How to calculate inverse of an element using MPC and secret sharing

Question

Suppose we have multiple parties ${P_i}$ each holding a secret share $(y_i,k_i)$ of the secret $(y,k)$. How can we calculate the inverse value $(y+k)^{-1}$ using multiparty computation such that these secret shares are never revealed?

Answer 1

First, to simplify notation, note that computing $(y+k)^{-1}$ is equivalent to computing $z^{-1}$, where $z=y+k$, which can be locally computed by the parties. Furthermore, I assume that $z \in \mathbb{F}_p$ is a field element (e.g., integers mod $p$).

Now, to compute $z^{-1} \in \mathbb{F}_p$, each party can (locally) sample a random share of a value $r \in \mathbb{F}_p \setminus \{0\}$. Then, the parties reveal $zr$ to compute $(zr)^{-1} \in \mathbb{F}_p$, in the clear. Finally, the parties use their shares of $r$ to engage in a MPC protocol for multiplication to compute $(zr)^{-1}\cdot r = z^{-1} \in \mathbb{F}_p$. A similar protocol was described in Algorithm for computing modular inverse in MPC under a different context.