# Why blinding integer $r$ needs to be co-prime with $N$ in blind RSA signature

According to blind RSA signature wiki the random value $$r$$ in $$m^\prime=mr^e \bmod N$$ needs to be relatively prime to $$N$$.

• Why is this a requirement?

A follow up a related question: after $$s$$ (unblinded signature) and $$m$$ (original message) are revealed, can anyone (and not only the signing authority) check the validity of the signature by checking $$s^e=m \bmod N$$?

Well, if $$r$$ has (say) $$p$$ as a factor, then it makes the unblinding step rather difficult.
The original $$m$$ has $$N = pq$$ possible, as does $$m^d$$; if $$r$$ is a multiple of $$p$$, then $$mr^e$$ has only $$q$$ possible values; if we pass it to the signer, it'll come back to us with a value $$m'^d$$ that also has only $$q$$ possible values. We cannot map $$m'^d$$ back to $$m^d$$ (as a single $$m'^d$$ value would map back to $$p$$ possible $$m^d$$ values, and we don't know which one it is.
A follow up a related question: after $$s$$ (unblinded signature) and $$m$$ (original message) are revealed, can anyone (and not only the signing authority) check the validity of the signature by checking $$s^e=m \bmod N$$?