  Weak coin flipping (WCF) is a fundamental cryptographic primitive for
two-party secure computation, where two distrustful parties need to remotely
establish a shared random bit whilst having opposite preferred outcomes. It is
the strongest known primitive with arbitrarily close to perfect security
quantumly while classically, its security is completely compromised (unless one
makes further assumptions, such as computational hardness). A WCF protocol is
said to have bias $\epsilon$ if neither party can force their preferred outcome
with probability greater than $1/2+\epsilon$. Classical WCF protocols are shown
to have bias $1/2$, i.e., a cheating party can always force their preferred
outcome. On the other hand, there exist quantum WCF protocols with arbitrarily
small bias, as Mochon showed in his seminal work in 2007 [arXiv:0711.4114]. In
particular, he proved the existence of a family of WCF protocols approaching
bias $\epsilon (k)=1/(4k+2)$ for arbitrarily large $k$ and proposed a protocol
with bias $1/6$. Last year, Arora, Roland and Weis presented a protocol with
bias $1/10$ and to go below this bias, they designed an algorithm that
numerically constructs unitary matrices corresponding to WCF protocols with
arbitrarily small bias [STOC'19, p.205-216]. In this work, we present new
techniques which yield a fully analytical construction of WCF protocols with
bias arbitrarily close to zero, thus achieving a solution that has been missing
for more than a decade. Furthermore, our new techniques lead to a simplified
proof of existence of WCF protocols by circumventing the non-constructive part
of Mochon's proof. As an example, we illustrate the construction of a WCF
protocol with bias $1/14$.
