  In a multi-party fair coin-flipping protocol, the parties output a common
(close to) unbiased bit, even when some adversarial parties try to bias the
output. In this work we focus on the case of an arbitrary number of corrupted
parties. Cleve [STOC 1986] has shown that in any such $m$-round coin-flipping
protocol, the corrupted parties can bias the honest parties' common output bit
by $\Theta(1/m)$. For more than two decades, the best known coin-flipping
protocol was the one of Awerbuch et al. [Manuscript 1985], who presented a
$t$-party, $m$-round protocol with bias $\Theta(t/\sqrt{m})$. This was changed
by the breakthrough result of Moran et al. [TCC 2009], who constructed an
$m$-round, two-party coin-flipping protocol with optimal bias $\Theta(1/m)$.
Haitner and Tsfadia [STOC 2014] constructed an $m$-round, three-party
coin-flipping protocol with bias $O(\log^3m / m)$. Still for the case of more
than three parties, the best known protocol remained the
$\Theta(t/\sqrt{m})$-bias protocol of Awerbuch et al.
  We make a step towards eliminating the above gap, presenting a $t$-party,
$m$-round coin-flipping protocol, with bias $O(\frac{t^4 \cdot 2^t \cdot
\sqrt{\log m}}{m^{1/2+1/\left(2^{t-1}-2\right)}})$ for any $t\le \tfrac12
\log\log m$. This improves upon the $\Theta(t/\sqrt{m})$-bias protocol of
Awerbuch et al., and in particular, for $t\in O(1)$ it is an $1/m^{\frac12 +
\Theta(1)}$-bias protocol. For the three-party case, it is an $O(\sqrt{\log
m}/m)$-bias protocol, improving over the $O(\log^3m / m)$-bias protocol of
Haitner and Tsfadia.
  Our protocol generalizes that of Haitner and Tsfadia, by presenting an
appropriate recovery protocol for the remaining parties to interact in, in the
case that some parties abort or are caught cheating. We prove the fairness of
the new protocol by presenting a new paradigm for analyzing fairness of
coin-flipping protocols.
