Computational Number Theory and Asymmetric Cryptography

Objectives
Asymmetric crypto-systems as public key ciphers, digital signatures, authentication protocols are deployed and used worldwide in security protocols in retail trade, banking, payments over the Internet, access control, and generally in any kind of digital communication and constitute a security basis for the modern society. The systems are called asymmetric as, roughly speaking, only one of the parties in communication possesses a secret key.

Many of asymmetric crypto systems are based on one of the following hard computational problems: solving non-linear equation systems over finite fields, computing discrete logarithms in finite fields and on elliptic curves, integer factorisation and various computational problems from lattices as finding a shortest non-zero vector in a large dimension lattice. Those problems are within Algebra and Number Theory. Solving the hard problem breaks a relevant crypto-system and undermines the security of the applications.

The course gives introduction to Computational methods in Algebra and Number Theory with focus on known approaches to solve the above problems and analysis of relevant asymmetric crypto-systems.

Some of them (as HFE) are broken, some (RSA, DSA) are widely used and some (as NTRU) have potential to be deployed in the future if quantum computers come in use and conventional crypto-systems as RSA, DSA get broken.

Content
The course incorporates four chapters.

• solving systems of linear and non-linear equations over finite fields, analysis of HFE (Hidden Field Equation) crypto-system.
• basic methods for computing discrete logarithms and factoring integers, analysis of RSA (Rivest-Shamir-Adleman) crypto-system and DSA(Digital Signature Algorithm).
• arithmetic and algorithms in elliptic curves.
• lattice reduction algorithms, analysis of NTRU crypto-system.

Knowledge

On completion of the course the student should have the following learning outcomes defined in terms of knowledge, skills and general competence.

The student should have knowledge of

• computational methods in Algebra and Number Theory,
• mathematical foundations for security of modern cryptography,
• asymmetric crypto-systems based on hard computational problems from Algebra and Number Theory,
• analysis and applications of asymmetric crypto-systems.

Skills

The student is able to

• solve common computational problems in Algebra and Number Theory,
• explain main cryptography applications of asymmetric crypto-systems,
• digest and explain how asymmetric crypto-systems work,
• locate issues in security protocols relevant to asymmetric cryptography.

General competence

The student

• is familiar with new ideas and innovation processes,
• can exchange opinions with others with relevant background and participate in discussions concerning the development of good practice.

Autumn.