# Can Secure Multi-party Computation be used to reconstruct secret shares?

When using Shamir's Secret Sharing is it possible to use Secure Multi-Party Computation to reconstruct the original secret without revealing one's secret share?

When searching I have found results talking about using secret sharing as part of a Multi-Party Computation scheme. But that is not my question.

## 1 Answer

Sure you can.

Suppose there are $t$ shares $(x_1,y_1),\ldots,(x_t,y_t)$ hold by $t$ different parties. Recall that in Shamir secret sharing, the secret is the free coefficient in the polynomial. Then to reconstruct the secret, the parties need to compute jointly:

$$L(0) = {\sum_{j=1}^{t}} y_j \cdot l_j(0)$$

where

$$l_j(0) =\prod_{\begin{smallmatrix}1\le m\le t\\ m\neq j\end{smallmatrix}} \frac{x_m}{x_m-x_j}$$

Usually $x_1,\ldots,x_t$ are public, so all parties can compute all $l_j(0)$ locally. Then what they need to do is to share $y_j$ among them and compute $L(0)$ securely using generic secret sharing based MPC.