While reading this document I came across the following problem. Assume you have $n$ clients. The clients need to generate random integers in $\mathbb{Z}_p$, say $T_i$ for $i \in \{1, \ldots, n\}$, such that $\sum_{i=1}^n T_i = 0$ in $\mathbb{Z}_p$. This integers are then used as secret keys, so if $n-2$ clients cooperate, they still should not be able to obtain the secret key of the other two.

This is a bit modified version of the problem appearing in this document (page 15) in SetUp procedure. There it is not explained how to do it.

The closest answer to this question I found is in this document (section 6.1), but is not exactly what I am searching for. Is there a known solution I am missing?

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    $\begingroup$ I've seen this referred to as pseudorandom zero sharing here and here. Though in those it was done in the shamir secret sharing scheme. $\endgroup$ – mikeazo Nov 5 '18 at 13:34

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