Secret Sharing and Secure Computing from Monotone Formulae

We present a construction of log-depth formulae for various threshold functions based on atomic threshold gates of constant size. From this, we 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. Our formulas can also be used to construct multiparty protocols from protocols for a constant number of parties. In particular we implement black-box multiparty computation over non-Abelian groups in a way that is much simpler than previously known and we also show how to get a protocol in this setting that is efficient and actively secure against a constant fraction of corrupted parties, a long standing open problem. Finally, we show a negative result on usage of our scheme for pseudorandom secret sharing as defined by Cramer, Damgård and Ishai

Efficient Multiparty Protocols via Log-Depth Threshold Formulae

We put forward a new approach for the design of efficient multiparty protocols: 1. Design a protocol for a small number of parties (say, 3 or 4) which achieves security against a single corrupted party. Such protocols are typically easy to construct as they may employ techniques that do not scale well with the number of corrupted parties. 2. Recursively compose with itself to obtain an efficient n-party protocol which achieves security against a constant fraction of corrupted parties. The second step of our approach combines the player emulation technique of Hirt and Maurer (J. Cryptology, 2000) with constructions of logarithmic-depth formulae which compute threshold functions using only constant fan-in threshold gates. Using this approach, we simplify and improve on previous results in cryptography and distributed computing. In particular: - We provide conceptually simple constructions of efficient protocols for Secure Multiparty Computation (MPC) in the presence of an honest majority, as well as broadcast protocols from point-to-point channels and a 2-cast primitive. - We obtain new results on MPC over blackbox groups and other algebraic structures. The above results rely on the following complexity-theoretic contributions, which may be of independent interest: - We show that for every integers j,k such that m = (k-1)/(j-1) is an integer, there is an explicit (poly(n)-time) construction of a logarithmic-depth formula which computes a good approximation of an (n/m)-out-of-n threshold function using only j-out-of-k threshold gates and no constants. - For the special case of n-bit majority from 3-bit majority gates, a non-explicit construction follows from the work of Valiant (J. Algorithms, 1984). For this special case, we provide an explicit construction with a better approximation than for the general threshold case, and also an exact explicit construction based on standard complexity-theoretic or cryptographic assumptions