Cryptanalysis of RSA Factorization - Library(ISI Kolkata) - Indian ...

Chapter 2: Mathematical Preliminaries 12
and we shall use the first two in this thesis.
Definition 2.1 (Small o Notation). Given two functions f(n) and g(n), we say
that f(n) ∈ o(g(n)) if, for any constant c > 0, there exists a constant N > 0 such
that 0 ≤ f(n) < cg(n) for all n ≥ N.
Definition 2.2 (Big O Notation). Given two functions f(n) and g(n), we say that
f(n) ∈ O(g(n)) if there exist constants c > 0 and N > 0 such that 0 ≤ f(n) ≤
cg(n) for all n ≥ N.
In other words, we say that f(n) ∈ o(g(n)) if limsup f(n)
g(n)
= 0, and f(n) ∈
O(g(n)) if limsup f(n)
g(n)
is some finite constant. For example, n 2 + 1 ∈ o(n 3 ),
and n 2 + 1 ∈ O(n 2 ), where n is a positive integer. But n 2 + 1 ∉ o(n 2 ), as
n
lim 2 +1
n→∞ = 1. From the discussion so far, one may observe that if f(n) ∈
n 2
O(g(n)), then f is asymptotically bounded above by g, up to a constant factor.
However, f(n) ∈ o(g(n)) indicates that f is asymptotically dominated by g. Thus,
the Small o is a stricter condition compared to Big O. In formal terms, we have
o(g(n)) ⊆ O(g(n)). One may refer to [27] for further technical details regarding
asymptotic notations.
Throughout this thesis, we shall denote the bitsize of an integer N by l N and
it is defined as the number of bits in N. That is, l N = ⌈log 2 N⌉ when N is not a
power of 2 and l N = log 2 N+1, when N is a power of 2. We say that an algorithm
A is a polynomial time algorithm if its running time is polynomial in the bitsize of
its input. For an example, consider the algorithm for schoolbook multiplication.
It is a polynomial time algorithm, because for any two integer inputs a,b, we can
find a×b in time O(loga·logb), which is polynomial (quadratic) in the combined
input bitsize of loga+logb.
2.2 RSA Cryptosystem
RSA was designed by R. Rivest, A. Shamir and L. Adleman in 1977, when they
were at the Massachusetts Institute of Technology (MIT). It was published in
1978 and over the last three decades, RSA has become the most popular public
key cryptosystem. One of the main reasons behind the soaring popularity of
RSA is the simplicity of its theory and computations. It is widely used in modern
electroniccommerceprotocolsandisbelievedtobesecureprovidedtheparameters