It's basically a kind of MCG (multiplicative congruential generator) but instead of the last entry in the series being the parameter I used the current ID. That means the result is not as random because the distance between consecutive ID is basically constant. The reason I did this is to save computation time (exponentiation would work, but 128 bit won't be enough for numbers that size).
The condition for p and m is that their GCD is 1. I chose m to be a power of 2 so it's prime factors are only 2s. For p I chose a prime that is about 25 % of m. For that I hardcoded a list of possible primes candidates into the program.
To make it more efficient the program calculates m to be the greatest power of 2 that fits in the output space.
For example: Let the character set be "0-9" (10 characters) and the length of the result is 4, the output space would be 10^4. The biggest power of 2 in that space is 2^13. That's m. The p would be 2039 (about 25 % of m).
(Note to myself: Maybe just using a big Mersenne prime (like 2^61-1) would also work. That would maximize the useable output space efficiency, because it's guaranteed that x^y with y > 1 is not a prime, thus m can be size of the output space.)
As for the decoding we have to solve the congruence equation x == p * i (mod m) for i. We can do that by multiplying x the multiplicative inverse of p and m.