The Welch map x -> gx-1+c is similar to the discrete exponential map x -> gx, which is used in many cryptographic applications including the ElGamal signature scheme. This paper analyzes the number of solutions to the Welch equation: gx-1+c = x (mod pe) where p is a prime, and looks at other patterns of the equation that could possibly exploited in a similar cryptographic system. Since the equation is modulo pe, where p is a prime number, p-adic methods of analysis are used in counting the number of solutions modulo pe. These methods include: p-adic interpolation, Hensel's lemma and Chinese Remainder Theorem.
Mann, Abigail and Yeoh, Adelyn, "Deconstructing the Welch Equation using p-adic Methods" (2014). Mathematical Sciences Technical Reports (MSTR). 150.