Deconstructing the Welch Equation Using $p$-adic Methods

The Welch map x → g 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: g ≡ 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 Chinese Remainder Theorem.