Boneh-Durfee Attack

and if we decide to consider $x = \frac{N+1}{2}$and $y = \frac{p+q}{2}$, we will have:

At this point, finding $d$is equivalent to find the 2 small solutions $x$and $y$to the congruence

now let $s = -\frac{p+q}{2}$and $A = \frac{N+1}{2}$this will preserve the scomposed $\phi(N)$subtraction

consider $e=N^{\alpha}$(with any $\alpha$), we deduct that $\alpha$must be really closed to $1$because $e$is in the same order of the length of $N$(so $e=N^{\sim 1^{-}}$), we will get