Solving [formula omitted] in [formula omitted] and an alternative proof of a conjecture on the differential spectrum of the related monomial functions

Abstract

This article determines all the solutions in the finite field F24n of the equation x23n+22n+2n−1+(x+1)23n+22n+2n−1=b. Specifically, we explicitly determine the set of b's for which the equation has i solutions for any positive integer i. Such sets, which depend on the number of solutions i, are given explicitly and expressed nicely, employing the absolute trace function over F2n, the norm function over F24n relatively to F2n and the set of (2n+1)st roots of unity in F24n. The equation considered in this paper comes from an article by Budaghyan et al. ([2]) in which the authors have investigated novel approaches for obtaining alternative representations for functions from the known infinite APN families. In particular, they have been interested in determining the differential spectrum of some power functions among them is the one F(x)=x23n+22n+2n−1 defined over F24n. The problem of the determination of such spectrum has led to a conjecture (Conjecture 27 in the preprint (2020) [2] for which an updated version will appear in 2022 at the IEEE Transactions Information Theory) stated by Budaghyan et al.As an immediate consequence of our results, we prove that the above equation has 22n solutions for one value of b, 22n−2n solutions for 2n values of b in F24n and has at most two solutions for all remaining points b, leading to complete proof of the conjecture raised by Budaghyan et al. We highlight that the recent work of Li et al., in [9] gives the complete differential spectrum of F and also gives an affirmative answer to the conjecture of Budaghyan et al. However, we emphasize that our approach is interesting and promising by being different from Li et al. Indeed, on the opposite to their article, our technique allows to determine ultimately the set of b's for which the considered equation has solutions as well as the solutions of the equation for any b in F24n.