  The Welch map $x \rightarrow g^{x-1+c}$ is similar to the discrete
exponential map $x \rightarrow g^x$, which is used in many cryptographic
applications including the ElGamal signature scheme. This paper analyzes the
number of solutions to the Welch equation: $g^{x-1+c} \equiv x \pmod{p^e}$
where $p$ is a prime and $g$ is a unit modulo $p$, and looks at other patterns
of the equation that could possibly be exploited in a similar cryptographic
system. Since the equation is modulo $p^e$, where $p$ is a prime number,
$p$-adic methods of analysis are used in counting the number of solutions
modulo $p^e$. These methods include: $p$-adic interpolation, Hensel's lemma and
Chinese Remainder Theorem.
