Deconstructing the Welch Equation using p-adic Methods

Authors

Abigail Mann, Rose-Hulman Institute of TechnologyFollow
Adelyn Yeoh, Mount Holyoke CollegeFollow

Document Type

Article

Publication Date

7-29-2014

First Advisor

Joshua Holden

Second Advisor

Margaret Robinson

Abstract

The Welch map x -> g^x-1+c is similar to the discrete exponential map x -> 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 = x (mod p^e) 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 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.

Comments

MSTR 14-04

Recommended Citation

Mann, Abigail and Yeoh, Adelyn, "Deconstructing the Welch Equation using p-adic Methods" (2014). Mathematical Sciences Technical Reports (MSTR). 150.
https://scholar.rose-hulman.edu/math_mstr/150