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It depends on how you make shares out of a polynomial. Consider for example Shamir secret sharing, using which a party can share a secret $s \in F$ (where $F$ is a finite field) with $n$ parties by doing the following: Construct a polynomial $f(x) = s + a_1x+a_2x^2+\dots + a_nx^t$, for some $t$, where $a_i \in_R F$ for all $1 \leq i \leq n$. Send the value ...

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Just to complement prof. Lindell's answer, although one cannot have a formal description of the case in which round complexity will matter more than communication or computation, a colleague of mine did some estimations two years ago, for a paper we were writing on round-efficient primitives for zero-knowledge. It's just a particular case, but having figures ...

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First note that all polynomial-time functions can be securely computed with a constant number of rounds (Yao and BMR families) and all can be securely computed with protocols that have rounds dependent on the depth of the circuit computing the function (GMW families). The question is when is one type better than another. The answer to this question is not ...

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