  The main component of (constructive) recognition algorithms for black box
groups of Lie type in computational group theory is the construction of
unipotent elements. In the existing algorithms unipotent elements are found by
random search and therefore the running time of these algorithms is polynomial
in the underlying field size $q$ which makes them unfeasible for most practical
applications \cite{guralnick01.169}. Meanwhile, the input size of recognition
algorithms involves only $\log q$. The present paper introduces a new approach
to construction of unipotent elements in which the running time of the
algorithm is quadratic in characteristic $p$ of the underlying field and is
polynomial in $\log q$; for small values of $p$ (which make a vast and
practically important class of problems), the complexity of these algorithms is
polynomial in the input size.
  For $\psl_2(q)$, $\qpone$, we present a Monte-Carlo algorithm which
constructs a root subgroup $U$, the maximal torus $T$ normalizing $U$ and a
Weyl group element $w$ which conjugates $U$ to its opposite. Moreover, we
extend this result and construct Steinberg generators for the black box
untwisted classical groups defined over a field of odd size $q=p^k$ where
$\qpone$. Our algorithms run in time quadratic in characteristic $p$ of the
underlying field and polynomial in $\log q$ and the Lie rank $n$ of the group.
  The case $\qmone$ requires the use of additional tools and is treated
separately in our next paper \cite{suko12B}. Further, and much stronger results
can be found in \cite{suko12E,suko12F}.
