dissertacao.pdf

1.5 Mathematical Basis
In this section we start by presenting definitions for some of the symbols used
throughout the work. Then we present the definitions necessary to understand
what is the efficiency of an algorithm. Next we introduce the reader to modular
arithmetic, the basis of RSA, along with some useful theorems and algorithms.
Finally we present some advanced results used in the attacks against RSA with
low exponents.
1.5.1 Notation
We will use the following symbols whenever a, b, N are positive integers, p a
prime number, m = p α1
1 pα2 2 ...pαn n is an odd integer with prime factors p1, ..., pn
and ( a
b ) is referred to as the Legendre symbol.
Symbol Definition
Table 1: Notation
(a,b) greatest common divisor between a and b
lcm(a,b) least common multiple of a and b
φ(N) number of positive integers smaller than and co-
prime to N
λ(N) smallest positive integer m such that
a
p
a m ∼ = 1 (mod N) ∀a ∈ ZN : (a, N) = 1
⎧
⎪⎨
⎪⎩
0 , if p|a
1 , if a is a quadratic residue modulo p
−1 , if a is a quadratic nonresidue modulo p
( a
a
m ) = [ p1 ]α1 [ a
p2 ]α2 ...[ a
pn ]αn
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