  Weak coin flipping is an important cryptographic primitive -- it is the
strongest known secure two-party computation primitive that classically becomes
secure only under certain assumptions (e.g. computational hardness), while
quantumly there exist protocols that achieve arbitrarily close to perfect
security. This breakthrough result was established by Mochon in 2007
[arXiv:0711.4114]. However, his proof relied on the existence of certain
unitary operators which was established by a non-constructive argument.
Consequently, explicit protocols have remained elusive. In this work, we give
exact constructions of related unitary operators. These, together with a new
formalism, yield a family of protocols approaching perfect security thereby
also simplifying Mochon's proof of existence. We illustrate the construction of
explicit weak coin flipping protocols by considering concrete examples (from
the aforementioned family of protocols) that are more secure than all
previously known protocols.
