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Problem 43 (Affine Encryption Key Introduction)
If we let $m=31$, then we can use the function $f(x)=5x+12\pmod m$ to encrypt the message "save them" by (1) swapping the letters to the numbers (19,1,22,5,0,20,8,5,13) and then (2) applying $f(x)$ to each letter to obtain the encrypted numbers (14, 17, 29, 6, 12, 19, 21, 6, 15).
- Find $c,d\in \mathbb{Z}_{31}$ so that the inverse of $f$ is $f^{-1}(x)=cx+d$ where $c,d\in \mathbb{Z}_{31}$.
- If instead we let $m=27$, then find the inverses of $f(x)=2x+17\pmod{27}$ or explain why it cannot be done.
- Give an example of nonzero $a,b\in \mathbb{Z}_{27}$ so that $f(x)=ax+b\pmod{27}$ is not invertible.
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