《Curriculum of the Mathematics Theory in Information Security》Course Syllabus

1. Course Objectives and Basic Requirements

This course introduces the mathematical theory used in computer science and technology especially information security technology, main contents of the course include three parts: number theory, algebra and information theory. Students can grasp the basic mathematical theory used in information security technology, learn the basic method to analyze and solve problems, so that they can lay a solid foundation for research and practice of information security. Prerequisites of the curriculum are linear algebra and probability theory.

2. Main Content

Part Ⅰ: Number Theory

Chapter 1 The divisibility of integers, including: the concept of integer and Euclidean Division, integer representation, the greatest common factor and the generalized Euclidean Division, the further properties of integers and the minimum common multiple, prime number and arithmetic basic theorems, prime theorem.

Chapter 2 Congruence, including: the concept and property of congruence, residual class and complete residue class, simplify surplus system and Euler function, Euler theorem and Fermat theorem, mode repeat square method.

Chapter 3 Congruence expression, including: the basic concept of congruence expression and linear congruence, Chinese remainder theorem, the solution of higher order congruence, congruence of prime modulus.

Chapter 4 Quadratic congruence and quadratic residue, including: general quadratic congruence, quadratic residue and quadratic non-residue, Legendre symbol, proof of quadratic reciprocity law, Jacobi symbol.

Chapter 5 Original root and index, including: exponent and its basic properties, conditions of original root existence, index and n-degree residue.

Chapter 6 Primality testing, including:  pseudoprime, Euler pseudoprime, strong pseudoprime.

Part Ⅱ:Algebra

Chapter 7 Group, including: group, homomorphism and isomorphism, quotient group.

Chapter 8 Structure of group, including: cycle group, permutation group.

Chapter 9 Ring and Field, including: ring and homomorphism, ideal polynomial ring.

Part Ⅲ: Information theory

Chapter 10 Model of communication system, including: model of communication system,  the central issue and development of information theory

Chapter 11 Information content and entropy, including: the non- average information content of discrete variable, average self information content of discrete set-- entropy, average mutual information of discrete set, mutual information and entropy of continuous random variable, convex function, convexity of mutual information.

Chapter 12 Channel coding, including: linear block code, generated matrix, check matrix, the special linear block codes, syndrome and minimum Hamming distance decoding, cycle code, BCH code, Reed-Solomon code.

3. Teaching materials

1. Chen Gongliang, The Mathematics Foundation in Information Security, Tsinghua University press, 2006.

2. Wang Yumin, Li Hui, Liang Chuanjia, Information Theory and Coding Theory, Higher Education Press, 2005.

    4. Reference

3. Pan Chengdong, Pan Chengbiao, Elementary Number Theory, Peking University press, 2003.

4. Yan S Y, Number Theory for Computing , Second Edition, Springer-Verlag, 2002.

   5. Course Evaluation (Tentative)

Assignments                    30%

Final exam                         70%