let's suppose that k=(α,β) in a Afffine Cypher on Z26
Prove that k is an involutary key iff
n | (a^2 -1) & β(α+1) is congruent to 0 (mod n)
I know that an involutary key is a key where
the encryption function and
decryption function is equal but I have no idea how I can use that to show that
n | (a^2 - 1) and that β(α+1) is congruent to 0 (mod n)
I'm hopefully lost.. I'm not even sure how they got to the end statement to begin with. If i set ek(x) = dk(x) I don't get anywhere close to what I need to prove.
I will show you half of this problem. All algebra is occurring mod $26$ (or $n$, whatever you like). If encryption occurs by $e(x) = \alpha x + \beta$ then decryption occurs by $d(x) = \alpha^{-1}x - \alpha^{-1}\beta$. This you already know, as $d$ is just $e^{-1}$ and you can do that algebra.
Your problem is asking for $d=e$, so equate the coefficients and get to work: $$ \alpha = \alpha^{-1} \quad \text{ and } \quad \beta = -\alpha^{-1}\beta. $$