Unit of study_

# MATH2988: Number Theory and Cryptography Adv

## Overview

This unit of study is an advanced version of MATH2088, sharing the same lectures but with more advanced topics introduced in the tutorials and computer laboratory sessions.

### Unit details and rules

Unit code MATH2988 Mathematics and Statistics Academic Operations 6 MATH2068 or MATH2088 MATH1902 or MATH1904 or (a mark of 65 or above in MATH1002 or MATH1004 or MATH1064) None None Yes

## Assessment

Type Description Weight Due Length
Supervised exam

Exam
Examination
60% Formal exam period 2 hours
Outcomes assessed:
Assignment Assignment 1
Assignment
10% Week 06
Due date: 07 Sep 2023 at 23:59

Closing date: 15 Sep 2023
See Canvas
Outcomes assessed:
Tutorial quiz Quiz 1
Quiz
10% Week 08
Due date: 21 Sep 2023 at 23:59

Closing date: 21 Sep 2023
45 minutes
Outcomes assessed:
Assignment Assignment 2
Assignment
10% Week 11
Due date: 19 Oct 2023 at 23:59

Closing date: 27 Oct 2023
See Canvas
Outcomes assessed:
Tutorial quiz Quiz 2
Quiz
10% Week 12
Due date: 26 Oct 2023 at 23:59

Closing date: 26 Oct 2023
45 minutes
Outcomes assessed:

### Assessment summary

• Quizzes: The quizzes test students’ understanding of the basic concepts and computational methods from the lectures and tutorials in weeks 1–6 for quiz 1, and in weeks 7–11 for quiz 2. University-approved non-programmable calculators may be used.
• Assignments: The assignments will be comprised of a theoretical element, testing students’ understanding of the number theory developed in lectures and tutorials (including ability to write correct proofs), and a computorial element, testing students’ understanding of the cryptographic concepts and methods developed in lectures and computer labs.
• Final exam: The exam will test the learning outcomes attained in lectures, tutorials and computer labs. University-approved non-programmable calculators may be used.

Detailed information for each assessment can be found on Canvas.

### Assessment criteria

The University awards common result grades, set out in the Coursework Policy 2014 (Schedule 1).

As a general guide, a high distinction indicates work of an exceptional standard, a distinction a very high standard, a credit a good standard, and a pass an acceptable standard.

Result name

Mark range

Description

High distinction

85 - 100

Represents complete or close to complete mastery of the material.

Distinction

75 - 84

Represents excellence, but substantially less than complete mastery.

Credit

65 - 74

Represents a creditable performance that goes beyond routine knowledge and understanding, but less than excellence.

Pass

50 - 64

Represents at least routine knowledge and understanding over a spectrum of topics and important ideas and concepts in the course.

Fail

0 - 49

When you don’t meet the learning outcomes of the unit to a satisfactory standard.

### Late submission

In accordance with University policy, these penalties apply when written work is submitted after 11:59pm on the due date:

• Deduction of 5% of the maximum mark for each calendar day after the due date.
• After ten calendar days late, a mark of zero will be awarded.

The Current Student website  provides information on academic integrity and the resources available to all students. The University expects students and staff to act ethically and honestly and will treat all allegations of academic integrity breaches seriously.

We use similarity detection software to detect potential instances of plagiarism or other forms of academic integrity breach. If such matches indicate evidence of plagiarism or other forms of academic integrity breaches, your teacher is required to report your work for further investigation.

You may only use artificial intelligence and writing assistance tools in assessment tasks if you are permitted to by your unit coordinator, and if you do use them, you must also acknowledge this in your work, either in a footnote or an acknowledgement section.

Studiosity is permitted for postgraduate units unless otherwise indicated by the unit coordinator. The use of this service must be acknowledged in your submission.

## Learning support

### Simple extensions

If you encounter a problem submitting your work on time, you may be able to apply for an extension of five calendar days through a simple extension.  The application process will be different depending on the type of assessment and extensions cannot be granted for some assessment types like exams.

### Special consideration

If exceptional circumstances mean you can’t complete an assessment, you need consideration for a longer period of time, or if you have essential commitments which impact your performance in an assessment, you may be eligible for special consideration or special arrangements.

Special consideration applications will not be affected by a simple extension application.

### Using AI responsibly

Co-created with students, AI in Education includes lots of helpful examples of how students use generative AI tools to support their learning. It explains how generative AI works, the different tools available and how to use them responsibly and productively.

