  We develop a new algebraic technique that solves the following problem: Given
a black box that contains an arithmetic circuit $f$ over a field of
characteristic $2$ of degree~$d$. Decide whether $f$, expressed as an
equivalent multivariate polynomial, contains a multilinear monomial of degree
$d$.
  This problem was solved by Williams \cite{W} and Bj\"orklund et. al.
\cite{BHKK} for a white box (the circuit is given as an input) that contains
arithmetic circuit. We show a simple black box algorithm that solves the
problem with the same time complexity.
  This gives a simple randomized algorithm for the simple $k$-path problem for
directed graphs of the same time complexity\footnote{$O^*(f(k))$ is
$O(poly(n)\cdot f(k))$} $O^*(2^k)$ as in \cite{W} and with reusing the same
ideas from \cite{BHKK} with the above gives another algorithm (probably not
simpler) for undirected graphs of the same time complexity $O^*(1.657^k)$ as in
\cite{B10,BHKK}.
