  For a map (function) $F(x):\ftwo^n\rightarrow\ftwo^n$ and a given $y$ in the
image of $F$ the problem of \emph{local inversion} of $F$ is to find all
inverse images $x$ in $\ftwo^n$ such that $y=F(x)$. In Cryptology, such a
problem arises in Cryptanalysis of One way Functions (OWFs). The well known
TMTO attack in Cryptanalysis is a probabilistic algorithm for computing one
solution of local inversion using $O(\sqrt N)$ order computation in offline as
well as online for $N=2^n$. This paper proposes a complete algorithm for
solving the local inversion problem which uses linear complexity for a unique
solution in a periodic orbit. The algorithm is shown to require an offline
computation to solve a hard problem (possibly requiring exponential
computation) and an online computation dependent on $y$ that of repeated
forward evaluation $F(x)$ on points $x$ in $\ff_{2^n}$ which is polynomial time
at each evaluation. However the forward evaluation is repeated at most as many
number of times as the Linear Complexity of the sequence $\{y,F(y),\ldots\}$ to
get one possible solution when this sequence is periodic. All other solutions
are obtained in chains $\{e,F(e),\ldots\}$ for all points $e$ in the Garden of
Eden (GOE) of the map $F$. Hence a solution $x$ exists iff either the former
sequence is periodic or a solution occurs in a chain starting from a point in
GOE. The online computation then turns out to be polynomial time $O(L^k)$ in
the linear complexity $L$ of the sequence to compute one possible solution in a
periodic orbit or $O(l)$ the chain length for a fixed $n$. Hence this is a
complete algorithm for solving the problem of finding all rational solutions
$x$ of the equation $F(x)=y$ for a given $y$ and a map $F$ in $\ff_{2^n}$.
