# Generate unique pair $g^k, E(k^{-1})$ for each group

Let say I have n computers, with some t-threshold encryption scheme. I want to have n public shares (known to every participant) such that any t of them generates a pair: $$g^k, E(k^{-1})$$ that is unique, such that $$E(x)$$ is the encrypted value of x. and $$k^{-1}$$ is computed in some cyclic group of power $$q$$. Also no member can know the value of $$k$$ or $$k^{-1}$$.

I have a solution that requires communication:

each share is $$(g^{k_i}, E(k_i), E(p_i), E(c_i))$$. The $$k_i$$, $$p_i$$ and $$c_i$$ are private, the shares are public.

When a group $$Is$$ of size $$t$$ wants to generate $$g^k$$ and $$E(k^{-1})$$ they compute

$$g^k = \prod_{i\in Is}g^{k_i}$$, $$E(k) = \sum_{i\in Is}E(k_i)$$, $$E(p) = \sum_{i\in Is}E(p_i)$$, $$E(c) = \sum_{i\in Is}E(c_i)$$

$$E(k)*E(p) + q*E(c)= E(kp+qc)$$ So we decrypt that and each party receives $$kp+qc$$. Remember, $$q$$ is the power of the group and public. So we can have kp by computing $$kp = (kp+qc) \space mod \space q$$. After that, we can compute $$E(k^{-1})=E({(kp)}^{-1}*p)={(kp)}^{-1}*E(p)$$.

To use this, we require an homomorphic encryption with level-1 multiplication. Is there anyway I could do this without communication? Is there any encryption scheme that support that properties?