Few years ago I wrote a brief paper regarding the RSA cryptosystem and the attack M.J. Wiener published in 1990. I tried to create a self-contained work, emphasising the power of playing with continued fractions using Legendre theorem. The attack states that when there is a small private exponent, the RSA module \(N\) could be factorized in \(\mathcal{O}\big(log_2(N)\big)\), let’s take a peak on how it works using an example.

There are three numbers you need to know when we talk about the RSA cryptosystem: the module \(N\) plus the exponents \(e\) and \(d\). So, given the public key \((e,N)\) of an RSA with a small private exponent \(d\)

\[(e,N) = (58549809,2447482909)\]

the expansion of \(\frac{e}{N}\) and its convergents will be

\[\frac{e}{N} = \bigl[\,0;\,41,\,1,\,4,\,23,\,\dots\,\bigr]\] \[\{ c_i \}_{i{\leq}15} = \biggl\{0,\,\frac{1}{41},\,\frac{1}{42},\,\frac{5}{209},\,\frac{116}{4849},\,\dots\biggr\}\]

Using the search algorithm derived from Wiener Theorem, we notice that the fourth convergent \(c_3{=}\frac{5}{209}\) is the right candidate \(\widetilde{\phi}\) for the Euler function \(\phi(N)\):

\[\biggl\lfloor \frac{e}{c_3} \biggr\rfloor = \biggl\lfloor \frac{58549809}{\frac{5}{209}} \biggr\rfloor = 2447382016\]

Now that we found a possible value for \(\phi\), we won: the two prime factors of the RSA module have to be \(60317\) and \(40577\). It was too quick, wasn’t it? Take a look at the paper here or in the window below, it will satisfy any doubts and show you some magic in Wolfram Mathematica.

No PDF support, click clack and download the thing.

I put some effort on the appearance too: all the work is written in \(\LaTeX\) with the Tufte-LaTeX package, plus few modifications here and there allowed me to squeeze all the content in about 20 pages.