## Weekly schedule

WK Topic Learning activity Learning outcomes
Week 01 1. Introduction to number theory and cryptography; 2. Mathematical induction Lecture (1 hr)
1. Divisibility; 2. Greatest common divisors; 3. Euclidean algorithm Lecture (1 hr)
1. Extended Euclidean algorithm; 2. Prime and composite numbers Lecture (1 hr)
Week 02 1. Fundamental theorem of arithmetic; 2. Factorisation; 3. Fermat method Lecture (1 hr)
1. Congruencies; 2. Complete and reduced systems of residues Lecture (1 hr)
1. Inverses; 2. Powers and order in modular arithmetic Lecture (1 hr)
Week 03 1. Basic concepts of cryptography; 2. Classical cryptosystems Lecture (1 hr)
Statistical attacks on classical cryptosystems Lecture (1 hr)
1. Euler–Fermat Theorem; 2. RSA theorem Lecture (1 hr)
Week 04 Relating congruences with different moduli Lecture (1 hr)
1. Chinese Remainder Theorem; 2. Computing big powers in modular arithmetics Lecture (1 hr)
Computation of k’th root modulo a number Lecture (1 hr)
Week 05 1. Multiplicative functions; 2. Euler’s phi function; 3. Mobius and Liouville functions Lecture (1 hr)
Sum and number of divisors Lecture (1 hr)
1. Classification of even perfect numbers; 2. More on Euler phi-function Lecture (1 hr)
Week 06 Relating different multiplicative functions Lecture (1 hr)
Mobius inversion formula Lecture (1 hr)
The RSA public key cryptosystem Lecture (1 hr)
Week 07 1. Computational complexity; 2. Elementary bit operations Lecture (1 hr)
1. Complexity of multiplication; 2. Karatsuba method Lecture (1 hr)
Big-O notation computational complexity of Euclidean algorithm Lecture (1 hr)
Week 08 1. Complexity of power algorithm; 2. Complexity of primality checks Lecture (1 hr)
Pollard’s rho factorisation algorithm Lecture (1 hr)
Polynomial congruences Lecture (1 hr)
Week 09 Primitive roots and discrete logarithms Lecture (1 hr)
1. Diffie–Hellman key exchange protocol; 2. Elgamal cryptosystem Lecture (1 hr)
Week 10 1. Safe primes; 2. Finding primitive roots Lecture (1 hr)
Applications of primitive roots Lecture (1 hr)
Modular Lagrange interpolation formula Lecture (1 hr)
Week 11 1. Secret sharing; 2. Baby-step/giant-step algorithm for discrete logarithms Lecture (1 hr)
Pohlig–Hellman algorithm for discrete logarithms Lecture (1 hr)
Week 12 Square roots modulo a prime Lecture (1 hr)
1. Square root problem modulo a composite number; 2. Rabin’s public key cryptosystem Lecture (1 hr)

### Study commitment

Typically, there is a minimum expectation of 1.5-2 hours of student effort per week per credit point for units of study offered over a full semester. For a 6 credit point unit, this equates to roughly 120-150 hours of student effort in total.

The textbook for the course is

• R. Howlett, Number Theory and Cryptography, School of Mathematics and Statistics, University of Sydney, 2019.

Although lecture notes will be posted, you will probably need a copy of the textbook, which is available from Kopystop. The changes from previous years’ editions are minimal, so second-hand copies will be fine.

## Learning outcomes

Learning outcomes are what students know, understand and are able to do on completion of a unit of study. They are aligned with the University's graduate qualities and are assessed as part of the curriculum.

At the completion of this unit, you should be able to:

• LO1. understand and use the basic terminology of number theory and cryptography
• LO2. carry out simple number-theoretic computations either with a calculator or using MAGMA
• LO3. apply standard number-theoretic algorithms
• LO4. understand and use some classical and number-theoretic cryptosystems
• LO5. apply standard methods to attack some classical cryptosystems
• LO6. understand the theory underlying number-theoretic algorithms and cryptosystems, including the general properties of primes, prime factorisation, modular arithmetic, divisors and multiplicative functions, powers and discrete logarithms.

The graduate qualities are the qualities and skills that all University of Sydney graduates must demonstrate on successful completion of an award course. As a future Sydney graduate, the set of qualities have been designed to equip you for the contemporary world.

 GQ1 Depth of disciplinary expertise Deep disciplinary expertise is the ability to integrate and rigorously apply knowledge, understanding and skills of a recognised discipline defined by scholarly activity, as well as familiarity with evolving practice of the discipline. GQ2 Critical thinking and problem solving Critical thinking and problem solving are the questioning of ideas, evidence and assumptions in order to propose and evaluate hypotheses or alternative arguments before formulating a conclusion or a solution to an identified problem. GQ3 Oral and written communication Effective communication, in both oral and written form, is the clear exchange of meaning in a manner that is appropriate to audience and context. GQ4 Information and digital literacy Information and digital literacy is the ability to locate, interpret, evaluate, manage, adapt, integrate, create and convey information using appropriate resources, tools and strategies. GQ5 Inventiveness Generating novel ideas and solutions. GQ6 Cultural competence Cultural Competence is the ability to actively, ethically, respectfully, and successfully engage across and between cultures. In the Australian context, this includes and celebrates Aboriginal and Torres Strait Islander cultures, knowledge systems, and a mature understanding of contemporary issues. GQ7 Interdisciplinary effectiveness Interdisciplinary effectiveness is the integration and synthesis of multiple viewpoints and practices, working effectively across disciplinary boundaries. GQ8 Integrated professional, ethical, and personal identity An integrated professional, ethical and personal identity is understanding the interaction between one’s personal and professional selves in an ethical context. GQ9 Influence Engaging others in a process, idea or vision.

### Outcome map

GQ1 GQ2 GQ3 GQ4 GQ5 GQ6 GQ7 GQ8 GQ9

## Responding to student feedback

This section outlines changes made to this unit following staff and student reviews.

No changes have been made since this unit was last offered

### Work, health and safety

We are governed by the Work Health and Safety Act 2011, Work Health and Safety Regulation 2011 and Codes of Practice. Penalties for non-compliance have increased. Everyone has a responsibility for health and safety at work. The University’s Work Health and Safety policy explains the responsibilities and expectations of workers and others, and the procedures for managing WHS risks associated with University activities.

General laboratory safety rules

• No eating or drinking is allowed in any laboratory under any circumstances

• A laboratory coat and closed-toe shoes are mandatory