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The authors present a construction of log-depth formulae for various threshold functions based on atomic threshold gates of constant size. From this, they build a new family of linear secret sharing schemes that are multiplicative, scale well as the number of players increases and allows to raise a shared value to the characteristic of the underlying field without interaction. Some of these schemes are in addition strongly multiplicative. Their formulas can also be used to construct multiparty protocols from protocols for a constant number of parties.
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