  The estimation of phase transitions in random boolean Constraint Satisfaction
Problems (CSP) is based on two fundamental tools: the first and second moment
methods. While the first moment method on the number of solutions permits to
compute upper bounds on any boolean CSP, the second moment method used for
computing lower bounds proves to be more tricky and in most cases gives only
the trivial lower bound 0. In this paper, we define a subclass of boolean CSP
covering the monotone versions of many known NP-Complete boolean CSPs. We give
a method for computing non trivial lower bounds for any member of this
subclass. This is achieved thanks to an application of the second moment method
to some selected solutions called characteristic solutions that depend on the
boolean CSP considered. We apply this method with a finer analysis to establish
that the threshold $r_{k}$ (ratio : #constrains/#variables) of monotone
1-in-k-SAT is $\log k/k\leq r_{k}\leq\log^{2}k/k$.
