In homomorphic encryption scheme FV, I can multiply an encrypted polynomial for any integer scalar, obtaining the same effect on the plain polynomial. For example, given the plain polynomial $\sum_ia_ix^i$, and its encrypted version $\sum_ib_ix^i$, I could multiply the latter by $c$ and decipher it, to obtain $\sum_i(ca_i)x^i$. Is there a way to do the same with squaring? Can I perform some operation on $\sum_ib_ix^i$ and then decipher it, to get $\sum_ia_i^2x^i$? I already tried with $\sum_ib_i^2x^i$, but it just dosen't work.
About the reasons I'm doing this, it's because I have to use a polynomial activation function in a neural network that uses homomorphic encryption, as explained in this paper