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Mathematics of Cryptography and Communications

Mathematics of Cryptography and Communications

MSc
  • Option 1 year full time or 2 years part time
  • Year of entry 2021
  • Campus Egham

The course

Excited by the role of mathematics in securing the modern electronics and communications that we all rely on? This intensive MSc programme explores the mathematics behind secure information and communications systems, in a department that is world renowned for research in the field.

You will learn to apply advanced mathematical ideas to cryptography, coding theory and information theory, by studying the relevant functions of algebra, number theory and combinatorial complexity theory and algorithms. In the process you will develop a critical appreciation of the challenges that mathematicians face in facilitating secure information transmission, data compression and encryption. You will learn to use advanced cypher systems, correcting codes and modern public key crypto-systems. As part of your studies you will have the opportunity to complete a supervised dissertation in an area of your choice, under the guidance of experts in the field who regularly publish in internationally competitive journals and work closely with partners in industry.

We are a lively, collaborative and supportive community of mathematicians and information security specialists, and thanks to our relatively compact scale we will take the time to get to know you as an individual. You will be assigned a personal advisor to guide you through your studies.

Mathematicians who can push the boundaries and stay ahead when it comes to cryptography and information security are in demand, and the skills you gain will open up a range of career options and provide a solid foundation if you wish to progress to a PhD. These include transferable skills such as familiarity with a computer-based algebra package, experience of carrying out independent research and managing the writing of a dissertation.

Core Modules

  • You will carry out a detailed study into a topic of your choosing in mathematics, analysing information from a range of sources. You will submit a written report of between 8,000 and 16,000 words in length.
  • In this module you will develop an understanding of the mathematical and security properties of both symmetric key cipher systems and public key cryptography. You will look at the concepts of secure communications and cipher systems, and learn how to use statistical information and the concept of entropy. You will consider the main properties of Boolean functions, their applications and use in cryptographic algorithms, and the structure of stream ciphers and block ciphers. You will examine how to construct keystream generators, and how to manipulate the concept of perfect secrecy. You will also analyse the concept of public key cryptography, including the details of the RSA and ElGamal cryptosystems.

  • To investigate the problems of data compression and information transmission in both noiseless and noisy environments.Entropy: Definition and mathematical properties of entropy, information and mutual information. Noiseless coding: Memoryless sources: proof of the Kraft inequality for uniquely decipherable codes, proof of the optimality of Huffman codes, typical sequences of a memory less source, the fixed-length coding theorem.Ergodic sources: entropy rate, the asymptotic equipartition property, the noiseless coding theorem for ergodic sources. Lempel-Ziv coding.Noisy coding: Noisy channels, the noisy channel coding theory, channel capacity.Further topics, such as hash codes, or the information-theoretic approach to cryptography and authentication.

     

  • In this module you will develop an understanding of the theory of error-correcting codes, employing the methods of elementary enumeration, linear algebra and finite fields. You will learn how to calculate the probability of error or the necessity of retransmission for a binary symmetric channel with given cross-over probability, with and without coding. You will& prove and apply various bounds on the number of possible code words in a code of given length and minimal distance, and use Hadamard matrices to construct medium-sized linear codes of certain parameters. You will also consider how to reduce a linear code to standard form, find a parity check matrix, and build standard array and syndrome decoding tables.
  • In this module you will develop an understanding of the mathematical ideas that underpin public key cryptography, such as discrete logarithms, lattices and elliptic curves. You will look at the RSA and Rabin cryptosystems, the hard problems on which their security relies, and attacks on them. You will consider finite fields, elliptic curves, and the discrete logarithm problem. You will examine security notions and attack models relevant for modern theoretical cryptography, such as indistinguishability and adaptive chosen ciphertext-attack. You will also gain experience in implementing cryptosystems and cryptanalytic methods using software such as Mathematica.

