• Class Number 8588
  • Term Code 3060
  • Class Info
  • Unit Value 6 units
  • Mode of Delivery In Person
    • Dr James Borger
    • Dr James Borger
  • Class Dates
  • Class Start Date 27/07/2020
  • Class End Date 30/10/2020
  • Census Date 31/08/2020
  • Last Date to Enrol 03/08/2020
SELT Survey Results

The need to protect information being transmitted electronically (such as the widespread use of electronic payment) has transformed the importance of cryptography. Most of the modern types of cryptosystems rely on (increasingly more sophisticated) number theory for their theoretical background. This course introduces elementary number theory, with an emphasis on those parts that have applications to cryptography, and shows how the theory can be applied to cryptography.

Number theory topics will be chosen from: the Euclidean algorithm, highest common factor, prime numbers, prime factorisation, primality testing, congruences, the Chinese remainder theorem, diophantine equations, sums of squares, Euler's function, Fermat's little theorem, quadratic residues, quadratic reciprocity, Pell's equation, continued fractions.

Cryptography topics will be chosen from: symmetric key cryptosystems, including classical examples and a brief discussion of modern systems such as DES and AES; public key systems such as RSA and discrete logarithm systems; cryptanalysis (code breaking) using some of the number theory developed.

Note: Graduate students attend joint classes with undergraduates but are assessed separately.

Learning Outcomes

Upon successful completion, students will have the knowledge and skills to:

On satisfying the requirements of this course, students will have the knowledge and skills to:

1. Solve problems in elementary number theory
2. Apply elementary number theory to cryptography
3. Develop a deeper conceptual understanding of the theoretical basis of number theory and cryptography

Examination Material or equipment


Staff Feedback

Students will be given feedback in the following forms in this course:

  • written comments
  • verbal comments
  • feedback to whole class, groups, individuals, focus group etc

Student Feedback

ANU is committed to the demonstration of educational excellence and regularly seeks feedback from students. Students are encouraged to offer feedback directly to their Course Convener or through their College and Course representatives (if applicable). The feedback given in these surveys is anonymous and provides the Colleges, University Education Committee and Academic Board with opportunities to recognise excellent teaching, and opportunities for improvement. The Surveys and Evaluation website provides more information on student surveys at ANU and reports on the feedback provided on ANU courses.

Class Schedule

Week/Session Summary of Activities Assessment
1 Prime numbers, uniqueness of factorization, Euclid's algorithm
2 Diophantine equations, arithmetic modulo N
3 Linear Diophantine equations, modular inverses, Chinese remainder theorem
4 Wilson's theorem, Fermat's Little Theorem, Pollard's algorithm, Euler's theorem
5 Primality testing, pseudoprimes, Miller's test, Rabin's test, cryptosystems
6 Public key cryptography, RSA, primitive roots, multiplicative functions
7 Existence of primitive roots modulo primes and odd prime powers
8 Discrete logarithms, primality test and Miller's test revisited
9 Continued fractions, Diophantine approximation
10 Applications of Dirichlet's theorem, continued fractions and Diophantine equations, solutions to Pell's equation
11 quadratic residues, Legendre symbol, quadratic reciprocity
12 Guest lectures and supplementary topics

Assessment Summary

Assessment task Value Due Date Return of assessment Learning Outcomes
Assignment 1 10 % 21/08/2020 28/08/2020 1,2,3
Assignment 2 10 % 04/09/2020 11/09/2020 1,2,3
Assignment 3 10 % 02/10/2020 09/10/2020 1,2,3
Assignment 4 10 % 16/10/2020 23/10/2020 1,2,3
Assignment 5 10 % 30/10/2020 06/11/2020 1,2,3
Final examination 50 % 05/11/2020 03/12/2020 1,2,3

* If the Due Date and Return of Assessment date are blank, see the Assessment Tab for specific Assessment Task details


ANU has educational policies, procedures and guidelines, which are designed to ensure that staff and students are aware of the University’s academic standards, and implement them. Students are expected to have read the Academic Misconduct Rule before the commencement of their course. Other key policies and guidelines include:

Assessment Requirements

The ANU is using Turnitin to enhance student citation and referencing techniques, and to assess assignment submissions as a component of the University's approach to managing Academic Integrity. For additional information regarding Turnitin please visit the Academic Integrity . In rare cases where online submission using Turnitin software is not technically possible; or where not using Turnitin software has been justified by the Course Convener and approved by the Associate Dean (Education) on the basis of the teaching model being employed; students shall submit assessment online via ‘Wattle’ outside of Turnitin, or failing that in hard copy, or through a combination of submission methods as approved by the Associate Dean (Education). The submission method is detailed below.

Moderation of Assessment

Marks that are allocated during Semester are to be considered provisional until formalised by the College examiners meeting at the end of each Semester. If appropriate, some moderation of marks might be applied prior to final results being released.


Students are expected to attend lectures and contribute to discussions. When this is not possible students are expected to listen to the audio recording(s) of all lectures


Final examination, 3 hours, no permitted materials. Please check the ANU final Examination Timetable to confirm the date, time and location of the end of semester exam.

Assessment Task 1

Value: 10 %
Due Date: 21/08/2020
Return of Assessment: 28/08/2020
Learning Outcomes: 1,2,3

Assignment 1

Questions on the material presented in lecture in weeks 1-3.

