the course: |
Elliptic curves are especially important in number theory. They also find applications in elliptic curve cryptography (ECC) and integer factorization. The aim of the course is to introduce the theory of elliptic curves and the application areas of elliptic curves . Moreover, to help student to see the connection between the theory of elliptic curves and some important areas of mathematics such as number theory and cryptography. In this course students will learn the basics of elliptic curves so that they can develop their knowledge on the theory of elliptic curves by themself. |
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Learning outcomes
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Upon successful completion of this course, students will be able to:
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Recognize elliptic curves and explain the equations which define these curves.
Contribution to Program Outcomes
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Define and manipulate advanced concepts of Mathematics in a specialized way
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Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
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Understand relevant research methodologies and techniques and their appropriate application within his/her research field,
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Develop mathematical, communicative, problem-solving, brainstorming skills.
Method of assessment
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Homework assignment
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Apply the j-invariant of elliptic curves, Tate modul, endomorphism ring and automorphism group.
Contribution to Program Outcomes
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Define and manipulate advanced concepts of Mathematics in a specialized way
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Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
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Understand relevant research methodologies and techniques and their appropriate application within his/her research field,
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Acquire scientific knowledge and work independently,
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Work effectively in multi-disciplinary research teams
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Continuously develop their knowledge and skills in order to adapt to a rapidly developing technological environment
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Develop mathematical, communicative, problem-solving, brainstorming skills.
Method of assessment
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Written exam
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Homework assignment
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Detect some special transformations between elliptic curves, isogenies, the algorithmic aspects of elliptic curves and elliptic curve cryptography.
Contribution to Program Outcomes
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Define and manipulate advanced concepts of Mathematics in a specialized way
-
Relate mathematics to other disciplines and develop mathematical models for multidisciplinary problems
-
Understand relevant research methodologies and techniques and their appropriate application within his/her research field,
-
Acquire scientific knowledge and work independently,
-
Work effectively in multi-disciplinary research teams
-
Continuously develop their knowledge and skills in order to adapt to a rapidly developing technological environment
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Develop mathematical, communicative, problem-solving, brainstorming skills.
Method of assessment
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Written exam
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Homework assignment
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Contents
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| Week 1: |
Algebraic varieties |
| Week 2: |
Algebraic curves |
| Week 3: |
Elliptic curves, Weierstrass Equations, the group law |
| Week 4: |
Other equations for elliptic curves: Legendre equations, cubic and quartic equations |
| Week 5: |
j-invariant and Isomorphisms |
| Week 6: |
j-invariant and Isomorphisms (continued) |
| Week 7: |
Isogenies, Endomorphism ring, Automorphism group, Torsion points |
| Week 8: |
Weil pairing, Elliptic curves over finite fields |
| Week 9: |
Elliptic curves over complex numbers |
| Week 10: |
Algorithmic aspects of elliptic curves, Double-and -Add Algorithms |
| Week 11: |
Counting the number of points on elliptic curves |
| Week 12: |
Elliptic curve cryptography |
| Week 13: |
Elliptic curve cryptography |
| Week 14: |
Elliptic curve cryptography |
| Week 15*: |
- |
| Week 16*: |
Final Exam. |
| Textbooks and materials: |
The Arithmetic of Elliptic Curves, 2nd Edition, Joseph H. Silverman.
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| Recommended readings: |
Elliptic Curves, 2nd Edition, Lawrence C. Washington. |
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* Between 15th and 16th weeks is there a free week for students to prepare for final exam.
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Assessment
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| Method of assessment |
Week number |
Weight (%) |
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| Mid-terms: |
5 |
30 |
| Other in-term studies: |
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0 |
| Project: |
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0 |
| Homework: |
2,3,4,6,7,8,9,10,11,12 |
30 |
| Quiz: |
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0 |
| Final exam: |
16 |
40 |
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Total weight: |
(%) |
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Workload
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| Activity |
Duration (Hours per week) |
Total number of weeks |
Total hours in term |
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| Courses (Face-to-face teaching): |
3 |
14 |
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| Own studies outside class: |
4 |
14 |
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| Practice, Recitation: |
0 |
0 |
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| Homework: |
6 |
10 |
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| Term project: |
0 |
0 |
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| Term project presentation: |
0 |
0 |
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| Quiz: |
0 |
0 |
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| Own study for mid-term exam: |
10 |
1 |
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| Mid-term: |
2 |
1 |
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| Personal studies for final exam: |
15 |
1 |
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| Final exam: |
2 |
1 |
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Total workload: |
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Total ECTS credits: |
* |
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* ECTS credit is calculated by dividing total workload by 25.
(1 ECTS = 25 work hours)
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