Optional Modules

There are a number of optional course modules available during your degree studies. The following is a selection of optional course modules that are likely to be available. Please note that although the College will keep changes to a minimum, new modules may be offered or existing modules may be withdrawn, for example, in response to a change in staff. Applicants will be informed if any significant changes need to be made.

  • In this module you will develop an understanding of the basic theory of field extensions. You will learn how to classify finite fields and determine the number of irreducible polynomials over a finite field. You will consider the fundamental thorem of Galois theory and how to compute in a finite field. You will also examine the applications of fields.
  • In this module you will develop an understanding of the principles of quantum superposition and quantum measurement. You will look at the many applications of quantum information theory, and learn how to manipulate tensor-product states and use the concept of entanglement. You will consider a range of problems involving one or two quantum bits and how to apply Grover's search algorithm. You will also examine applications of entanglement such as quantum teleportation or quantum secret key distribution, and analyse Deutsch's algorithm and its implications for the power of a quantum computer.
  • In this module you will develop an understanding of the fundamental principles of algorithm design, including basic data-structures and asymptotic notation. You will look at how algorithms are designed to meet desired specifications, and consider the importance of algorithmic efficiency. You will also examine fundamental problems such as sorting numbers and multiplying matrices.
  • In this module you will develop an understanding of the autoregressive conditionally heteroscedastic family of models in time series and the ideas behind the use of the BDS test and the bispectral test for time series. You will consider the partial differential equation for interest rates and its assumptions, and model forward and spot rates. You will consider the validity of various linear and non-linear time series occurring in finance, and apply stochastic calculus to interest rate movements and credit rating. You will also examine how to model the prices for Asian and barrier options.
  • In this module you will develop an understanding of the standard techniques and concepts of combinatorics, including methods of counting and the principle of inclusion and exclusion. You will perform simple calculations with generation functions, and look at Ramsey numbers, calculating upper and lower bounds for these. You will consider how to calculate sets by inclusion and exclusion, and examine how to use simple probabilistic tools for solving combinatorial problems.
  • In this module you will develop an understanding of the major methods used for testing and proving primality and for the factorisation of composite integers. You will look at the mathematical theory that underlies these methods, such as the theory of bionary quadratic forms, elliptic curves, and quadratic number fields. You willl also analayse the complexity of the fundamental number-theoretic algorithms.
  • In this module you will develop an understanding of the different classes of computational complexity. You will look at the formal definition of algorithms and Turing machines, and the concept of computational hardness. You will consider how to deduce cryptographic properties of related algorithms and protocols, and see how not all language are computable. You will examine the low-level complexity classes and prove that simple languages exist in each, and use complexity theoretic techniques as a method of analysing communication services.
  • In this module you will develop an understanding of the mathematical theory underlying the main principles and methods of statistics, in particular, parametric estimation and hypotheses testing. You will learn how to formulate statistical problems in rigorous mathematical terms, and how to select and apply appropriate tools of mathematical statistics and advanced probability. You will construct mathematical proofs of some of the main theoretical results of mathematical statistics and consider the asymptotic theory of estimation.

  • In this module you will develop an understanding of what it means for knots and links to be equivalent. You will look at the properties of a metric space, and learn how to determine whether a given function defines a metric. You will consider how topological spaces are defined and how to verify the axioms for given examples. You will examine the concepts of subspace, product spaces, quotient spaces, Hausdorff space, homeomorphism, connectedness and compactness, and the notions of Euler characteristic, orientability and how to apply these to the classification of closed surfaces.
  • In this module you will develop an understanding of the principal methods of the theory of stochastic processes, and probabilistic methods used to model systems that exhibit random behaviour. You will look at methods of conditioning, conditional expectation, and how to generate functions, and examine the structure and concepts of discrete and continuous time Markov chains with countable state space. You will also examine the structure of diffusion processes.

You will initially choose 8 courses from the list of available options, of which you specify 6 courses during the second term that will count towards your final award. You will also complete a core research project under the supervision of one of our academic staff.There is a strong focus on small group teaching throughout the programme.

