In this thesis, we study the concepts of deﬁciency and ambiguity for a bijection on a ﬁnite Abelian group, in particular, for permutation polynomials over ﬁnite ﬁelds. Our focus is on all normalized permutation polynomials of degree ≤ 5. We obtain explicit numbers for deﬁciency and ambiguity of eleven classes of polynomials in Table A.0.1. We veriﬁed them by using SAGE. However, we can not do much for the remaining two cases, except one observation on a relation between ambiguity and deﬁciency in a special case for one of the remaining unsolved classes.