PROBLEMS MOTIVATED BY CRYPTOLOGY: COUNTING FIXED POINTS AND TWO-CYCLES OF THE DISCRETE LAMBERT MAP
In this thesis, we begin with a brief introduction to some relevant number theory and to digital signature schemes (DSS). We explain how information about the discrete Lambert map (DLM)  relates to DSS security. Next we introduce results from p-adic analysis. We summarize the results from previous work on the DLM and extend these results to p = 2. In the main part of this thesis we explain our results counting fi xed points and two-cycles of the DLM. That is, for a fi xed prime p and a nonzero integer g where p does not divide g and e is a positive integer, we will count the number of fi xed points or solutions to xgx ≡x (mod p^e) and the number of two cycles or simultaneous solutions to xgx ≡y (mod p^e) and ygy ≡x (mod p^e) where x and y range through appropriate sets of integers. This work is a continuation of work started by Holden and Robinson in  and their students from the 2014 Mount Holyoke summer REU program: Anne Waldo and Caiyun Zhu , Yu Liu , and Abigail Mann and Adelyn Yeoh .