# Mathematics

## ECM3726 - Cryptography (2015)

MODULE TITLE CREDIT VALUE Cryptography 15 ECM3726 Prof Andreas Langer (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 0 11 weeks 0
 Number of Students Taking Module (anticipated) 116
DESCRIPTION - summary of the module content

Cryptography is the science of encryption. In this module, you will learn to formulate encryption as a mathematical problem. Cryptography can be defined as the conversion of data into a scrambled code that can be deciphered and sent across a public or private network.

Cryptography uses two main styles or forms of encrypting data: symmetrical and asymmetrical, and all good encryption schemes use the concept of a key which may take on any one of a number of values. The collection of all possible values is called the keyspace and encryptions and decryptions are represented as functions.

You will focus on two encryption algorithms: so called symmetric and public key algorithms, then you will concentrate on public key algorithms where you cannot deduce one key from another key. Encryption algorithms have huge commercial value; indeed, every day, millions of financial and business transactions are carried out over the internet; and security of internet transactions relies on the highest levels of encryption.

Prerequisite module: ECM2712 or equivalent

AIMS - intentions of the module

The aim of this module is to apply elementary number theory to problems in the real world where it is important to transmit information in a secret way. For example, cryptography is used in bank accounts, and is traditionally applied in military science. Note that there is another application of number theory and linear algebra called coding theory - but this is not the same as cryptography.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:
1 formulate encryption as a mathematical problem;
2 demonstrate an understanding of public key cryptography, other practical crypto-systems, some useful impractical ones and some peculiar cryptosystems like the hidden monomial or the combinatorial-algebraic cryptosystem;
3 articulate the  complexity of computations in cryptography.
Discipline Specific Skills and Knowledge:
4 reveal a grasp of the  number-theoretic and algebraic aspects of cryptography.
Personal and Key Transferable/ Employment Skills and  Knowledge:
5 show an appreciation of how concrete problems typically require abstract theories for their solution;
6 display a comprehension of how to apply algorithms and why they work.

SYLLABUS PLAN - summary of the structure and academic content of the module

- introduction to cryptography, history of cryptography Symmetric and anti-symmetric cryptosystems; some simple examples;

- one way functions;

- number theoretic cryptography: introduction to congruences and prime numbers; Euler’s totient function, the integer factorisation and discrete logarithm problems;

- the RSA cryptosystem (Rivest Shamir Adleman), the ElGamal cryptosystem;

- possible attacks and their computational complexity;

- algebraic cryptography: a short review of group and field theory;

- hidden monomial cryptosystem;

- several group-theoretic cryptostems;

- a view of elliptic curve cryptography.

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study Placement / Study Abroad 33 117 0
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 Category Hours of study time Description Scheduled learning and teaching activities 33 Lectures/example classes Guided independent study 117 Lecture and assessment preparation; private study

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Not applicable

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams 20 80
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Written exam – closed book 80 2 hours All Exam mark
Coursework – three example sheets 20   All Written

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-reassessment
All above Written exam (100%) All August Ref/Def period

RE-ASSESSMENT NOTES

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

ELE – http://vle.exeter.ac.uk