dissertacao.pdf

Another very important procedure for RSA regarding congruences is Mod-
ular Exponentiation. Given integers b, e, N, suppose we wish to calculate the
integer c satisfying:
c ∼ = b e
(mod N) (6)
A straightforward method is to simply calculate b e and then its remainder when
divided by N. This algorithm’s complexity is O(e) so it is infeasible for large
values of e. An alternative method, more efficient, is described below:
Algorithm 1. (Exponentiation by repeated squaring and multiplica-
tion). Given integers b, e, N, to compute the modular exponentiation
c ∼ = b e (mod N) we proceed as follows:
1. write e in its binary representation: (em−1, em−2, ..., e1, e0)2 such that
e = i=m−1 i=0 ei2i .
2. set c = 1.
3. from i = m − 1 to i = 0:
4. return c.
1. c = c 2 (mod N).
2. If ei = 1, then c = cb (mod N).
The complexity of this algorithm is O(m) = O(log(e)). The operations
in step 3 can be efficiently done implementing a technique called Montgomery
Reduction[34].
We now introduce one definition used in the surprisingly efficient determin-
istic primality checking algorithm presented in Chapter 2.
Definition 10. Given an integer a and an integer N such that (a, N) = 1, the
multiplicative order of a modulo N, denoted oN(a), is the smallest positive
integer k such that
test.
a k ∼ = 1 (mod N) (7)
And a definition used in the application of the Solovay-Strassen primality
Definition 11. Given an integer q and an integer n, we say that q is a
quadratic residue modulo n if there exists an integer x such that:
x 2 ∼ = q (mod n) (8)
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