Determine the number of the rational zeros of any given linearized polynomial is one of the vital problems in finite field theory, with applications in modern symmetric cryptosystems. But, the known general theory for this task is much far from giving the exact number when applied to a specific linearized polynomial. The first contribution of this paper is a better general method to get a more precise upper bound on the number of rational zeros of any given linearized polynomial over arbitrary finite field. We anticipate this method would be applied as a useful tool in many research branches of finite field and cryptography. Really we apply this result to get tighter estimations of the lower bounds on the second-order nonlinearities of general cubic Boolean functions, which has been an active research problem during the past decade. Furthermore, this paper shows that by studying the distribution of radicals of derivatives of a given Boolean function one can get a better lower bound of the second-order nonlinearity, through an example of the monomial Boolean functions \(g_{\mu }=Tr(\mu x^{2^{2r}+2^{r}+1})\) defined over the finite field \({\mathbb F}_{2^{n}}\).

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线性多项式的有理零个数与三次布尔函数的二阶非线性

确定任何给定的线性多项式的有理零的数目是有限域理论中的重要问题之一，其在现代对称密码系统中的应用。但是，用于此任务的已知一般理论与应用于特定线性化多项式时给出准确的数字相去甚远。本文的第一个贡献是一种更好的通用方法，它可以在任意有限域上获得任何给定线性化多项式的有理零的个数的更精确上限。我们预计该方法将在有限领域和密码学的许多研究领域中用作有用的工具。的确，我们将这个结果用于对一般三次布尔函数的二阶非线性的下界进行更严格的估计，这在过去十年中一直是一个活跃的研究问题。此外，\（g _ {\ mu} = Tr（\ mu x ^ {2 ^ {2r} + 2 ^ {r} +1}）\）在有限字段\（{\ mathbb F} _ {2 ^ {n }} \）。