Inverting bijective polynomial maps over finite fields

We study the problem of inverting a bijective polynomial map F: Fqⁿ→ Fqⁿover a finite field Fq. Our interest mainly stems from the case where F encodes a permutation given by some cryptographic scheme. Given y⁽⁰⁾∈ Fqⁿ, we are able to compute the value x⁽⁰⁾∈ Fqⁿfor which F(x⁽⁰⁾) = y⁽⁰⁾holds in time O(Ln^O(1)δ⁴) up to logarithmic terms. Here L is the cost of the evaluation of F and δ is a geometric invariant associated to the graph of the polynomial map F, called its degree.