On the higher-order nonlinearity of a Boolean bent function class (Constructed via Niho power functions)

Abstract

The r th-order nonlinearity of a Boolean function plays a vital role in analyzing the security of stream and block ciphers. In this paper, we present a lower bound on the $\left (\frac{m}{2}\right )$th-order nonlinearity of a bent Boolean function class of the form $g(x)=T{r_{1}^{n}}\left (\alpha _{1}x^{d_{1}}+\alpha _{2}x^{d_{2}}\right )$, where $\alpha _{1}, \alpha _{2}\in \mathbb{F}_{2^{n}}, n=2m$ and d₁, d₂ are Niho exponents, which was constructed by Dobbertin et al. (Construction of bent functions via Niho power functions, J. Comb. Theory. Ser. A, 113, 779–798 in 8). We compared the computed lower bound for the second-order nonlinearity with the lower bound obtained by Gode and Gangopadhyay (IACR Crypt. ePrint Archieve 2009, 502, 15) for the class of Boolean functions $g_{\mu }(x)=T{r_{1}^{n}}(\mu x^{2^{i}+2^{j}+1}), i>j$, for n > 2i(n≠i + j and n≠ 2i − j) when n = 8 and i = 3. We also compared the computed lower bound for the third-order nonlinearity with the lower bound obtained by Singh (Int. J. Eng. Math. 2014, 1–7, 22) for the class of Boolean functions $f_{\lambda }(x)=T{r_{1}^{n}}\left (\lambda x^{d}\right )$, for all $x\in \mathbb{F}_{{2}^{n}}$, $\lambda \in \mathbb{F}_{2^{n}}^{*}$, where d = 2ⁱ + 2^j + 2^k + 1, where i, j and k are integers such that i > j > k ≥ 1 and n > 2i when n = 12 and i = 5, and Gode and Gangopadhyay (Cryptogr. Commun. 2, 69–83, 15) for the class $f_{\mu }(x)=T{r_{1}^{n}}\left (\mu x^{k}\right )$, where k = 2^2d − 2^d + 1 with 1 ≤ d < n, n > 10 and $\mu \in \mathbb{F}_{{2}^{n}}$, for d= 3 and for all $x\in \mathbb{F}_{2^{n}}$ when n = 12. It is then shown that our computed lower bound is better than the lower bounds obtained by Gode et al. and Singh for other classes of Boolean functions for the provided values of n and i.