Decisional composite residuosity assumption

The decisional composite residuosity assumption (DCRA) is a mathematical assumption used in cryptography. In particular, the assumption is used in the proof of the Paillier cryptosystem.

Informally, the DCRA states that given a composite $$n$$ and an integer $$z$$, it is hard to decide whether $$z$$ is an $$n$$-residue modulo $$n^2$$. I.e. whether there exists a $$y$$ such that


 * $$z \equiv y^n \pmod{n^2}. \, $$