Abstract
Permutation polynomials (PPs) and their inverses have applications in cryptography, coding theory and combinatorial design theory. In this paper, we make a brief summary of the inverses of PPs of finite fields, and give the inverses of all PPs of degree $\leq 6$ over finite fields $\mathbb{F}_{q}$ for all $q$ and the inverses of all PPs of degree $7$ over $\mathbb{F}_{2^n}$. The explicit inverse of a class of fifth degree PPs is the main result, which is obtained by using Lucas' theorem, some congruences of binomial coefficients, and a known formula for the inverses of PPs of finite fields.
Highlights
F OR a prime power q, let Fq denote the finite field with q elements, F∗q = Fq \{0}, and Fq [x] the ring of polynomials over Fq
A polynomial f ∈ Fq [x] is called a permutation polynomial (PP) of Fq if it induces a bijection from Fq to itself
It is interesting to note that the explicit formulae of inverses of low degree PPs have been neglected in the literature
Summary
It is interesting to note that the explicit formulae of inverses of low degree PPs have been neglected in the literature. This motivates us to give a short review of the progress in this topic and find explicit expressions of inverses of all classes of PPs of degree ≤ 7 in [12], [22], [38].
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