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

π

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

π

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

j

,

k

 ∈ ℕ such that

$m \triangleq \frac{k-1}{j-1}$

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.