For the Substitution Cipher, for functions such as $$f(x) = x^k \pmod{26}$$ or $$f(x) = x^k + k \pmod{26}$$ (functions that consist of x^k), why do the values of k=5 or k=7 or k > 10 have one-to-one mapping? I know 5 and 7 are relatively prime to 26, however, what about 3 or 9? Also why do it map when k > 10?
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The map $x\mapsto x+a ~\pmod n$ is a permutation for any constant $a$.
The map $x\mapsto x^k \pmod n$ is a permutation if and only if )$\gcd(k,\phi(n))=1,$ as pointed out in the other answer.
So $x\mapsto x^k+k\pmod n$ is also a permutation.
kodlu
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Unlike $k\cdot x$, $x^k$ works in the multiplicative group rather than additive group.
So (by Fermat's little theorem IIRC) you need to factor 26, subtract 1 from each factors, then multiply them back together (like RSA). which means $k$ must be coprime to 12.
By this logic, 3 and 9 obviously don't work; and it's probably because you've not tried every $k>10$ to determine validity.