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

139 7.6 Improved Results for Larger k
comparison of our work with that of [40].
Very recently, M. Ritzenhofen [107] presented a distinct lattice based approach
for the problem of implicit factorization with shared MSBs and LSBs of
p 1 ,p 2 ,...,p k . The strategy of [107, Theorem 6.1.7] works when β ≤ 1−α− k
k−1 α.
Similar to our comparison with that of [86] for LSB case, it can be noted that our
method requires less number of bits to be shared compared to [107].
We have explained our results for the MSB case as well as LSB case and
compared with state of the art literature. The experimental results in both the
cases are of similar quality using our techniques. Similar results are achieved
in our case if one considers sharing of MSBs and LSBs together in the primes
p 1 ,p 2 ,...,p k . Thus, we do not repeat these results.
7.6 Improved Results for Larger k
In [128, Section 5.2], the authors studied the EPACDP for analyzing the security
of their scheme. Initially this strategy has been analyzed in [73,74]. Based on the
idea presented in [128], we get Theorem 7.14. The result in Theorem 7.14 below
is not exactly presented in a similar form in [128].
In case of EPACDP, one can write
a 1 = gq 1 ,
ã 2 = gq 2 − ˜x 2 ,
.
ã k = gq k − ˜x k .
Let us construct the matrix
⎛
M =
⎜
⎝
⎞
2 ρ ã 2 ã 3 ... ã k
0 −a 1 0 ... 0
. . .
...
⎟
. ⎠
0 0 0 ... −a 1