MATH2088: Semester 2, 2022

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Units MATH2088 Semester 2 2022 [Normal day]

Unit of study_

MATH2088: Number Theory and Cryptography

Semester 2, 2022 [Normal day] - Camperdown/Darlington, Sydney

Overview

Cryptography is the branch of mathematics that provides the techniques for confidential exchange of information sent via possibly insecure channels. This unit introduces the tools from elementary number theory that are needed to understand the mathematics underlying the most commonly used modern public key cryptosystems. Topics include the Euclidean Algorithm, Fermat's Little Theorem, the Chinese Remainder Theorem, Mobius Inversion, the RSA Cryptosystem, the Elgamal Cryptosystem and the Diffie-Hellman Protocol. Issues of computational complexity are also discussed.

Unit details and rules

Unit code	MATH2088
Academic unit	Mathematics and Statistics Academic Operations
Credit points	6
Prohibitions ?	MATH2068 or MATH2988
Prerequisites ?	MATH1002 or MATH1902 or MATH1004 or MATH1904 or MATH1064 or (a mark of 65 or above in MATH1014)
Corequisites ?	None
Assumed knowledge ?	None
Available to study abroad and exchange students	Yes

Teaching staff

Coordinator	Dzmitry Badziahin, dzmitry.badziahin@sydney.edu.au
Tutor(s)	Van Nguyen, van.nguyen@sydney.edu.au
	Albie Ewins, aewi0252@uni.sydney.edu.au
	Mengfan Lyu, mengfan.lyu@sydney.edu.au

Type	Description	Weight	Due	Length
Final exam (Take-home short release)	Exam Examination	60%	Formal exam period	2 hours
Outcomes assessed:
Assignment	Assignment 1 Assignment	10%	Week 06 Due date: 08 Sep 2022 at 23:59 Closing date: 15 Sep 2022	See Canvas
Outcomes assessed:
Tutorial quiz	Quiz 1 Short answer questions	10%	Week 08	45 minutes
Outcomes assessed:
Assignment	Assignment 2 Assignment	10%	Week 11 Due date: 20 Oct 2022 at 23:59 Closing date: 27 Oct 2022	See Canvas
Outcomes assessed:
Tutorial quiz	Quiz 2 Short answer questions	10%	Week 12	45 minutes
Outcomes assessed:
= Type D final exam ?

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.

For more information see sydney.edu.au/students/guide-to-grades.

For more information see guide to grades.

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.

Academic integrity

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
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)
	1. Big-O notation 2. 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)
Week 09	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)
	Quadratic residues;	Lecture (1 hr)
Week 12	Square roots modulo a prime	Lecture (1 hr)
Week 12	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.

Required readings

The textbook for this 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.

Outcomes
Graduate qualities

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.

Graduate qualities

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

Learning outcomes	Graduate qualities
Learning outcomes

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

Additional information

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
Follow safety instructions in your manual and posted in laboratories
In case of fire, follow instructions posted outside the laboratory door
First aid kits, eye wash and fire extinguishers are located in or immediately outside each laboratory
As a precautionary measure, it is recommended that you have a current tetanus immunisation. This can be obtained from University Health Service: unihealth.usyd.edu.au/

Disclaimer

The University reserves the right to amend units of study or no longer offer certain units, including where there are low enrolment numbers.

To help you understand common terms that we use at the University, we offer an online glossary.

Current students

Overview

Overview

Unit details and rules

Teaching staff

Assessment

Assessment

Assessment summary

Assessment criteria

Late submission

Academic integrity

Learning support

Learning support

Simple extensions

Special consideration

Using AI responsibly

Weekly schedule

Weekly schedule

Study commitment

Required readings

Learning outcomes

Learning outcomes

Graduate qualities

Outcome map

Responding to student feedback

Responding to student feedback

Additional information

Additional information

Work, health and safety

Disclaimer

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