Fixed Points of Rational Functions Satisfying the Carlitz Property

Recent research within the field of cryptography has suggested that S-boxes should be chosen to contain few fixed points, motivating analysis of the fixed points of permutations. This thesis presents a novel means of obtaining fixed points for all functions satisfying a property put forth by L. Carlitz. We introduce an algorithm which cyclically generates fixed points for three such classes of functions, the most renowned of which are Rédei rational functions. Further, we provide an explicit expression for the fixed points of all Rédei functions over Fq.