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

55 3.3 A New Class of Weak Keys
9456331382787 and
2671883975804894456278842490580443758950128272095600125097638973022749
4745205672316.
Note that these numbers are clearly greater than N 1 4 (in contrary to the bound
presented in [96] where 1 ≤ Y < X < 2 −1 4N 1 4) as N is a 1000 bit integer here.
Nowweshowthatthetechniqueof[96]willnotworkhere. WecalculatetheCF
e
expansion of and study all the convergents Y with denominator X < 4N 4.
1 N X 2−1
ExceptX = Y = 1, no eX−1 isaninteger. WhenX = Y = 1, wehave eX−1 = e−1.
Y
Y
Thus in this case, (p−u)(q−v) = e−1. As given in [96, Lemma 4], one needs to
satisfy the condition |(p−u)(q − v) −N| < 2 −1 2N 1 2. Thus, in this example, one
needs to satisfy |e−1−N| < 2 −1 2N 1 2, which is not true.
Then we attempted different cases with e = N α having varying α, given the
same p,q as in Example 3.3. The experimental results are as shown in Table 3.2
where each run to find (p−u)(q −v) requires less than 15 minutes.
α 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.875
γ 0.274 0.336 0.399 0.464 0.529 0.596 0.659 0.726 – –
Table 3.2: The numerical values of γ given α found by experiment when N is of
1000 bits and p,q are as in Example 3.3. The lattice has the parameters m =
7,t = 3,w = 60.
Note that compared to Table 3.1, the results in Table 3.2 gives slightly lower
values of γ. Further, we do not get the solutions for these p,q values when α =
1.8,1.875 as 1+γ−α becomes very close to zero and hence X does not exist given
the bound of Y. The maximum e for which we get a valid X is of size 1774 bits in
this example. Thus the maximum value of α for which our method works in this
example is 1.774. The value of γ in this example is 0.778 as Y is a 778-bit integer.
3.3 A New Class of Weak Keys
The problem with the idea of [96] is that one needs to factorize (p−u)(q −v) in
order to attack RSA, and this is only possible when the factors of either (p−u)
or (q−v) are relatively small. In this section we present a new class of weak keys