Assessment is carried out through a variety of methods, including coursework, examinations and the main project. End-of-year examinations in May or June will count for 66.7% of your final award, while the dissertation will make up the remaining 33.3% and has to be submitted by September.

2:1

Mathematics as a main field of study and good marks in relevant courses.

Normally we require a UK 2:1 (Honours) or equivalent in relevant subjects but we will consider high 2:2 or relevant work experience. Candidates with professional qualifications in an associated area may be considered. Where a ‘good 2:2’ is considered, we would normally define this as reflecting a profile of 57% or above. Exceptionally, at the discretion of the course director, qualifications in other subjects (for example, physics or computer science) or degrees of lower classification may be considered.

International & EU requirements

English language requirements

All teaching at Royal Holloway (apart from some language courses) is in English. You will therefore need to have good enough written and spoken English to cope with your studies right from the start.

The scores we require
  • IELTS: 6.5 overall. No subscore lower than 5.5.
  • Pearson Test of English: 61 overall. No subscore lower than 51.
  • Trinity College London Integrated Skills in English (ISE): ISE III.
  • Cambridge English: Advanced (CAE) grade C.

Country-specific requirements

For more information about country-specific entry requirements for your country please see here.

By the end of this programme you will have an advanced knowledge and understanding of all the key mathematical principles and applications that underpin modern cryptography and communications. You will have advanced skills in coding, algebra and number theory, and be able to synthesise and interpret information from multiple sources with insight and critical awareness. You will have learnt to formulate problems clearly, to undertake independent research and to express your technical work and conclusions clearly in writing. You will also have valuable transferable skills such as advanced numeracy and IT skills, time management, adaptability and self-motivation.

Graduates from this programme have gone on to carry out cutting-edge research in the fields of communication theory and cryptography, as well as to successful careers in industries such as: information security, IT consultancy, banking and finance, higher education and telecommunications. Our mathematics postgraduates have taken up roles such as: Principal Information Security Consultant at Abbey National PLC; Senior Manager at Enterprise Risk Services, Deloitte & Touche; Global IT Security Director at Reuters; and Information Security Manager at London Underground.

The campus Careers team will be on hand to offer advice and guidance on your chosen career. The University of London Careers Advisory Service runs regular, tailored sessions for mathematics students, on finding summer internships or vacation employment and getting into employment.

Home (UK) students tuition fee per year*: £8,100

EU and International students tuition fee per year**: £17,200

Other essential costs***: There are no single associated costs greater than £50 per item on this course.

How do I pay for it? Find out more about funding options, including loans, grants, scholarships and bursaries.

* and ** These tuition fees apply to students enrolled on a full-time basis. Students studying on the standard part-time course structure over two years are charged 50% of the full-time applicable fee for each study year.

All postgraduate fees are subject to inflationary increases. This means that the overall cost of studying the programme via part-time mode is slightly higher than studying it full-time in one year. Royal Holloway's policy is that any increases in fees will not exceed 5% for continuing students. For further information see tuition fees see our terms and conditions. Please note that for research programmes, we adopt the minimum fee level recommended by the UK Research Councils for the Home tuition fee. Each year, the fee level is adjusted in line with inflation (currently, the measure used is the Treasury GDP deflator). Fees displayed here are therefore subject to change and are usually confirmed in the spring of the year of entry. For more information on the Research Council Indicative Fee please see the RCUK website.

** For EU nationals starting a degree in 2021/22, the UK Government has recently confirmed that you will not be eligible to pay the same fees as UK students. This means you will be classified as an international student. At Royal Holloway, we wish to support those students affected by this change in status through this transition. For eligible EU students starting their course with us in September 2021, we will award an automatic fee reduction which brings your fee into line with the fee paid by UK students. This will apply for the duration of your course.

*** These estimated costs relate to studying this particular degree programme at Royal Holloway. Costs, such as accommodation, food, books and other learning materials and printing, have not been included.

 

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