The absence of an efficient algorithm to solve the Discrete Logarithm Problem is often exploited in cryptography. While exponentiation with a modulus, bx≡ a (mod m), is extremely fast with a modern computer, the inverse is decidedly not. At the present time, the best algorithms assume that the inverse mapping is completely random. Yet there is at least some structure, such as the fact that b1≡ b (mod m). To uncover additional structure that may be useful in constructing or refining algorithms, statistical methods are employed to compare mappings, x ≡ bx (mod m), to random mappings. More concretely, structure will be defined by representing the mappings as functional graphs and using parameters from graph theory such as cycle length. Since the literature for random permutations is more extensive than other types of functional graphs, only permutations produced from the experimental mappings are considered.

Author Bio

Andrew Hoffman, from Fort Wayne, Indiana, is a member of Wabash College㤼㸲s class of 2010. He is a double major in mathematics and philosophy with a minor in music. This work was completed through the 2009 Rose-Hulman Research Experience for Undergraduates in Computational Number Theory. His future plans are to pursue a doctorate in mathematics.