Assessment Task 2

Value: 10 %
Due Date: 04/09/2020
Return of Assessment: 11/09/2020
Learning Outcomes: 1,2,3

Assignment 2

Questions on the material presented in lecture in weeks 4-5.

Assessment Task 3

Value: 10 %
Due Date: 02/10/2020
Return of Assessment: 09/10/2020
Learning Outcomes: 1,2,3

Assignment 3

Questions on the material presented in lecture in weeks 6-7.

Assessment Task 4

Value: 10 %
Due Date: 16/10/2020
Return of Assessment: 23/10/2020
Learning Outcomes: 1,2,3

Assignment 4

Questions on the material presented in lecture in weeks 8-9.

Assessment Task 5

Value: 10 %
Due Date: 30/10/2020
Return of Assessment: 06/11/2020
Learning Outcomes: 1,2,3

Assignment 5

Questions on the material presented in lecture in weeks 10-11.

Assessment Task 6

Value: 50 %
Due Date: 05/11/2020
Return of Assessment: 03/12/2020
Learning Outcomes: 1,2,3

Final examination

The date range in the Assessment Summary indicates the start of the end of semester exam period and the date official end of semester results are released on ISIS. Please check the ANU final Examination Timetable http://www.anu.edu.au/students/program-administration/assessments-exams/examination-timetable to confirm the date, time and location exam.

Academic Integrity

Academic integrity is a core part of the ANU culture as a community of scholars. At its heart, academic integrity is about behaving ethically, committing to honest and responsible scholarly practice and upholding these values with respect and fairness.

The ANU commits to assisting all members of our community to understand how to engage in academic work in ways that are consistent with, and actively support academic integrity. The ANU expects staff and students to be familiar with the academic integrity principle and Academic Misconduct Rule, uphold high standards of academic integrity and act ethically and honestly, to ensure the quality and value of the qualification that you will graduate with.

The Academic Misconduct Rule is in place to promote academic integrity and manage academic misconduct. Very minor breaches of the academic integrity principle may result in a reduction of marks of up to 10% of the total marks available for the assessment. The ANU offers a number of online and in person services to assist students with their assignments, examinations, and other learning activities. Visit the Academic Skills website for more information about academic integrity, your responsibilities and for assistance with your assignments, writing skills and study.

Online Submission

You will be required to electronically sign a declaration as part of the submission of your assignment. Please keep a copy of the assignment for your records. Unless an exemption has been approved by the Associate Dean (Education) submission must be through Turnitin.

Hardcopy Submission

For some forms of assessment (hand written assignments, art works, laboratory notes, etc.) hard copy submission is appropriate when approved by the Associate Dean (Education). Hard copy submissions must utilise the Assignment Cover Sheet. Please keep a copy of tasks completed for your records.

Late Submission

Individual assessment tasks may or may not allow for late submission. Policy regarding late submission is detailed below:

  • Late submission permitted. Late submission of assessment tasks without an extension are penalised at the rate of 5% of the possible marks available per working day or part thereof. Late submission of assessment tasks is not accepted after 10 working days after the due date, or on or after the date specified in the course outline for the return of the assessment item. Late submission is not accepted for take-home examinations.

Referencing Requirements

Accepted academic practice for referencing sources that you use in presentations can be found via the links on the Wattle site, under the file named “ANU and College Policies, Program Information, Student Support Services and Assessment”. Alternatively, you can seek help through the Students Learning Development website.

Returning Assignments

Assignments will be returned via Turnitin.

Extensions and Penalties

Extensions and late submission of assessment pieces are covered by the Student Assessment (Coursework) Policy and Procedure. Extensions may be granted for assessment pieces that are not examinations or take-home examinations. If you need an extension, you must request an extension in writing on or before the due date. If you have documented and appropriate medical evidence that demonstrates you were not able to request an extension on or before the due date, you may be able to request it after the due date.

Resubmission of Assignments

No resubmission.

Privacy Notice

The ANU has made a number of third party, online, databases available for students to use. Use of each online database is conditional on student end users first agreeing to the database licensor’s terms of service and/or privacy policy. Students should read these carefully. In some cases student end users will be required to register an account with the database licensor and submit personal information, including their: first name; last name; ANU email address; and other information.
In cases where student end users are asked to submit ‘content’ to a database, such as an assignment or short answers, the database licensor may only use the student’s ‘content’ in accordance with the terms of service – including any (copyright) licence the student grants to the database licensor. Any personal information or content a student submits may be stored by the licensor, potentially offshore, and will be used to process the database service in accordance with the licensors terms of service and/or privacy policy.
If any student chooses not to agree to the database licensor’s terms of service or privacy policy, the student will not be able to access and use the database. In these circumstances students should contact their lecturer to enquire about alternative arrangements that are available.

Distribution of grades policy

Academic Quality Assurance Committee monitors the performance of students, including attrition, further study and employment rates and grade distribution, and College reports on quality assurance processes for assessment activities, including alignment with national and international disciplinary and interdisciplinary standards, as well as qualification type learning outcomes.

Since first semester 1994, ANU uses a grading scale for all courses. This grading scale is used by all academic areas of the University.

Support for students

The University offers students support through several different services. You may contact the services listed below directly or seek advice from your Course Convener, Student Administrators, or your College and Course representatives (if applicable).

Dr James Borger

Research Interests

Dr James Borger

Dr James Borger

Research Interests

Dr James Borger

Responsible Officer: Registrar, Student Administration / Page Contact: Website Administrator / Frequently Asked Questions