UTSA logo Number Theory
MAT 4253
Fall, 2002
Instructor: J. Iovino
 

The course has been scheduled for M,W, 12 noon-1:15 pm. Room MS 2.02.02 (1604 campus).

Brief description

Number theory is the queen of mathematics and mathematics is the queen of all sciences, so number theory is the science with double crown.

For centuries, number theory was regarded as the "purest" part of mathematics, the only one where one could find areas with absolutely no real-world applications. Hardy, the great English number theorist, took pride on this; one did number theory for its sheer beauty, not for utilitarian purposes.

Today, number theory is the source of one of the most important applications of mathematics: cryptography and data security.

The first part of the course is a brief introduction to classical elementary number theory. The second part will focus on applications to cryptography, e.g., classical ciphers, block and stream ciphers, public key cryptosystems, and cryptographic protocols.

Prerequisites

Foundations of Analysis (MAT 3213) or consent of instructor.

Textbook

Keneth Rosen, Elementary Number Theory and its Applications, Fourth Edition, Addison Wesley Longman, 1999.

The review of the fourth edition of this book by the American Mathematical Society (MathSciNet) starts with the sentence

"This exemplary undergraduate number theory text keeps getting better."

and ends by stating

"The new edition stands as an excellent text for the new millennium."

Content

  • Primes and Greatest Common Divisors: Euclidean algorithm, The Fundamental Theorem of Arithmetic, factorization methods and Fermat numbers, linear Diophantine equations. (Chapter 3).
  • Congruences: Linear congruences, the Chinese Remainder Theorem, polynomial congruences. (Sections 4.1-4.3.)
  • Special Congruences: Wilson's Theorem, Fermat's Little Theorem, pseudoprimes, Euler's Theorem. (Chapter 6.)
  • Multiplicative Functions: The Euler phi-function, the sum and number of divisors. (Sections 7.1-7.2.)
  • Cryptography: Character ciphers, block and stream ciphers, exponentiation ciphers, public-key cryptography, Knapsack ciphers, cryptographic protocols and applications. (Chapter 8.)

    • Schedule

    For an outline of the material covered each day of the academic semester, click here.

    Evaluation

    There will be three problem sets. The students will have two weeks to work on each set. Each problem set will be worth 25% of the grade. Towards the end of the course, every student will choose a section of the book to present orally. The presentation will be worth 25% of the grade.

    How to contact the instructor

    Office: SB 4.01.34 (Directions: Go to the fourth floor of the Science Building and as you get off the elevator follow the arrows to the Mathematics Department office. I am right across the hall from the Department Chair's office.)

    Telephone: (210) 458-5531

    Email: iovino@math.utsa.edu

    Office hours (Fall 2002): M,W 10:11:30 am, or by appointment.

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