Piecewise constructions of inverses of cyclotomic mapping permutations
Yanbin Zheng, Yuyin Yu, Dingyi Pei · arXiv (Cornell University) · 2015
Given a permutation polynomial $f(x)$ of a small finite field $\mathbb{F}_q$, the inverse $f^{-1}(x)$ over $\mathbb{F}_q$ could be determined by using the Lagrange Interpolation Formula. But for a large $q$, finding the explicit expression of $f^{-1}(x)$ is usually a hard problem. A piecewise interpolation formula for the inverses of permutation polynomials of arbitrary finite fields is introduced, which extends the Lagrange Interpolation Formula. The explicit inverses of cyclotomic mapping permutations are constructed to demonstrate the formula. The explicit inverses of permutation polynomials of the form $x^rh(x^{(q-1)/d})$ are presented.