On the number of the rational zeros of linearized polynomials and the second-order nonlinearity of cubic Boolean functions

Abstract

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 ${\mathbb F}_{2^{n}}$ .