Completely characterizing a class of permutation quadrinomials

Abstract

In the present article, we provide a complete characterization of permutations on the finite field F4m of shape fϵ_(X):=ϵ1X‾q+1+ϵ2X‾qX+ϵ3X‾Xq+ϵ4Xq+1, definitively (where q=2k, Q=2m, m is odd, (m,k)=1, X‾=XQ). Our achievement firstly extends the very recent literature (a long series of interesting recent (2019-2022) results derived in (at least) seven articles) about the bijectivity on FQ2 of fϵ_. It also gives a complete proof of the conjecture (Conjecture 19) posed in the recent literature by Li et al. (2021) [24] about the characterization of permutations fϵ_ with 4-uniform BCT (which can be viewed, thanks to their nice cryptographic properties, as promising candidates as S-boxes to be used in designing secure block ciphers in symmetric cryptography). To derive our results about the quadrinomials fϵ_, we shall keep the essence of our successful approach initiated by Kim et al. in [23] by relating the problem of characterization of the bijectivity of fϵ_ over FQ2 to the equation Xq+1+X+a=0 but also by performing algebraic developments with new nature by considering some rational functions over finite fields. Besides, we shall employ novel algebraic techniques and derive new auxiliary results (some of them have their independent interest) going beyond the recent results given in [23]. Notably, we present a complete proof of the bijectivity of fϵ_ over FQ2 without any restriction. Specifically, our results are valid when the coefficients ϵi (i∈{1,2,3,4}) lie in FQ2 and for any value of k≥1 (at the opposite of the results in [23] valid with the condition that the coefficients ϵi (i∈{1,2,3,4}) lie in the finite field FQ while covering all values of k≥1). Finally, we emphasize that very recently, Göloğlu has presented in (Göloğlu, 2022 [13]), an alternative novel approach providing a complete description of permutations of the form mentioned above using biprojective polynomials.