Unit 1 Introduction to Number Theory

8 lecture hours

• Introduction-Divisibility - Greatest common divisor – Primes- Prime numbers – Cardinality of Primes
• Fundamental theorem of arithmetic - Mersenne primes - Fermat numbers
• Fermat’s and Euler’s Theorem
• Testing for Primality
• The Chinese Reminder Theorem
• Quadratic Congruence
• Exponentiation and Logarithms
• Discrete Logarithms

Unit 2 Algebraic Structures and Finite Fields

6 lecture hours

• Groups – Cyclic groups
• Co sets
• Modulo groups - Primitive roots – Discrete logarithms The Euclidean Algorithm
• Modular Arithmetic
• Algebraic Structures-Groups
• Rings and Fields
• Future Fields of the Form GF(2”)
• Polynomial Arithmetic
• Finite Fields of the Form GF(2n)

Unit 3 Pseudorandom Number Generation and Stream Ciphers

5 lecture hours

• Principles of Pseudorandom Number Generation
• Principles of Pseudorandom Number Generation using a Block Cipher
• Stream Ciphers, RC4, True Random Number Generators

Unit 4 Discrete Mathematics for Cryptography

4 lecture hours

• Cryptography and Modular Arithmetic
• Inverses & GCDs
• The RSA Cryptosystems
• Mathematical Induction
• Recurrences and Induction
• Recurrences and Selection

Unit 5 Coding Theory 05 15

• Introduction - Basic concepts: codes
• minimum distance
• equivalence of codes
• Linear codes - Generator matrices and parity-check matrices - Syndrome decoding – Hamming codes - Hadamard Code - Goppa codes

Unit 6 Probability Theory

4 lecture hours

• Introduction – Concepts of Probability - Conditional Probability - Baye’s Theorem - Random Variables – discrete and continuous- central Limit Theorem-Stochastic Process- Markov Chain

Unit 7 Cryptographic Hash Functions

4 lecture hours

• Application of Cryptographic Hash Functions
• Two Simple Hash Functions
• Requirements and Security
• Hash Functions Based on Cipher Block Chaining
• Secure Hash Functions (SHA)