# Shamir's Secret Sharing of modular inverse

Given a secret $K \bmod q$ which is shared among entities $E_1,…,E_n$ using polynomial Shamir's Secret Sharing, how can the inverse of $k$ be shared without revealing $k$ and $k^{-1}$?

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I'm assuming $q$ is prime.
Note that $k^{-1}\equiv k^{q-2}\bmod{q}$.
We can compute $k^{q-2}\bmod{q}$ via a multiparty computation protocol such as the one outlined in this paper ($\S3.1)$. Or, since $q$ is likely a public parameter, you can even do the simple square-and-multiply algorithm (for example, VIFF provides this capability).