课程名称 (Course Name) ：Applied Algebra

课程代码 (Course Code）：X033536

学分/学时 (Credits/Credit Hours)：3/48

开课时间 (Course Term )：春spring

开课学院（School Providing the Course）: 电子信息与电气工程学院  seiee

任课教师（Teacher）: 刘胜利

课程讨论时数（Course Discussion Hours）:   0 小时(Hours)

课程实验数（Lab Hours）:       0小时(Hours)

课程内容简介(Course Introduction):

“Applied algebra” is a succeeding course of “Abstract Algebra”, which was taught by Prof. Zheng dong in the fall semester. In preceding course of “Abstract Algebra”, the theory of group, ring and finite field is introduced. “Applied algebra” focuses on a special algebraic structure, namely “Finite field” and its applications to cryptography and error-correction codes. This course is a fundamental course in the subject of cryptography and information security. With this course, the students will learn the mathematical background of cryptography and coding theory.

教学大纲(Course Teaching Outline):

The content of the course includes

1.    Introduction of Finite field

2.   Euclidean Domain and the Unique Factorization Theorem for Euclidean Domain

3.   Building Finite Fields from Euclidean Domain

4.   Properties of Finite Fields

5.         Factoring Polynomials over Finite Fields

6.         The concept of Linear Feedback Shift Register (LFSR) and applications to LFSR

7.         The application to cryptography

8.         The application to error-correction codes

课程进度计划(Course Schedule):

Week 1: Integral Domain, Euclidean Domain, and examples. 整环，欧基里德环的基本概念与实例

Week 2: The concept of greatest common divisor in Euclidean domain. Unique Factorization Theorem, Eulidean Algorithms.欧氏环中的最大公约数概念，唯一 分解定理，欧基里德算法

Week 3: How to construct fields form Euclidean domains.从欧氏环构造域

Week 4: Algebraic structures of a Finite Field and properties有限域的构造和结构及性质

Week 5: Order of elements in a finite field, and the Guass algorithm of primitive roots.有限域中元素的阶，求本原元的高斯算法

Week 6: Minimal polynomials, primitive polynomials, and reducible polynomials. 极小多项式，本原多项式，不可约多项式

Week 7: Existence and Uniqueness of Finite fields.有限域的存在性和唯一性

Week 8:Mobuis Formula and applications. Mobuis公式及其应用

Week 9: May festival放假

Week 10: Cyclotomic polynomials and properties. 分圆多项式概念和性质

Week 11: Factoring cyclotomic polynomials over finite fields.分圆多项式在有限域上的分解

Week 12: Exercises.习题课

Week 13: Special polynomials and their factorizations特殊多项式在有限域上的分解

Week 14: Factoring any polynomials over finite fields.任意多项式在有限域上的分解

Week 15: The concept of LFSR and the relation to polynomials. LFSR概念与多项式的关系

Week 16: m sequences and primitive polynomials. m序列与本原多项式

课程考核要求(Course Assessment Requirements)：

Final Test(70%)+homework/attendance(30%)

参考文献(Course References)：

[1]      R. J. McEliece, “Finite Fields for Computer Scientists and Engineers”, Kluwer international series in engineering and computer science. Kluwer Academic Publishers, 1988.
Copies of the material will be supplied

[2]      Gong Guang, “Sequence analysis”, http://www.comsec.uwaterloo.ca/ ggong/CO739x/course.html

[3]      Wenbo Mao, “Modern Cryptography: principles and practice”

[4]      The slides and some references can be download from “ftp://ftp.cs.sjtu.edu.cn/liu-sl/应用代数”