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.

Author Bio

Abigail Mann is a mathematics and computer science major at Rose-Hulman Institute of Technology. She completed her REU at Mount Holyoke College in July 2014, and has presented her research at the REU-Mini Conference at Yale University as well as an undergraduate research poster competition at Rose-Hulman. Outside of class, she enjoys playing violin and piano.

Adelyn Yeoh is a mathematics major at Mount Holyoke College, and she is an international student from Malaysia. She has presented this research at the 2014 REU-Mini Conference at Yale University. This paper is the culmination of her research during the REU at Mount Holyoke College in the summer of 2014. She hopes to pursue a masters in applied mathematics after graduation.