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inverse element in Paillier cryptosystem
So your idea is that we are still considering the problem of whether there exists a x such that ax=1 mod N, where a here is defined as r^n and N here is defined as n^2 in the Paillier system. Because we know a has the multiplicative inverse only when gcd(a,N)=1, it turns out we know that r^n has the multiplicative inverse only when gcd(r^n,n^2)=1.