If the plaintext is made of both letters (a to z) and digits (0 to 9), how do you find the key domain of the multiplication cipher?


You have 36 possible "characters" here.

If a single character is encrypted by E(C) = (c * k) % 36 then possible keys k are numbers that are coprime to 36, ie. gcd(k,36)=1. Furthermore it makes not much sense to consider numbers not between 1 and 36, because of the modulo.

As 36=2*2*3*3, the possible keys are basically all numbers not multiples of 2 and/or 3. That is:
5 7 11 13 17 19 23 25 29 31 35
Technically 1 too, but this would be no change from plaintext

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