We investigate the Coppersmith technique for finding solutions of
a univariate modular equation within a range given by range parameter U.
This technique converts a given equation to an algebraic equation vi
a lattice reduction algorithm, and the choice of the lattice is crucial
for the performance of the technique. This paper provides a way to analyze
a general type of limitation of this lattice construction. Our analysis
bounds the possible range of U from above that is asymptotically equal
to the bound given by the original result of Coppersmith.  It means
that Coppersmith has already given the best lattice construction. To show
our result, we establish a framework for the technique by following the
reformulation of Howgrave-Graham, and derive a condition, which we call
the lattice condition, for the technique to work. We then provide a way
to analyze a bound of U for achieving the lattice condition. Technically,
we show that (i) the original result of Coppersmith achieves an optimal
bound for U when constructing a lattice in a standard way. We then show
evidence supporting that (ii) a nonstandard lattice construction is
generally difficult. We also report on computer experiments
demonstrating the tightness of our analysis.