dissertacao.pdf

Definition 8. Given an integer N > 0, the modular ring ZN is the set:
ZN = {[0], [1], ..., [N − 1]} (3)
(which will also be defined as ZN = {0, 1, ..., N − 1})
along with the modular operations defined above. The division operation is
an extension of the multiplication operation: to divide the equivalence class x
by the equivalence class y, we multiply x by the inverse of y (mod N), according
to the next definition.
In the ring ZN we can easily define inverses, which will be necessary to create
RSA keys:
Definition 9. Let a be an element of the modular ring ZN. The inverse of
a modulo N is the integer x satisfying:
which we will refer to as a −1 (mod N).
ax ∼ = 1 (mod N) (4)
It is important to note that a −1 (mod N) exists if and only if (a, N) = 1.
When this is the case, the Extended Euclidean Algorithm (explained in the next
section) will provide us with the values x, y such that ax + yN = 1. From the
last equation we get:
ax ∼ = 1 (mod N) ⇔ ax − 1 = kN ⇔ ax − kN = 1 (5)
We know a and N so x, the inverse of a, is given by the Extended Euclidean
Algorithm (along with the value of k , which can be discarded). If (a, N) = 1
then there are no integer solutions x, k for this equation, and therefore there is
no inverse of a modulo N. However, if given a modulus N and an integer a,
we find out that a does not have an inverse modulo N, then we can compute a
factor of N by simply computing (a, N).
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