# MAT 448 Cryptography

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### Course Description

Congruencies and number theory, history and early cryptosystems, cryptographic data structures, public key cryptography, additional cryptosystems such as DES, AES, and elliptic curve cryptography. Computer implementations will also be covered, as will any needed additional mathematical topics (e.g., finite fields).

3 units credit.

Additional comments (not in the official course description): The objective of this course is to study the central and contemporary notions of cryptography, along with the necessary mathematical concepts. The topics will be studied both from theoretical perspectives and from a practical viewpoint. The core topics covered will be: Basic Number Theory, Basic Cryptography, DES and AES, Public-Key Cryptography. Related history of the subject will be outlined throughout. If time permits, additional topics from the following might be covered: True and Probabilistic Primality Testing, Different methods of factoring, Internet security, and Elliptic Curve Cryptography.

### Prerequisites

MAT 271 with a grade of "C" or better. CSC 115 or CSC 121 with a grade of "C" or better are recommended.

### Text

Texts are chosen by the instructor. For example:

A. Stanoyevitch, Introduction to Cryptography, with Mathematical Foundations and Computer Implementations, Chapman & Hall (2011)

### Course Requirements, Tentative Schedule of Class Meetings and Topics, Readings, Assignments and Due dates, Exams

A schedule of class meetings, topics, assignments, due dates, exam dates, etc. will be provided by instructor. See your class syllabus.

Here is an example course outline, based on the above text.

 Week Topics 1 Overview of the subject with introductions to some key historical cryptographic systems and events 2 Solving Problems and Proving Theorems involving Divisibility and Primes. Unique Factorization 3 Working with Congruencies and Modular Arithmetic 4 The evolution of codemaking until the computer era 5 Vector and matrices of modular integers Exam #1 6 Working with vectors and strings on computing platforms 7 The evolution of codebreaking until the computer era (including the exciting cryptographic breakthroughs that help end World War II earlier than it would have otherwise ended) 8 Representation of integers in different bases 9 The data encryption standard (DES) cryptosystem 10 Computer Implementations of DES Exam #2 11 Topics in Number Theory 12 Public Key Cryptography Digital Signatures: A means for authentication and nonrepudiation 13 Finite Fields in General and GF(28) in particular 14 The advanced encryption standard (AES) cryptosystem 15 Review for final

The final exam is given at the date and time announced in the Schedule of Classes.

### Learning Objectives

By the end of this course, students will

• Write proofs of theorems about basic properties of numbers including even numbers, odd numbers, divisibility, and prime numbers.
• Write proofs of theorems in modular arithmetic and arithmetic in other bases.
• Compute and write encryptions in various cryptosystems.
• Demonstrate through explanation and solving problems how one might go about attacking cryptosystems.
• Analyze the security and any vulnerabilities of an assortment of cryptosystems.
• Write and run computer programs for encryption and decryption in complex contemporary cryptosystems.

### Computers and Calculators, Computer Literacy

Most instructors encourage the use of machines, calculators computers, phones etc., for analyzing data. The use of machines may be restricted during examinations or at certain other times. Ask your instructor for the policy in your class.

Students are not expected to be programmers or to know any particular computer language before starting this class. Some instructors may expect students to be able to access information on the internet, or to use calculators, or to learn to use particular software with instruction. Basic skill in algebra and the use of mathematical symbols, order of operations etc., and the willingness to read and follow instruction manuals and help files will suffice.

Students' grades are based on homework, class participation, short tests, and scheduled examinations covering students' understanding of the topics covered in this course. The instructor determines the relative weights of these factors and the grading scale. See the syllabus for your particular class.

### Location of Class Meetings

Classes meet on the dates and room announced in the official Schedule of Classes. This is a traditional, face-to-face class.

### Attendance Requirements

Attendance policy is set by the instructor.

### Policy on Due Dates, Make-Up Work, Missed Exams, and Extra-Credit Assignments

Due dates and policy regarding make-up work and missed exams are set by the instructor. Instructors may, or may not, choose to offer extra credit assignments. If extra credit assignments are offered they will be available to all students.

The mathematics department does not tolerate cheating. Students who have questions or concerns about academic integrity should ask their professors or the counselors in the Student Development Office, or refer to the University Catalog for more information. (Look in the index under "academic integrity".)

### Accomodations for Students with Disabilities

Cal State Dominguez Hills adheres to all applicable federal, state, and local laws, regulations, and guidelines with respect to providing reasonable accommodations for students with temporary and permanent disabilities. If you have a disability that may adversely affect your work in this class, I encourage you to register with Disabled Student Services (DSS) and to talk with me about how we best can help you. All disclosures of disabilities will be kept strictly confidential. Please note: you must register with DSS to arrange an no accommodation. For information call (310) 243-3660 or send an email message to dss@csudh.edu or visit the DSS website http://www4.csudh.edu/dss/contact-us/index or visit their office WH D-180

### Behavioral Expectations

We all are adults so behavior rarely is an issue. Just follow the Golden Rule: "do unto others as you would have them do unto you" then everything will be fine.

The university must maintain a classroom environment that is suitable for learning, so anyone who insists on disrupting that environment will be expelled from the class.

Revision history:

Prepared by A. Stanoyevitch spring 2011. Revised 3/4/2011, 1/10/15 (G. Jennings)