I try to create a valid decryption function for a modified the Goldwasser-Micali scheme
$pk = (N,p,q)$ where N is a Blum-Number $(p = q = 3 \bmod 4)$
$Enc_{pk}(m) = (-1)^{m} \cdot r^{2} \bmod N \text{ where } r \in_{R} \mathbb{Z}^{*}_{N}$
Now when I want to decrypt a ciphertext I can check if $m = 0$ by checking if $c \in QR_{N}$ because $$ c = (-1)^{0} \cdot r^{2} \bmod N = 1 \cdot r^{2} \bmod N $$
but when $m = 1$ then we got $$ c = -1 \cdot r^{2} \bmod N $$ now the Law of quadratic reciprocity states
The first supplement to the law of quadratic reciprocity is that if p ≡ 1 (mod 4) then −1 is a quadratic residue modulo p, and if p ≡ 3 (mod 4) then −1 is a nonresidue modulo p.
so we know $c \in QNR_{N}$ but for Goldwasser-Micali I need $c \in QNR^{+1}_{N}$ is there a way to proof that in this case $c$ has the Jacobi symbol $+1$ ? Or is this the wrong direction?
