MATHS 328 : Algebra and Applications
2023 Semester One (1233) (15 POINTS)
This course deals with the applications of modern algebra, number theory and combinatorics, to areas such as information theory, cryptography, secret sharing and error-correction. It attracts a broad student audience, from engineering to computer science, who would like to understand in detail how modern information technologies work. In this course students learn to use computational algebra package GAP, which many find useful in their careers.
Capabilities Developed in this Course
|Capability 1:||Disciplinary Knowledge and Practice|
|Capability 2:||Critical Thinking|
|Capability 3:||Solution Seeking|
|Capability 4:||Communication and Engagement|
|Capability 5:||Independence and Integrity|
|Capability 6:||Social and Environmental Responsibilities|
- Display a high level of knowledge of number-theoretic and algebraic concepts in the foundation of modern applied algebra. (Capability 1 and 3)
- Understand complexity of algorithms and the role of complexity in modern cryptography. (Capability 2)
- Demonstrate high level of understanding of RSA and ElGamal cryptosystems, secret sharing schemes and error-correcting codes. (Capability 1, 2 and 3)
- Use competently GAP computational algebra package for computing in various algebraic systems and solving algebraic problems computationally. (Capability 1, 3 and 4)
- Engage in group discussions and critical interactions about the role of cryptography in the society. (Capability 2, 4, 5 and 6)
|Final Exam||45%||Individual Examination|
|Assessment Type||Learning Outcome Addressed|
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- Numbers (2 weeks) Division with remainder, divisibility, primes, prime factorisation. Greatest common divisor, extended Euclidean algorithm, Chinese remainder theorem. Euler's totient function. Congruences, Little Fermat's and Euler's theorems. Integers mod n. Binary representation of integers.
- Complexity (0.5 weeks) Big-oh, little-oh notation. Order of magnitude. Growth. Time complexity function of an algorithm. The time complexity of several number-theoretic algorithms. One-way functions, trapdoor functions.
- Cryptography (1.5 weeks) Classical secret-key cryptography: one-time pad, affine cryptosystem, Hill's cryptosystem. Modern public-key cryptography. RSA cryptosystem. Applications of cryptography to digital signature schemes, friend-or-foe identification, passwords.
- Groups and Elliptic Curves (2 weeks) Permutation groups. Cycle decomposition. Orders of permutations. General groups. Orders of elements. Isomorphism. Subgroups. Elliptic curves. Group of points on an elliptic curve. Applications of elliptic curves to cryptography.
- Fields (1 week) Fields. Vector spaces over finite fields. Finite fields. Multiplicative group of a finite field is cyclic. Discrete logarithms and ElGamal public-key cryptosystem.
- Introduction to Polynomials and Secret Sharing (1 week) Operations over polynomials. Division with remainder. Lagrange interpolation. Secret sharing schemes. Access structure. Shamir's threshold access secret sharing scheme.
- Polynomials and finite fields (1.5 weeks) Greatest common divisor. Irreducible polynomials. Factoring polynomials. Polynomials mod f(x). Finite fields.
- Error-correcting Codes (2.5 weeks) Codes. Hamming distance. Linear codes and matrix encoding techniques. Generator and parity check matrices. Polynomial codes. Hamming codes. Minimal annihilating polynomials and BCH codes. Non-binary codes. Reed-Solomon codes.
This course is a standard 15-point course and students are expected to spend 10 hours per week involved in each 15-point course that they are enrolled in. For each week of this course, you can expect 3 hours of lectures, a 1-hour tutorial, 3 hours of reading and thinking about the content, and 3 hours of work on assignments and/or test preparation.
- Attendance is expected at scheduled activities including tutorials.
- Lectures will be available as recordings. Other learning activities including tutorials will not be available as recordings.
- The course will not include live online events.
- The activities for the course are scheduled as a standard weekly timetable.
Course materials are made available in a learning and collaboration tool called Canvas which also includes reading lists and lecture recordings (where available).
