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

To determine the dimension of null space of any given linearized polynomial is one of vital problems in finite field theory, with concern to design of modern symmetric cryptosystems. But, the known general theory for this task is much far from giving the exact dimension when applied to a specific linearized polynomial. The first contribution of this paper is to give a better general method to get more precise upper bound on the root number of any given linearized polynomial. We anticipate this result 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 being an active research problem during the past decade, with many examples showing great improvements. Furthermore, this paper shows that by studying the distribution of radicals of derivatives of a given Boolean functions one can get a better lower bound of the second-order nonlinearity, through an example of the monomial Boolean function $g_{\mu}=Tr(\mu x^{2^{2r}+2^r+1})$ over any finite field \GF{n}

On lower bounds of second-order nonlinearities of cubic bent functions constructed by concatenating Gold functions

In this paper we consider cubic bent functions obtained by Leander and McGuire (J. Comb. Th. Series A, 116 (2009) 960-970) which are concatenations of quadratic Gold functions. A lower bound of second-order nonlinearities of these functions is obtained. This bound is compared with the lower bounds of second-order nonlinearities obtained for functions belonging to some other classes of functions which are recently studied

On lower bounds on second--order nonliearities of bent functions obtained by using Niho power functions

In this paper we find a lower bound of the second-order nonlinearities of Boolean bent functions of the form $f(x) = Tr_{1}^{n}(\alpha_{1}x^{d_{1}} + \alpha_{2}x^{d_{2}})$,where $d_1$ and $d_2$ are Niho exponents. A lower bound of the second-order nonlinearities of these Boolean functions can also be obtained by using a result proved by Li, Hu and Gao (eprint.iacr.org/2010 /009.pdf). It is demonstrated that for large values of $n$ the lower bound obtained in this paper are better than the lower bound obtained by Li, Hu and Gao

Third-order nonlinearities of some biquadratic monomial Boolean functions

In this paper, we estimate the lower bounds on third-order nonlinearities of some biquadratic monomial Boolean functions of the form $Tr_1^n(\lambda x^d)$ for all $x \in \mathbb F_{2^n}$, where \lambda \in \BBF_{2^n}^{*}, \begin{itemize} \item [{(1)}]$d = 2^i + 2^j + 2^k + 1$, $i, j, k$ are integers such that $i > j > k \geq 1$ and $n > 2 i$. \item [{(2)}] $d = 2^{3\ell} + 2^{2\ell} + 2^{\ell} + 1$, $\ell$ is a positive integer such that $\gcd (i, n) = 1$ and $n > 6$. \end{itemize

A quantum algorithm to estimate the Gowers U2U_2 norm and linearity testing of Boolean functions

We propose a quantum algorithm to estimate the Gowers $U_2$ norm of a Boolean function, and extend it into a second algorithm to distinguish between linear Boolean functions and Boolean functions that are $\epsilon$-far from the set of linear Boolean functions, which seems to perform better than the classical BLR algorithm. Finally, we outline an algorithm to estimate Gowers $U_3$ norms of Boolean functions

On the Lower Bounds of the Second Order Nonlinearity of some Boolean Functions

The $r$-th order nonlinearity of a Boolean function is an important cryptographic criterion in analyzing the security of stream as well as block ciphers. It is also important in coding theory as it is related to the covering radius of the Reed-Muller code $\mathcal{R}(r, n)$. In this paper we deduce the lower bounds of the second order nonlinearity of the two classes of Boolean functions of the form \begin{enumerate} \item $f_{\lambda}(x) = Tr_1^n(\lambda x^{d})$ with $d=2^{2r}+2^{r}+1$ and $\lambda \in \mathbb{F}_{2^{n}}$ where $n = 6r$. \item $f(x,y)=Tr_1^t(xy^{2^{i}+1})$ where $x,y \in \mathbb{F}_{2^{t}}, n = 2t, n \ge 6$ and $i$ is an integer such that $1\le i < t$, $\gcd(2^t-1, 2^i+1) = 1$. \end{enumerate} For some $\lambda$, the first class gives bent functions whereas Boolean functions of the second class are all bent, i.e., they achieve optimum first order nonlinearity

The Good lower bound of Second-order nonlinearity of a class of Boolean function

In this paper we find the lower bound of second-order nonlinearity of Boolean function $f_{\lambda}(x) = Tr_{1}^{n}(\lambda x^{p})$ with $p = 2^{2r} + 2^{r} + 1$, $\lambda \in \mathbb{F}_{2^{r}}^{*}$ and $n = 5r$. It is also demonstrated that the lower bound obtained in this paper is much better than the lower bound obtained by Iwata-Kurosawa \cite{c14}, and Gangopadhyay et al. (Theorem 1, \cite{c12})