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To determine the dimension of null space of any given linearized polynomialis one of vital problems in finite field theory, with concern to design ofmodern symmetric cryptosystems. But, the known general theory for this task ismuch far from giving the exact dimension when applied to a specific linearizedpolynomial. The first contribution of this paper is to give a better generalmethod to get more precise upper bound on the root number of any givenlinearized polynomial. We anticipate this result would be applied as a usefultool in many research branches of finite field and cryptography. Really weapply this result to get tighter estimations of the lower bounds on the secondorder nonlinearities of general cubic Boolean functions, which has been beingan active research problem during the past decade, with many examples showinggreat improvements. Furthermore, this paper shows that by studying thedistribution of radicals of derivatives of a given Boolean functions one canget a better lower bound of the second-order nonlinearity, through an exampleof the monomial Boolean function $g_{\mu}=Tr(\mu x^{2^{2r}+2^r+1})$ over anyfinite field $\GF{n}$.

Improved upper bound on root number of linearized polynomials and its application to nonlinearity estimation of Boolean functions