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Number Theory and Cryptology for Middle-Level Teachers
Target Audience: This course was designed for teachers of middle level mathematics.
Comfort with functions, algebraic representations and basic logic is strongly
recommended.
Graduate Credit Hours: 3
Usual Delivery Format: On-site summer course. Designed to be completed in one
week, meeting M – F from 8:00 – 5:00 (with one hour break for lunch).
Texts/Materials:
1. Course notebook containing handouts and problems (see Course Notebook
section )
2. The Mathematical Universe, by William W. Dunham; Chapters A, F and P are
assigned as readings.
3. NOVA movies N is a Number, The Proof, and Decoding Nazi Secrets
Description: This course focuses on basic number theory results which are needed to
understand the number theoretic RSA cryptography algorithm (an encryption algorithm
which is in use today to secure information sent via the internet). As the number theory
results are developed, connections to middle level curricula are emphasized and proofs
are carefully selected so that those which are included in the course are particularly
relevant and accessible to middle level teachers. This portion of the course promotes a
deep understanding of the integers and their properties in connection with the operations
of multiplication and division. Elementary ciphers (methods for encoding and decoding)
are included to introduce the nature of cryptology in preparation for understanding the
RSA method. The cryptology related activities are readily adaptable as enrichment
activities for middle level students. The connection of number theory to the RSA
encryption algorithm allows the participants to see and understand a very relevant, realworld application of mathematics.
Course Goals: The goals of the course are to introduce the teacher participants to: (1)
the basic results of elementary number theory, (2) the rigor of mathematical definitions,
reasoning and proof, (3) the application of number theory to cryptology, (4) and the
connections between Number Theory and the middle school curriculum.
Topics:
Day Topics
1
Course Introduction
Section 1: Integers and Divisibility
Section 2: Primes and Factorization
2
Introduction to Cryptology (substitution and transposition ciphers)
Section 3: Linear Diophantine Equations
3
Section 4: Congruence
Section 5: Linear Congruence Equations
4
5
Section 6: Fermat’s and Wilson’s Theorems
Section 7: Euler Phi-Function
Other Number Bases
RSA Public Key Cryptography
References:
1. Dudley, Underwood, Elementary Number Theory, Second Edition, W. H.
Freeman and Company, 1978.
2. Walker, Judy (University of Nebraska – Lincoln), The Joy of Numbers
Instructional Style: The course is designed in an interactive-lecture style (similar to a
Socratic method) with problem sessions, examples and activities designed for cooperative
groups distributed consistently throughout.