Cryptography Reference
In-Depth Information
Chapter 3
Discrete Mathematics
In this chapter, we begin the part on the mathematical fundamentals by discussing
the aspects of discrete mathematics that are relevant for contemporary cryptography.
More specifically, we introduce algebraic basics in Section 3.1, elaborate on integer
and modular arithmetic in Sections 3.2 and 3.3, introduce elliptic curves in Section
3.4, and conclude with some final remarks in Section 3.5. Note that this chapter
is intentionally kept short, and that many facts are stated without a proof. There
are many (introductory) topics on discrete mathematics and algebra that contain the
missing proofs, put the facts into perspective, and provide much more background
information (e.g., [1-5]). Most importantly, Victor Shoup's topic about number
theory and algebra [6] is electronically available
1
and is recommended reading for
anybody interested in discrete mathematics.
3.1
ALGEBRAIC BASICS
The term
algebra
refers to the mathematical field of study that deals with sets of
elements (e.g., sets of numbers) and operations on these elements.
2
The operations
must satisfy specific rules (called
axioms
). These axioms are defined abstractly, but
most of them are motivated by existing mathematical structures (e.g., the set of
integers with the addition and multiplication operations).
1
http://shoup.net/ntb
2
For the purpose of this topic, we assume familiarity with set theory at a basic level.
