Undergraduate Programme and Module Handbook 2010-2011 (archived)

# Module MATH2591: Elementary Number Theory and Cryptography II

## Department: Mathematical Sciences

### MATH2591: Elementary Number Theory and Cryptography II

Type | Open | Level | 2 | Credits | 20 | Availability | Available in 2010/11 | Module Cap | None. | Location | Durham |
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#### Prerequisites

- Core Mathematics A (MATH1012)

#### Corequisites

- None.

#### Excluded Combination of Modules

- Mathematics for Engineers and Scientists (MATH1551), Single Mathematics A (MATH1561), Single Mathematics B (MATH1571), Foundation Mathematics (MATH1641)

#### Aims

- To provide and introduction to the basics of number theory and the application of these ideas in cryptography.

#### Content

- Review of basic features of integers.
- Congruences and modular arithmetic
- Quadratic reciprocity
- Applications to cyptography
- Diophantine equations
- Elliptic curves
- Counting prime numbers.

#### Learning Outcomes

Subject-specific Knowledge:

- By the end of the module students will: be able to solve a range of predictable and unpredictable problems in Number Theory and Cryptography.
- have an awareness of the abstract concepts of theoretical mathematics in the field of Number Theory and application of these ideas in cryptography.
- have a knowledge and understanding of fundamental theories of these subjects demonstrated through one or more of the following topic areas: Fundamental theorem of arithmetic, modular arithmetic and chinese remainder theorem.
- Cryptography, public key systems.
- Diophantine equations, irrationality.
- Elliptic curves, applications in cryptography.
- Prime number theorem.

Subject-specific Skills:

- In addition students will have the ability to undertake and defend the use of alternative mathematical skills in the following areas with minimal guidance: Abstract reasoning.

Key Skills:

#### Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

- Lecturing demonstrates what is required to be learned and the application of the theory to practical examples.
- Weekly homework problems provide formative assessment to guide students in the correct development of their knowledge and skills.
- Tutorials provide active engagement and feedback to the learning process.
- The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

#### Teaching Methods and Learning Hours

Activity | Number | Frequency | Duration | Total/Hours | |
---|---|---|---|---|---|

Lectures | 42 | 2 per week | 1 Hour | 42 | |

Tutorials | 10 | Fortnightly for 20 weeks | 1 Hour | 10 | ■ |

Problems Classes | 10 | Fortnightly for 20 weeks | 1 Hour | 10 | |

Preparation and Reading | 138 | ||||

Total | 200 |

#### Summative Assessment

Component: Examination | Component Weighting: 100% | ||
---|---|---|---|

Element | Length / duration | Element Weighting | Resit Opportunity |

Written examination | 3 hours | 100% | Yes |

#### Formative Assessment:

One written assignment to be handed in every third lecture in the first 2 terms. Normally each will consist of solving problems from a Problems Sheet and typically will be about 2 pages long. Students will have about one week to complete each assignment.

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University