# Generalized Benaloh cryptosystem with $r=2$

Benaloh cryptosystem requires $$\gcd(r, (q-1))=1$$ which is impossible if $$q>2$$ (since it needs to be a large prime) and $$r=2$$. This confuses me, since Benaloh is referred to as an "extension" or "generalization" of Goldwasser-Micali cryptosystem, but even though they're extremely close, Benaloh doesn't seem to work at $$r=2$$ when Goldwasser-Micali specifically works at $$r=2$$. Is there a way to make changes to these cryptosystems so that we can "generalize" Goldwasser-Micali in a way it also works for $$r=2$$ (i.e. works for all prime numbers* $$r$$)?

• [*] I said prime since Benaloh's original cryptosystem doesn't work for composite $$r$$s, as pointed out and fixed by Fousse et al. Prime $$r$$ is sufficient for my uses, but I'd like to generalize for all primes including $$2$$.