Unit 1 Finite Fields

8 lecture hours

• Cyclic groups
• Rings and Fields
• Modular Arithmetic
• Properties of Modular Arithmetic
• The Euclidean Algorithm- for GCD, Finite Fields of the Form GF(p)
• Multiplicative inverse of GF(p)
• Polynomial Arithmetic
• Finite Fields of the Form GF(2n)-modular polynomial arithmetic
• finding multiplicative inverse
• Extended Euclid algorithm

Unit 2 Introduction to Number Theory -Divisibility - Greatest common divisor

8 lecture hours

• –Prime numbers – Fermat’s and Euler’s Theorem
• Euler Totient function
• Testing for Primality-Miller Rabin Algorithm
• The Chinese Reminder Theorem
• Discrete Logarithms –Power of integer modulo n, Logarithm for modular arithmetic

Unit 3 Elliptic Curve Arithmetic

7 lecture hours

• Abelian groups
• Elliptic curve over Real Numbers
• Geometric Description of Addition
• Algebraic Description of Addition
• Elliptic curve over Zp, Elliptic curve over GF(2m)

Unit 4 Random Number Generation

6 lecture hours

• Use of random numbers
• Pseudorandom Number Generation (PRNGs)
• Linear Congruential Generators
• Cryptographically Generated Random numbers
• Blum Blum Shub Generator
• True Random Number Generators

Unit 5 Coding Theory

• 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. 8 15

Unit 6 Probability Theory

• Introduction – Concepts of Probability - Conditional Probability - Baye’s Theorem - Random Variables – discrete and continuous- central Limit Theorem-Stochastic Process- Markov Chain. 8 15