MATH 311

Math 311: Mathematical Cryptography

Prerequisites: Either MATH 171 (Calculus I) or MATH 169 (Calculus I with Integrated Precalculus II); MATH 260 (Linear Algebra) recommended.

Credit Hours: (3)

Exploration of the mathematical principles underlying modern cryptography and data security. Topics include modular arithmetic, number theory, linear algebra, symmetric and public-key encryption, digital signatures, and hash functions. Emphasis is placed on the mathematical structures and algorithms that ensure confidentiality, integrity, and authenticity in digital communication. Given the fact that cryptographic algorithms are highly computational in nature, the use of a programming language and/or symbolic manipulator will be implemented throughout the course to demonstrate realistic examples.

 

Content

1. Introductory Concepts
   
   1. References to important historical ciphers.
   2. Modular arithmetic and its applications to classical mathematical ciphers.
   3. Introduction to cryptanalysis
2. Symmetric Key Cryptosystems
   
   1. Stream ciphers
   2. Pseudorandom key generation and LFSR bit generation.
   3. Perfect Secrecy and the One-Time Pad
3. Block Ciphers
   
   1. Hill Cipher and matrix cryptanalysis
   2. The Data Encryption Standard (DES)
   3. The Advanced Encryption Standard (AES)
4. Public Key Cryptosystems
   
   1. Background number theory including prime numbers, Euclidean Algorithm and modular inverses, and modular exponentiation.
   2. RSA cryptosystem.
   3. Primality testing and RSA cryptanalysis.
5. Discrete Logarithms
   
   1. Diffie Helman Key Exchange
   2. The ElGamal Public Key Cryptosystem
   3. Methods for solving discrete logarithms.
6. Message Authentication
   
   1. Hash Functions
   2. RSA Digital Signatures and other signature schemes
   3. Hashing and Signing
7. Other topics if time permits

Applications of linear algebra, abstract algebra, and number theory will be discussed as appropriate throughout the course.

 

Detailed Description of Conduct of Course

Most instructors use the lecture-discussion method. Some may require students to work together in small groups.

 

Student Learning Outcomes

After having completed this course, students will

1. Have a better knowledge of how core topics in mathematics involving linear algebra, abstract algebra, number theory, probability, and statistics can be applied in a practical setting.
2. Have a better understanding of some of the current methods being used for transferring information securely.

 

Assessment Measures

Graded tasks may include tests, quizzes, homework exercises, class participation, and attendance.

 

Other Course Information

This course is intended as an elective for majors and minors in mathematics and other related fields.

 

Review and Approval

November 20, 2025