The paper “Residuosity Problem and Its Applications to Cryptography” considers the exponent to be an odd integer.
When $k = 2$, it is called the quadratic residuosity problem (mod $n$, where $n$ is composite) which is hard and can be solved if the factorization of n is known.
What happens to the $k$th residuosity problem if $k$ is an even integer $> 2$?
