# Why should the smudge noise be used?

Consider a threshold FHE scheme based on RLWE like this: Refer to this paper

$$\textbf{Initialization:}$$

Every party generates his own secret key $$s_i$$, then uses the common polynomial $$a$$ to generate his public key $$pk_i=(p_{i,0},p_{i,1})=(-as_i+e_i,a)$$. Therefore, the ideal public key is $$pk = (p_0,p_1)=(\sum p_{i,0}, a)$$.

$$\textbf{Online:}$$

The party $$P_i$$ has his own data $$x_i$$, he encrypts it under the $$pk$$ and sends the ciphertext to some receiver. The receiver computes some function and then sends the result $$(c_0,c_1)$$ to every party.For the parties, they perform the partial decryption: $$h_i = c_1s_i+e_{smudge}$$.

My Question: Why smudge? Why not use a common noise $$\chi$$ distribution? Could you give a simple derivation?