I am studying an encryption scheme which is Elgamal-like where I think CRT can help optimise the encryption and decryption but I am not sure if I am applying CRT the correct way.
I have a cyclic group $G$ of order $n=p_1p_2q$, generator $g$ of $G$, an element $h=g^x$ of $G$ where $x$ is a random integer and the secret key, and a message $m\in\mathbb{Z}_{p_1p_2}^*$. This group $G$ has a cyclic subgroup $F$ of order $p_1p_2$ generated by $f$. Computing discrete logs in $F$ is easy which we use to recover our message $m$ as follows:
Encryption -- Compute ciphertext $(c_1,c_2) = (g^r,\ f^mh^r)$ where $r$ is a random integer
Decryption -- Compute $c_2/c_1^x = f^m = l$. Recover $m$ by computing $$m = l^{-1}\pmod{p_1p_2}$$.
I am applying CRT in the final step by computing $$l^{-1}\pmod{p_1}$$ and $$l^{-1}\pmod{p_2}$$ and then combining them both using CRT to recover $m$. I have a feeling this is stupid because I am not really optimising anything, infact making it more computation heavy as single inversion modulo is faster than two inversions modulo + CRT. This is also visible when I time these two methods -- single inversion modulo $p_1p_2$ is faster.
So how should I properly apply CRT in these type of schemes?
