Algorithms for Big Numbers: Z_n and Orders

This lecture covers algorithms for big numbers, focusing on Z_n, the ring of residues modulo n, and orders in a group. It explains the multiplication algorithm, double-and-add method, and square-and-multiply technique. The instructor discusses the definition of a ring, group of units, and group and ring constructors. Examples of Z, Z[X], and modulo 9 reduction are provided. The lecture also delves into the Diffie-Hellman cryptography, modular arithmetic, and modular inversion. Various arithmetic operations with big numbers are explained, including addition, multiplication, Euclidean division, and the Euclid algorithm. The importance of structure property of Z, element order, group exponent, and orders in Zm are highlighted.