On Subspaces of Kloosterman Zeros and Permutations of the Form $$L_1(x^{-1})+L_2(x)$$

Permutations of the form \(F(x)=L_1(x^{-1})+L_2(x)\) with linear functions \(L_1,L_2\) are closely related to several interesting questions regarding CCZ-equivalence and EA-equivalence inverse function. In this paper, we show that F cannot be a permutation on binary fields if kernel \(L_1\) or \(L_2\) is large. A key step our proof an observation maximal size subspace V \(\mathbb {F}_{2^n}\) consists Kloosterman zeros, i.e. such \(K_n(v)=0\) for every \(v \in V\) where \(K_n(v)\) denotes sum v.