• Class Number 6232
• Term Code 3160
• Class Info
• Unit Value 6 units
• Mode of Delivery In Person
• COURSE CONVENER
• AsPr James Borger
• LECTURER
• AsPr James Borger
• Class Dates
• Class Start Date 26/07/2021
• Class End Date 29/10/2021
• Census Date 14/09/2021
• Last Date to Enrol 02/08/2021
SELT Survey Results

Number Theory and Cryptography (MATH3301)

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.

Honours Pathway Option (HPO):

Students who take the HPO will complete extra work of a more theoretical nature. The assignments will be replaced by alternative assignments and the final exam will contain alternative questions requiring deeper conceptual understanding.

## 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. (HPO only) Develop a deeper conceptual understanding of the theoretical basis of number theory and cryptography

## Examination Material or equipment

None.

Recommended student system requirements

ANU courses commonly use a number of online resources and activities including:

• video material, similar to YouTube, for lectures and other instruction
• two-way video conferencing for interactive learning
• email and other messaging tools for communication
• interactive web apps for formative and collaborative activities
• print and photo/scan for handwritten work
• home-based assessment.

To fully participate in ANU learning, students need:

• A computer or laptop. Mobile devices may work well but in some situations a computer/laptop may be more appropriate.
• Webcam
• Speakers and a microphone (e.g. headset)
• Reliable, stable internet connection. Broadband recommended. If using a mobile network or wi-fi then check performance is adequate.
• Suitable location with minimal interruptions and adequate privacy for classes and assessments.
• Printing, and photo/scanning equipment

## Staff Feedback

Students will be given feedback in the following forms in this course:
• Feedback to the whole class, to groups, to individuals, focus groups

## 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
12 Guest lectures and supplementary topics

## Assessment Summary

Assessment task Value Due Date Return of assessment Learning Outcomes
Assignment 1 10 % 20/08/2021 27/08/2021 1,2
Assignment 2 10 % 03/09/2021 10/09/2021 1,2
Assignment 3 10 % 01/10/2021 08/10/2021 1,2
Assignment 4 10 % 15/10/2021 22/10/2021 1,2
Assignment 5 10 % 29/10/2021 05/11/2021 1,2
Final examination 50 % 04/11/2021 02/12/2021 1,2

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

## Policies

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 ANU Online website Students may choose not to submit assessment items through Turnitin. In this instance you will be required to submit, alongside the assessment item itself, hard copies of all references included in the assessment item.

## 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.

## Participation

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.

This course has been adjusted for remote participants, there will be separate workshops allocated for in person and on campus students. Remote students are encouraged to identify themselves to the course convener before the start of semester

## Examination(s)

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.

Value: 10 %
Due Date: 20/08/2021
Return of Assessment: 27/08/2021
Learning Outcomes: 1,2

Assignment 1

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

Value: 10 %
Due Date: 03/09/2021
Return of Assessment: 10/09/2021
Learning Outcomes: 1,2

Assignment 2

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

Value: 10 %
Due Date: 01/10/2021
Return of Assessment: 08/10/2021
Learning Outcomes: 1,2

Assignment 3

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

Value: 10 %
Due Date: 15/10/2021
Return of Assessment: 22/10/2021
Learning Outcomes: 1,2

Assignment 4

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

Value: 10 %
Due Date: 29/10/2021
Return of Assessment: 05/11/2021
Learning Outcomes: 1,2

Assignment 5

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

Value: 50 %
Due Date: 04/11/2021
Return of Assessment: 02/12/2021
Learning Outcomes: 1,2

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.

## Online Submission

The ANU uses 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. While the use of Turnitin is not mandatory, the ANU highly recommends Turnitin is used by both teaching staff and students. For additional information regarding Turnitin please visit the ANU Online website.

## 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

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 5 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 The Course Convener may grant extensions 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.

No resubmission.

## Privacy Notice

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).

## Convener

 AsPr James Borger James.Borger@anu.edu.au

## Instructor

 AsPr James Borger 61252908 James.Borger@anu.edu.au