dissertacao.pdf

The first case refers to algorithms which we say run in time polynomial in
log(N). Such an algorithm is called efficient as its complexity function grows
in a rather controlled way. In opposition, algorithms belonging to the second
case are called inefficient: they quickly become infeasible by any computer or
even network of computers.
When referring to problems, we will call them hard if there is no known
algorithm running in polynomial time that solves it. In opposition, a problem
which can be solved for any input with an algorithm running in polynomial time
is said to be an easy problem.
1.5.3 Modular Arithmetic
Modular Arithmetic is a fundamental basis for cryptography in general and for
RSA in particular. For this reason we present a set of definitions and results
which will enable the reader to fully understand the operations behind RSA. We
will also present some of the most used algorithms within the RSA operations.
We start with the congruence relation:
Definition 6. Let N be a positive integer. We say that two integers a and
b are congruent modulo N if there exists an integer k such that a − b = kN.
We represent this relation with the following notation:
a ∼ = b (mod N) (2)
It should be noted that this is an equivalence relation. After choosing the
modulus N, we get a partition of the integers into N different classes since every
integer a must be congruent modulo N to an integer in the set {0, 1, ..., N − 1}.
When the modulus N is implicit, we define [a] = {b ∈ Z : a ∼ = b (mod N)},
called the equivalence class of a. We introduce two operations with these equiv-
alence classes, simple extensions of the usual addition and multiplication:
Definition 7. (Modular Operations): Given two classes of equivalence [x]
and [y] defined modulo N, we define the operations:
addition: [x] + [y] = [x + y]
multiplication: [x][y] = [xy]
Throughout the text we will stop referring to the equivalence class [x] as [x]
and instead write simply x. The modulus regarding which the equivalence class
is defined will also not be written, rather it will be explicit from its context.
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