dissertacao.pdf

1.5.2 Time Complexity
In cryptography, rather than the value of an integer, it is common to work with
its size in bits. Most of the algorithms presented in this work will depend on
the size of integers, so it is convenient to present firstly the following definition:
Definition 3. The size of N is the number of bits it takes to represent N in
base 2, namely log 2(N)
Throughout the rest of this work, when referring to the size of a number,
log(N) should be considered log 2(N). To describe an algorithm’s efficiency we
usually relate its running time to some class of functions, for which we use the
following notation:
Definition 4. Big-O notation: Given two functions f and g, we say that
f(n) ∈ O(g(n)) iff:
∃ constants c, n0 > 0 : ∀n ≥ n0 we have 0 ≤ f(n) ≤ cg(n) (1)
This means that, for sufficiently large values of n, the function f does not
exceed g.
The efficiency of the algorithms will be described in terms of the number of
basic operations it executes in function of the input given.
Definition 5. The running time of an algorithm A, TA(n) is the number
of elementary instructions it executes when the input data is n. Occasionally,
the running time can depend on more than one parameter. When the situation
will be clear, we will simply say that the running time of an algorithm A
is O(f), meaning that TA(n) ∈ O(f(n)).
It is obvious that each function does not belong to a single class of complex-
ity.For example, if T (n) ∈ O(f(n)), then obviously T (n) ∈ O(2f(n)). For this
reason, when saying that T (n) ∈ O(f(n)) we mean that f is the smallest known
function that satisfies this, unless otherwise stated.
Here are some simple cases for the complexity of an algorithm A, TA(n):
1. TA(n) is O(log(n) α ) means that the running time of A is polynomial in
the size of its input.
2. TA(n) is O(β log(n) ) means that the running time of A is exponential in
the size of its input.
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