dissertacao.pdf

1.5.5 Continued Fractions
The continued fraction expansion of a real number x is its representation as the
tuple [a0; a1, ..., an, · · · ], such that the following equality is satisfied:
1
x = a0 +
1
a1 +
1
a2 +
. .. +
1
an + 1
. ..
with the coefficient a0 being any integer and the coefficients a1, · · · , an, · · · pos-
itive integers. If x = p
q
is a rational number then the number of coefficients in
its continued fraction expansion is finite, namely n = log(q). This coefficients
are the quotients computed by the Euclidean Algorithm applied to p and q.
The value ci = [a0; a1 · · · ai, for i = 0, ..., n, called the i-th convergent of
x, can be seen as an approximation of the value of x = [a0; a1, · · · , an].
We now present one result about continued fractions which is the basis for
Wiener’s attack, presented in Chapter 3.
Theorem 4. Let α ∈ Q and c, d ∈ Z such that:
|α − c 1
| < .
d 2d2 Then c
d , as an irreducible fraction, is one of the convergents of the continued
fraction expansion of α.
1.5.6 Results from Coppersmith
In an article written in 1997[9], Coppersmith shows how to solve a modular
polynomial equation of degree k in a single variable x, as long as the solutions are
sufficiently small. The work of Coppersmith is based in the theory of Lattices,
namely in one algorithm developed by Lenstra, Lenstra and Lovasz, the LLL-
algorithm[26].
Theorem 5. Let p(x) be a monic integer polynomial of degree k and N a
positive integer of unknown factorization. We can find all integer solutions x0
of p(x0) ∼ = 0 (mod N), such that |x0| < N 1
k , in time polynomial in log(N) and
k.
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