Constant-Rounds, Almost-Linear Bit-Decomposition of Secret Shared Values

Abstract

Bit-decomposition of secret shared values – securely computing sharings of the binary representation – is an important primitive in multi-party computation. The problem of performing this task in a constant number of rounds has only recently been solved.

This work presents a novel approach at constant-rounds bit-decomposition. The basic idea provides a solution matching the big-$\mathcal{O}$-bound of the original while decreasing the hidden constants. More importantly, further solutions improve asymptotic complexity with only a small increase in constants, reducing it from $\mathcal O(\ell{\rm log}(\ell))$ to $\mathcal O({\ell}{\rm log}^*(\ell))$ and even lower. Like previous solutions, the present one is unconditionally secure against both active and adaptive adversaries.