MATH30120: Cryptography and Codes

• Prior knowledge of Elementary Number Theory at undergraduate level.

• None

• None

• To give a basic introduction to two topics in data transfer which rely on abstract mathematics: Error correcting Codes which are used widely in data transmission over noisy channels, Cryptography which is widely used in banking, internet browsing, and to ensure privacy on mobile networks.

• Introduction to codes: The Hamming distance, Error detection and correction, equivalence of codes
• Linear Codes, Dual codes and Decoding Methods
• Hamming Codes, Golay Codes,
• Linear Codes over cyclic fields, Cyclic Codes, BCH codes, Reed-Solomon Codes
• Introduction to open-key cryptography, notion of trapdoor function. The factorisation and discrete logarithm problems
• Diffie-Hellman key exchange scheme. RSA cryptosystem
• Elliptic curves over rational numbers and finite fields, Elliptic Curve Diffie-Hellman scheme
• Lenstra factoring algorithm

Subject-specific Knowledge:

• By the end of the module students will: be able to solve a range of predictable and unpredictable problems in Cryptography and Codes.
• have an awareness of the abstract concepts of theoretical mathematics in Codes and Cryptography.
• have a knowledge and understanding of fundamental theories of these subjects demonstrated through one or more of the following topic areas:
• Codes: Linear, Hamming, Cyclic, BCH, Reed-Solomon Codes
• Cryptography: open-key systems
• Elliptic curves, applications in cryptography.

Subject-specific Skills:

• In addition students will have specialised mathematical skills in the following areas which can be used in minimal guidance: Abstract Reasoning.

Key Skills:

• Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
• Assignments for self-study develop problem-solving skills and enable students to test and develop their knowledge and understanding.
• Formatively assessed assignments provide practice in the application of logic and high level of rigour as well as feedback for the students and the lecturer on students' progress.
• The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

Eight written or electronic assignments to be assessed and returned. Other assignments are set for self-study and complete solutions are made available to students.

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