this video has the extended GCD and also the very first introduction to fields

https://www.youtube.com/watch?v=baJp5VLjLMk&list=PLy_2fgXkPiZt3oJcHalAWy9uhQ3wd2fTV&index=10

https://www.youtube.com/watch?v=8lNaf7IprS8&list=PLy_2fgXkPiZt3oJcHalAWy9uhQ3wd2fTV&index=11

Integers and Cryptography

• In modern Ciphers, binary data are ciphered in the form of binary strings of data either bit-by-bit and byte-by-byte in stream ciphers or block of bits of fixed length in block ciphers.

• Binary strings (blocks of bits) are naturally integer numbers, an n bits binary string can be an integer number on the range of 0 to 2n-1 (000…00 → 111…11)

• In other case this binary string can be viewd as a binary polynomial coefficients vector (will be elaborated on at the end of this chapter)

• Integers and the basic arithmetic operations:

  • Addition (+)
  • Subtraction (-)
  • Multiplication (*)
  • Division (/)

  Can be utilized to build transforms that can be the base for ciphers

Two main problems

• The set of Integers is an infinite set {0,1,2,3,…………,∞{, hence operations such as (+) and (*) will result in a larger number of the inputs
• The (/) operation will not always result and integer and hence it is not defined on all integer number
• Solution: Finite Fileds

we have groups which include Rings which include fields and we will be working with Finite Fields so let’s discuss them…

Groups

A Group $(G, \cdot)$ is a set of elements $G$ paired with a binary operation $\cdot$ (this dot can be any operation like sum, mul, etc…) that obeys four fundamental mathematical rules (axioms):

• (A1) Closure: For any $a, b \in G$, the result $a \cdot b$ is also in $G$.
• (A2) Associativity: For all $a, b, c \in G$, $(a \cdot b) \cdot c = a \cdot (b \cdot c)$.
• (A3) Identity Element: There exists an element $e \in G$ such that for every $a \in G$, $a \cdot e = e \cdot a = a$.