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 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 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 the Chinese Remainder Theorem.
Josh Holden, Margaret Robinson
Mann, Abigail and Yeoh, Adelyn
"Deconstructing the Welch Equation Using 𝑝-adic Methods,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 16:
1, Article 1.
Available at: https://scholar.rose-hulman.edu/rhumj/vol16/iss1/1