# Efficient Multiparty Protocols via Log-Depth Threshold Formulae

(Extended Abstract)
• Gil Cohen
• Ivan Bjerre Damgård
• Yuval Ishai
• Jonas Kölker
• Peter Bro Miltersen
• Ran Raz
• Ron D. Rothblum
Conference paper

DOI: 10.1007/978-3-642-40084-1_11

Part of the Lecture Notes in Computer Science book series (LNCS, volume 8043)
Cite this paper as:
Cohen G. et al. (2013) Efficient Multiparty Protocols via Log-Depth Threshold Formulae. In: Canetti R., Garay J.A. (eds) Advances in Cryptology – CRYPTO 2013. Lecture Notes in Computer Science, vol 8043. Springer, Berlin, Heidelberg

## Abstract

We put forward a new approach for the design of efficient multiparty protocols:

1. 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. 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.

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© International Association for Cryptologic Research 2013

## Authors and Affiliations

• Gil Cohen
• 1
• Ivan Bjerre Damgård
• 2
• Yuval Ishai
• 3
• Jonas Kölker
• 2
• Peter Bro Miltersen
• 2
• Ran Raz
• 1
• Ron D. Rothblum
• 1
1. 1.Weizmann InstituteRehovotIsrael
2. 2.Aarhus UniversityAarhusDenmark
3. 3.TechnionHaifaIsrael