On permutation quadrinomials with boomerang uniformity 4 and the best-known nonlinearity

Motivated by recent works on the butterfly structure, particularly by its generalization introduced by Canteaut et al. (IEEE Trans Inf Theory 63(11):7575–7591, 2017), we first push further the study of permutation polynomials over binary finite fields by completely characterizing those permutations\(f_{\underline{\epsilon }}\) defined over the finite field \({\mathbb F}_{Q^2}\) (of order \(Q^2\)) having the following shape: where \(q=2^k\), \(Q=2^m\), m is odd, \(\mathrm {gcd}(m,k)=1\), \(\overline{X}=X^Q\) and \(\underline{\epsilon }=(\epsilon _1, \epsilon _2, \epsilon _3, \epsilon _4)\in {\mathbb F}_{Q}^4\). We shall provide an approach to handle the bijectivity of \(f_{\underline{\epsilon }}\) for any \(k\ge 1\). Notably, we show that the problem of finding conditions for bijectivity of the quadrinomial \(f_{\underline{\epsilon }}\) is closely related to the study of the famous equation \(X^{q+1}+X+a=0\) (*). We then reduce the initial problem into the problem of finding conditions for which an equation of the form (*) has a unique solution in \({\mathbb F}_{Q}\) for every \(a\in {\mathbb F}_{Q}\). In addition, as a crucial direct consequence our result, we prove the validity of the conjecture (Conjecture 19) proposed by Li et al. (Des Codes Cryptogr 89:737–761, 2021). We emphasize that our positive answer completely characterizes permutations with boomerang uniformity 4 from the butterfly structure, which leads to the view of the quadrinomial \(f_{\underline{\epsilon }}\) as excellent candidates to design block ciphers in symmetric cryptography. Despite a lot of attention regarding the considered conjecture, it remains unsolved on its whole when the coefficients lie in \({\mathbb F}_{Q}\). However, this article is the first which propose an approach that solves the enter conjecture by handling both sides of it involving equivalence simultaneously. We believe that our novel approach and its strength could benefit from proving the bijectivity of other families of polynomials over finite fields.