A description of the challenge was also provided along with the source code. Please have a look at cotan.pdf.

Challenge files: cotan.pdf, cotan.py.

The challenge name refers to the fact that we're somehow `emulating' the cotangent function in a finite field

cotangent

In real numbers, this is effectively finding given , however, the challenge is in

Consider how cotangent is related to the complex expressions:

Performing the substitution , we get and

in

Plugging in the values from the challenge, we have and (where )

Now we need to somehow express in . Since , we simply need to find a number such that , which is trivial for (here ). Both solutions would work here.

solving for , and

\begin{align*} u&=(2+i)/(2-i)\\\\ c&=i(u^n+1)/(u^n-1)\\\\ u^n&=(c+i)/(c-i)\\\\ \end{align*}

Now it's just a discrete log problem, which takes some time to solve (15mins on my old and broken laptop)