DGK Cryptosystem Encryption Speedup
As it turns out, this is just a beginner question and I think I managed to figure it out. Since $E(m,r) = g^mh^r \pmod n$, I can just split it in two: $E(m,r) = (g^m \pmod n) (h^r\pmod n)$ and then apply the CRT trick on both sides, after which I can multiply the results together to get back the original:

$E_{nonrand}(m,r) = (g^m \bmod p)*q*(q^{-1}\bmod p) + (g^m \bmod q)*p*(p^{-1}\bmod q) \pmod n$

$R_{h}(r) = (h^r \bmod p)*q*(q^{-1}\bmod p) + (h^r \bmod q)*p*(p^{-1}\bmod q) \pmod n$

$E(m,r) = E_{nonrand}(m,r)R_{h}(r) \pmod n$

Can anyone think of any reasons why I should not skip the (mod n) reductions for $E_{nonrand}(m,r)$ and $R_{h}(r)$? It looks to me like they are redundant...