论文标题
求解$ x^{2^{3n}+2^{2n}+2^{n} -1}+（x+1）
Solving $X^{2^{3n}+2^{2n}+2^{n}-1}+(X+1)^{2^{3n}+2^{2n}+2^{n}-1}=b$ in $GF{2^{4n}}$
论文作者
论文摘要
本文确定了有限字段$ gf {2^{2^{4n}} $的所有解具体来说，我们明确确定了$ b $的集合，该集合方程为任何正整数$ i $具有$ i $ solutions。这些集合取决于解决方案$ i $的数量，明确给出并表达得很好，在$ gf {2^{n}} $上采用了绝对跟踪功能，$ gf {2^{4n}} $上的norm函数在$ gf {2^{4n}} $。本文考虑的方程式来自Budaghyan等人的一篇文章。 \ cite {bccdk20}。 As an immediate consequence of our results, we prove that the above equation has $2^{2n}$ solutions for one value of $b$, $2^{2n}-2^n$ solutions for $2^n$ values of $b$ in $GF{2^{4n}}$ and has at most two solutions for all remaining points $b$, leading to complete proof of the conjecture raised by Budaghyan et al.我们强调，Li等人的最新工作，在\ cite {li-et-al-2020}中给出了$ f $的完整差异频谱，并且还为Budaghyan等人的猜想提供了肯定的答案。但是，我们强调的是，我们的方法与Li等人不同，这是有趣和有希望的。确实，与他们的文章相反，我们的技术允许最终确定$ b $的集合，该集合具有解决方案的解决方案以及对任何$ b $ in $ gf {2^{4n}} $中任何$ b $的解决方案。
This article determines all the solutions in the finite field $GF{2^{4n}}$ of the equation $x^{2^{3n}+2^{2n}+2^{n}-1}+(x+1)^{2^{3n}+2^{2n}+2^{n}-1}=b$. 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 $GF{2^{n}}$, the norm function over $GF{2^{4n}}$ relatively to $GF{2^{n}}$ and the set of $2^n+1$st roots of unity in $GF{2^{4n}}$. The equation considered in this paper comes from an article by Budaghyan et al. \cite{BCCDK20}. As an immediate consequence of our results, we prove that the above equation has $2^{2n}$ solutions for one value of $b$, $2^{2n}-2^n$ solutions for $2^n$ values of $b$ in $GF{2^{4n}}$ 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 \cite{Li-et-al-2020} 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 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 $GF{2^{4n}}$.
