For a project I am working on, I have access to a CSPRNG that outputs a random integer in the interval $[0, 2^n-1]$ for any integer $n$ greater than 0. I cannot use the zero values, so I have my RNG code in a
while loop that runs until the random number is
!==0. For small $n$ it is very likely that the while loop will run more than once, even more than twice (for $n=8$, the probability is $1/65536$). While this is not the bottleneck in my program, I want to eliminate the
while loop and replace it with a linear transformation on the the generated random integer to get the integers I need in $[1, 2^n-1]$.
Is there a transformation that can be performed on the interval $[0, 2^n-1]$ to yield integers in $[1, 2^n-1]$ while retaining the uniform distribution of the random numbers?
(Would this be better posted in math.se?)