Please remember that the recording of any class on a personal device requires the permission of the instructor.
The textbook for this course is Slinko, A. “Algebra for Applications: Cryptography, Secret Sharing, Error-Correcting, Fingerprinting. Compression.” Springer, 2nd ed. 2020. This book covers the material of the whole course and is available in electronic form through the Library.
The secondary textbook which covers about 2/3 of the course is Meijer, A. R. “Algebra for cryptologists.” Springer, 2016. It covers the material of the first part of the course.
During the course Class Representatives in each class can take feedback to the staff responsible for the course and staff-student consultative committees.
At the end of the course students will be invited to give feedback on the course and teaching through a tool called SET or Qualtrics. The lecturers and course co-ordinators will consider all feedback.
Your feedback helps to improve the course and its delivery for all students.
The University of Auckland will not tolerate cheating, or assisting others to cheat, and views cheating in coursework as a serious academic offence. The work that a student submits for grading must be the student's own work, reflecting their learning. Where work from other sources is used, it must be properly acknowledged and referenced. This requirement also applies to sources on the internet. A student's assessed work may be reviewed against online source material using computerised detection mechanisms.
Class representatives are students tasked with representing student issues to departments, faculties, and the wider university. If you have a complaint about this course, please contact your class rep who will know how to raise it in the right channels. See your departmental noticeboard for contact details for your class reps.
The content and delivery of content in this course are protected by copyright. Material belonging to others may have been used in this course and copied by and solely for the educational purposes of the University under license.
You may copy the course content for the purposes of private study or research, but you may not upload onto any third party site, make a further copy or sell, alter or further reproduce or distribute any part of the course content to another person.
All students are asked to discuss any impairment related requirements privately, face to face and/or in written form with the course coordinator, lecturer or tutor.
Student Disability Services also provides support for students with a wide range of impairments, both visible and invisible, to succeed and excel at the University. For more information and contact details, please visit the Student Disability Services’ website http://disability.auckland.ac.nz
If your ability to complete assessed coursework is affected by illness or other personal circumstances outside of your control, contact a member of teaching staff as soon as possible before the assessment is due.
If your personal circumstances significantly affect your performance, or preparation, for an exam or eligible written test, refer to the University’s aegrotat or compassionate consideration page https://www.auckland.ac.nz/en/students/academic-information/exams-and-final-results/during-exams/aegrotat-and-compassionate-consideration.html.
This should be done as soon as possible and no later than seven days after the affected test or exam date.
In the event of an unexpected disruption, we undertake to maintain the continuity and standard of teaching and learning in all your courses throughout the year. If there are unexpected disruptions the University has contingency plans to ensure that access to your course continues and course assessment continues to meet the principles of the University’s assessment policy. Some adjustments may need to be made in emergencies. You will be kept fully informed by your course co-ordinator/director, and if disruption occurs you should refer to the university website for information about how to proceed.
The delivery mode may change depending on COVID restrictions. Any changes will be communicated through Canvas.
Student Charter and Responsibilities
The Student Charter assumes and acknowledges that students are active participants in the learning process and that they have responsibilities to the institution and the international community of scholars. The University expects that students will act at all times in a way that demonstrates respect for the rights of other students and staff so that the learning environment is both safe and productive. For further information visit Student Charter https://www.auckland.ac.nz/en/students/forms-policies-and-guidelines/student-policies-and-guidelines/student-charter.html.
Elements of this outline may be subject to change. The latest information about the course will be available for enrolled students in Canvas.
In this course students may be asked to submit coursework assessments digitally. The University reserves the right to conduct scheduled tests and examinations for this course online or through the use of computers or other electronic devices. Where tests or examinations are conducted online remote invigilation arrangements may be used. In exceptional circumstances changes to elements of this course may be necessary at short notice. Students enrolled in this course will be informed of any such changes and the reasons for them, as soon as possible, through Canvas.