Affine Cipher: Why is the multiplicative inverse of an element mod $26$ is $a^{−1} ≡ a^{11}\mod 26$? Prove with Euler’s Theorem.

Question

The total question is: "For the affine cipher in Chapter $1$ the multiplicative inverse of an element modulo 26 can be found as $a^{−1} ≡ a^{11}\mod 26$. Derive this relationship by using Euler’s Theorem."

I could really use help figuring this out. I don't even know where to start, and my textbook is not very helpful.

Answer 1

Euler's Theorem

Let $\phi$ be Euler's Totient function. If $\gcd(a,n)=1$, then $$a^{\phi(n)}\equiv_n 1$$

Assume that $a$ has an inverse ($a,26$ are coprime), then by Eulers theorem$$a^{\phi(26)}\equiv_{26}1\\a^{\phi(26)-1}\equiv_{26}a^{-1}$$ We have $\phi(26)=\phi(2)\phi(13)=12$, so $$a^{11}\equiv_{26}a^{-1}$$