From: Christoph Fuerst <ch.fuerst@gmx.at>
Date: Sat, 25 Mar 2017 09:44:07 +0000 (+0100)
Subject: Structure, first contents on mathematics
X-Git-Url: http://git.risc.jku.at/gitweb/?a=commitdiff_plain;h=61f7de4432cd6f5bddc64c7e7daff6201a6c1f09;p=cfuerst%2Fformal-numbers.git

Structure, first contents on mathematics
---

diff --git a/report/formal.pdf b/report/formal.pdf
index 8ff3840..ccfdd39 100644
Binary files a/report/formal.pdf and b/report/formal.pdf differ
diff --git a/report/formal.tex b/report/formal.tex
index 626f3a6..a4dca18 100644
--- a/report/formal.tex
+++ b/report/formal.tex
@@ -1,5 +1,8 @@
 \documentclass[a4paper,10pt]{scrartcl}
 \usepackage[utf8]{inputenc}
+\usepackage{amsmath}
+\usepackage{amsthm}
+\usepackage{amssymb}
 \usepackage{listings}
 %opening
 \title{Formal Verification of Algorithms arising in Cryptography}
@@ -29,6 +32,23 @@ systems.
 \section{Introduction}
 
 \section{Mathematical and Cryptographic Preliminaries}
+One of the most fundamental notions in mathematics is without doubt
+the term \emph{divisor}, we say that \emph{$a$ divides $c$} if there is a number $b$
+such that $c = a\cdot b$. Among all divisors of two numbers $a$,$b$ there is
+a unique \emph{greatest} common divisor of $a$ and $b$, denoted by $\gcd(a,b)$ 
+that is computed by the Euclidean algorithm. The extended Euclidean algorithm 
+allows to compute numbers $s$ and $t$ such that $a\cdot s + b\cdot t = \gcd(a,b)$.
+\subsection{Finite Fields}
+An elementary notion in cryptographic mathematical theories, is the notion
+of a finite field. A field is a set $K$ where the elementary mathematical
+operations of addition, subtraction, multiplication and division are applicable.
+While the classic examples, the field of real numbers $\mathbb{R}$ and the
+complex numbers $\mathbb{C}$, are infinite fields, the focus in cryptographic
+applications is on \emph{finite fields}. A finite field with $p$ elements, where
+$p$ is prime, is usually written as $\mathbb{Z}_p = \mathbb{Z}/p\mathbb{Z}$,
+and consists of the elements $\{0,1,\ldots,p-1\}$. The number $p$ is called the
+\emph{characteristic} of $\mathbb{Z}_p$, in fact, the characteristic of every 
+finite field is a prime power.
 
 \section{The Formal Verification}